Duality for optimal consumption with randomly terminating income
DDUALITY AND DEEP LEARNING FOR OPTIMAL CONSUMPTION WITHRANDOMLY TERMINATING INCOME
ASHLEY DAVEY, MICHAEL MONOYIOS, AND HARRY ZHENG
Abstract.
We establish a rigorous duality theory for a finite horizon problem of optimal consump-tion in the presence of an income stream that can terminate randomly at an exponentially distributedtime, independent of the asset prices. We thus close a duality gap encountered by Vellekoop andDavis [35] in an infinite horizon version of this problem. Nearly all the classical tenets of dualitytheory are found to hold, with the notable exception that the marginal utility of initial wealth atzero is finite . The intuition is that the agent will receive some income, no matter how early itterminates, so is not infinitely penalised for having zero initial capital. We then solve the problemnumerically, with an additional terminal wealth objective, using deep learning. We transform theproblem with randomly terminating income into one that no longer depends on the jump componentbut has an additional inter-temporal wealth objective. We then numerically solve the second orderbackward stochastic differential equations (2BSDEs), in both the primal and dual dimensions, tofind the optimal control and tight lower and upper bounds for the value function.
MSC 2010 subject classifications:
Keywords and phrases:
Duality, utility from consumption, portfolio optimisation, supermartingaledeflator, terminating income, deep learning, primal and dual 2BSDEs, HJB equation1.
Introduction
In this paper we solve an optimal consumption problem in an otherwise complete financial market,in which incompleteness is generated by the presence of a randomly terminating income stream,which pays at a constant rate until some exponential random time, independent of the asset prices,at which time the income abruptly stops. We consider a finite horizon version of this problem, forwhich Vellekoop and Davis [35] considered the infinite horizon case in a Black-Scholes market.Despite the apparent simplicity of the modification of the classical Merton problem with deter-ministic (or indeed no) income, the problem in [35] proved a remarkably intractable one to solve andunderstand. Vellekoop and Davis implemented a differential-equation based dual approach, madean ansatz that the dual control was deterministic, or at least did not depend on the terminationtime, and also used some differential equation heuristics to argue that the derivative of the valuefunction at zero initial wealth was infinite. Ultimately, though, they encountered a duality gap, inthat their derived value function could not solve the Hamilton-Jacobi-Bellman (HJB) equation, in-dicating that the dual optimiser must indeed be stochastic and state-dependent in some way. Whatis more, their numerical solutions indicated that the value function derivative at zero initial wealthwas finite . The intuition behind this last feature is clear: even though the income can terminate verysoon after time zero, it cannot terminate immediately, except in the limiting case that the intensityof the exponential time approaches infinity. Thus, the agent is bound to receive some income, andthis means she is not penalised in an infinitely draconian manner (in terms of utility) for havingzero initial capital, so the marginal utility of wealth is finite at this point.All the above issues make this a still very interesting problem to analyse, and we do so in this paperfrom two distinct perspectives. First, we provide a fully rigorous duality characterisation of the finite
Date : November 3, 2020.Part of this work was carried out during visits by the second author to Universit´e Paris Diderot and ´Ecole Poly-technique. Many thanks to Huyˆen Pham and Nizar Touzi for generous hospitality, and we are grateful to them andto Chris Rogers for valuable discussions. We dedicate this paper to the memory of Mark H.A. Davis. a r X i v : . [ q -f i n . M F ] O c t ASHLEY DAVEY, MICHAEL MONOYIOS, AND HARRY ZHENG horizon pure consumption problem, that is, the finite horizon version of the Davis-Vellekoop [35]problem. We close the duality gap, and find that all the usual tenets of duality theory hold, withthe exception of the important feature that, as our analysis confirms, the marginal utility at zeroinitial wealth is indeed finite. Second, we numerically solve the problem (adding a terminal wealthobjective in the numerical examples) using a deep learning approach to the HJB equations andbackward stochastic differential equations (BSDEs) connected with the problem. The numericalresults confirm, in particular, the property of finite marginal utility at zero wealth.Our approach to the duality is largely classical, and rooted in some of the seminal techniquesinspired by Kramkov and Schachermayer [22, 23], as adapted recently to the classical (no income)infinite horizon consumption problem by Monoyios [26] in a general semimartingale framework underthe minimal no-arbitrage assumption of no unbounded profit with bounded risk (NUPBR). Here,there are important adaptations of the approach in [26] and of the classical Kramkov-Schachermayerapproach. Most notable is that we do not use the bipolar theorem of Brannath and Schacher-mayer [5], because the primal variable appearing in the budget constraint is the the differencebetween the consumption and income rates, so is not guaranteed to be non-negative, precluding anapplication of the bipolar theorem.Let us outline our method. We first establish a budget constraint on admissible consumptionplans (Lemma 2.1), where we initially use a set (cid:101) Y of discounted local martingale deflators as dualvariables. The budget constraint follows from the property that deflated wealth plus cumulativedeflated consumption over income is a supermartingale. We then define an enlarged set Y of dualvariables ( consumption deflators ), for which the same supermartingale property, and thus the budgetconstraint, holds by definition, and which will form the dual domain for the consumption problem.The budget constraint serves to fix the structure of the dual objective functional. Ultimately we shallrelate the domains Y and (cid:101) Y , and it turns out that the latter is dense (in an appropriate topology)in the former, so that the infima over both domains in the dual problem coincide. We later showthat the budget constraint is both a necessary and sufficient condition for admissible consumptionprocesses (in Lemma 7.2), by exploiting a version of the optional decomposition theorem due toStricker and Yan [34]. This ensures that the primal domain is closed in an appropriate topology.This, and convexity of this domain, is used later to show existence and uniqueness in the primalproblem.We then transport the primal and dual problems to abstract domains in a finite measure space,using a product measure µ := κ × P on [0 , T ] × Ω, where
T < ∞ is the time horizon, κ is a measurethat serves to discount consumption, and P is the underlying physical probability measure. Thebudget constraint is trivially re-cast in this abstract space. We establish some basic properties ofthe abstract primal and dual domains (in Proposition 4.1). As well as the aforementioned closedproperty of the primal domain, both domains are bounded in L ( µ ) and, in particular, the dualdomain is bounded in L ( µ ). This last property relies on the finiteness of the time horizon.For the dual side of the analysis, we show directly (in Lemma 7.8) that the abstract dual domainis closed with respect to the topology of convergence in measure µ , by exploiting Fatou convergenceof supermartingales. This is key to establishing existence and uniqueness in the dual problem.With these ingredients, we prove an abstract duality theorem (Theorem 4.2), from which theconcrete duality (Theorem 3.1) follows, including a characterisation of the optimal wealth process.We then show that the abstract dual domain formed from consumption deflators coincides with theclosure of the corresponding dual domain formed from discounted local martingale deflators (so thelatter is dense in the former) in Proposition 4.3 (in the abstract formulation) leading to the concreterealisation (in Theorem 3.2).Utility maximisation problems with an additional random endowment have proven to be some ofthe hardest problems in financial mathematics, and a variety of techniques have been used to obtainduality results for these problems. In some papers, such as Cvitani´c et al [6] (with a terminal wealthobjective) and Karatzas and ˇZitkovi´c [21] (with a consumption objective), the dual space was forced ANDOMLY TERMINATING INCOME 3 to incorporate a singular part, in the form of finitely additive set functions (often referred to, withan abuse of terminology, as finitely additive measures). In other papers, such as Hugonnier andKramkov [18], Mostovyi [28] and Mostovyi and Sˆırbu [29], finitely additive measures were avoidedby expanding the dimension of the value function to incorporate an additional variable describingthe number of units of the random endowment. It remains very much an open question as to therelation between these different approaches to the problem, though there must be one, since theycoincide when the number of units of the random endowment is fixed at one.In some problems, neither of the above features needs to be introduced to get a rigorous duality.This is the case in some early papers on the subject, such as He and Pag`es [17], for a consumptionproblem in a complete Brownian market with an otherwise replicable income stream, where theincompleteness comes from a borrowing constraint, which prevents the adoption of the classicalstrategy of borrowing against future income and implementing the optimal no-income strategy withthe increased initial capital. In our problem, the possibility of abrupt termination of the incomealso prevents the agent from borrowing against future receipts, as one could be left insolvent if theincome stops at a time when wealth has not recovered to be positive. As in [17], we also do not haveto introduce finitely additive measures, or to expand the dimension of the value function, in orderto obtain a rigorous duality. It is an open question to fully understand the relations between allthese rather different forms of dual characterisation of utility maximisation problems with randomendowment.Some early papers on optimal investment and consumption with random endowment, usually ina Brownian filtration, adopt a HJB equation perspective, such as Duffie and Zariphopoulou [9]. Insome of these works, notably El Karoui and Jeanblanc-Picqu´e [11], as well as He and Pag`es [17],when the incompleteness is induced by a borrowing constraint, the spatial derivative of the dualvalue function, which is negative, may reach zero at a finite value of the space variable, reflectingthe finite marginal utility at zero wealth, and a free boundary appears in the dual HJB equation.This was handled in [11] using methods akin to those for American options.In the second part of the paper, we study the randomly terminating income problem from a sto-chastic control perspective, and numerically solve the problem using a deep learning (DL) approach.It is natural to decompose the problem into post-income termination and pre-income terminationproblems. The optimal solution is well known for the post-terminating problem, so we only needto solve a pre-termination problem. To deal with the random termination, we integrate over therandom time distribution, transforming the problem into an equivalent new problem with non-terminating income, involving an additional inter-temporal wealth objective as well as the originalconsumption objective. We also incorporate a terminal wealth objective in this part of the paper.The transformed problem admits a dual control formulation with a dual cost functional that is anupper bound of the primal value function for any choice of dual state and control. This allows usto form bounds on the value function, and if said bound is tight, then we have found approximateoptimal state and control processes, in a way that would be impossible by simply trying to solvethe primal problem.We derive the primal and dual HJB equations and their associated 2BSDEs. We then apply DLto numerically solve the resulting equations. There has been much recent research in DL methods.E et al. [10] and Beck et al. [2] adopt deep BSDE methods to solve nonlinear PDEs. The BSDEsare solved by forward simulation, setting the unknown process as the output of a neural networkthat is optimised using the terminal condition of the BSDE as a loss function. A key benefit of DLis the reduced impact of dimensionality on the convergence rate of numerical algorithms, see Grohset al. [15]. This, together with rapid advances in computational power, allows us to solve controlproblems with very large dimensions, see Beck et al. [1].There is a growing literature in applying DL to solve stochastic control problems. Han [16]presents a generic stochastic control solver, in which the control process is a neural network thattakes the state process as input, and the loss function is the terminal cost (or gain) function. Thedifference between [16] and our approach is the global optimality condition used. To update the
ASHLEY DAVEY, MICHAEL MONOYIOS, AND HARRY ZHENG control using stochastic gradient descent, say, the processes must be simulated, then derivativestaken of the terminal cost. This derivative involves back propagation through discretisation, whichcan be computationally costly for a high control dimension or large discretisation. Additionally,our algorithm results in additional information on the value function and its derivatives. Hur´e etal. [19] propose several algorithms to combat these issues, using the dynamic programming principle(DPP) together with a value function approximation to localise the optimisation of the control.Their approach is related to ours, but does not use a BSDE representation.The remainder of the paper is structured as follows. In Section 2 we formulate the primal problem,budget constraint and the dual problem. In Section 3 we state the main duality theorem (Theorem3.1) and also Theorem 3.2, that the infima over consumption deflators and discounted martingaledeflators coincide. In Section 4 we state key properties of the abstract primal and dual domains(Proposition 4.1), the abstract duality theorem (Theorem 4.2) and the property that the dual domainbased on discounted local martingale deflators is dense in the domain based on consumption deflators(Proposition 4.3), which underlies Theorem 3.2. In Section 5 we prepare for numerical solutionsby formulating our stochastic control approach, leading to transformed primal and dual problems.Section 6 describes the primal and dual 2BSDEs and DL algorithms for solving these equations,with a numerical example for power utility. The duality results of Sections 3 and 4 are proven inSection 7. Section 8 concludes. 2.
Problem formulation
The financial market.
We consider an investment-consumption problem in the presence of arandomly terminating income stream. We have a complete stochastic basis (Ω , F , F := ( F t ) t ∈ [0 ,T ] , P )for some fixed horizon T < ∞ , with the filtration F satisfying the usual hypotheses of right-continuityand augmentation with P -null sets of F . On this space we have a complete, Brownian financialmarket, with d stocks driven by a d -dimensional ( P , F )-Brownian motion W , and a cash accountwith a non-negative bounded deterministic interest rate process r = ( r t ) t ∈ [0 ,T ] . The stock pricevector S = ( S , . . . , S d ) has dynamics given byd S t = diag( S t )[ b t d t + σ t d W t ] , with a bounded deterministic drift vector b : [0 , T ] → R d and a bounded and positive definiteinvertible volatility matrix σ : [0 , T ] → R d × d + .We have an exponentially distributed time τ ∼ Exp( η ) with parameter η ≥
0, and with τ, W independent. Define the c`adl`ag process N by(2.1) N t := { t<τ } , t ∈ [0 , T ] . We take F := ( F t ) t ∈ [0 ,T ] to be the P -augmentation of the filtration generated by ( W, N ).An agent with initial capital x ∈ R + trades the stock plus cash, consumes wealth at a non-negative adapted rate c = ( c t ) t ∈ [0 ,T ] , and receives income at some non-negative bounded adaptedrate f = ( f t ) t ∈ [0 ,T ] . Let a ≥ randomly terminating constantincome stream , for which f is given by f t := aN t = a { t<τ } , t ∈ [0 , T ] . Thus, the stochastic income stream pays at the constant rate a up to the random time τ , at whichpoint it abruptly terminates.The agent’s wealth process is X , which follows(2.2) d X t = d (cid:88) i =1 H it d S it + r t (cid:32) X t − d (cid:88) i =1 H it S it (cid:33) d t − c t d t + f t d t, X = x > , where the trading strategy H = ( H , . . . , H d ) is a predictable S -integrable process for the numberof shares of each stock held. When the wealth is strictly positive, with π i := H i S i /X, i = 1 , . . . , d ANDOMLY TERMINATING INCOME 5 the process for the proportion of wealth held in each stock and with π = ( π , . . . , π d ) the associatedvector, the wealth dynamics may also be written as(2.3) d X t = ( r t X t − c t + f t ) d t + X t π (cid:62) t σ t ( λ t d t + d W t ) , X = x > , where λ : [0 , T ] → R d is the stock’s d -dimensional market price of risk vector, satisfying σλ = ( b − r ),with denoting the d -dimensional unit vector. The consumption rate process c is assumed toalmost surely satisfy the minimal integrability condition (cid:82) T c s d s < ∞ , with appropriate integrabilityconditions on π ensured by the S -integrability of the trading strategy H .For any process P , let (cid:101) P denote its discounted incarnation, so that (cid:101) P := exp (cid:0) − (cid:82) · r s d s (cid:1) P . Interms of discounted quantities, the wealth process has decomposition (cid:101) X = (cid:101) X + F − C , where(2.4) (cid:101) X := x + ( H · (cid:101) S )is the discounted wealth process of a self-financing portfolio corresponding to strategy H , where( H · (cid:101) S ) ≡ (cid:82) · H s d (cid:101) S s denotes the stochastic integral and C := (cid:82) · (cid:101) c s d s, F := (cid:82) t · (cid:101) f s d s denote thenon-decreasing cumulative discounted consumption and income processes.As is known from Vellekoop and Davis [35], because the income stream terminates randomly, theagent is not able to follow the classical program of borrowing against the present value of futureincome (so allowing wealth to become negative) and using the optimal no-income strategy with aninitial wealth enlarged by the present value of future income. We shall therefore assume solvencyat all times, so X ≥ x >
0, we call the pair (
H, c )(or (
X, c ), or ( π, c )) an x -admissible investment-consumption strategy. Denote the x -admissibleinvestment-consumption strategies by H ( x ).If, for a consumption process c we can find a predictable S -integrable process H such that ( H, c ) ∈H ( x ) is an x -admissible investment-consumption strategy, then we say that c is an x -admissibleconsumption process or, briefly, an admissible consumption plan. Denote the set of x -admissibleconsumption plans by A ( x ):(2.5) A ( x ) := (cid:26) c ≥ ∃ H such that (cid:101) X := x + ( H · (cid:101) S ) + (cid:90) · ( (cid:101) f s − (cid:101) c s ) d s ≥ , a.s (cid:27) , x > . For x = 1 we write A ≡ A (1). It is easy to verify that A ( x ) is a convex set.For c ≡ f ≡
0, the wealth process is that of a self-financing portfolio, with discounted wealthprocess (cid:101) X as in (2.4). Define X ( x ) as the set of almost surely non-negative self-financing wealthprocesses with initial value x > X ( x ) := (cid:110) X : (cid:101) X = x + ( H · (cid:101) S ) ≥ , a.s. (cid:111) , x > . We write
X ≡ X (1), with X ( x ) = x X for x >
0, and we note that X is a convex set.2.2. The primal problem.
Let U : R + → R be a utility function, strictly concave, strictly in-creasing, continuously differentiable on R + and satisfying the Inada conditions(2.6) lim x ↓ U (cid:48) ( x ) = + ∞ , lim x →∞ U (cid:48) ( x ) = 0 . In numerical solutions (see Section 6.2) we shall take U ( · ) to be a constant relative risk aversion(CRRA) utility function of the power form.Let δ > , T ] in the presence of the terminating incomestream. The primal value function u : R + → R is defined by(2.7) u ( x ) = sup c ∈A ( x ) E (cid:20)(cid:90) T U ( c t ) d κ t (cid:21) , x > , This becomes transparent when one converts the problem to one involving an inter-temporal wealth objective,with a utility of inter-temporal wealth that requires a non-negative argument, as in Section 5.
ASHLEY DAVEY, MICHAEL MONOYIOS, AND HARRY ZHENG where κ : [0 , T ] → R + is discounted Lebesgue measure, given by(2.8) κ = 0 , d κ t = e − δt d t. For later use, define the positive process ζ = ( ζ t ) t ∈ [0 ,T ] as the reciprocal of ( d κ t / d t ) t ∈ [0 ,T ] :(2.9) ζ t := (cid:18) d κ t d t (cid:19) − = exp ( δt ) , t ∈ [0 , T ] . Our first goal will be to develop a rigorous dual characterisation for the problem in (2.7). Weshall later develop numerical solutions involving deep learning techniques for this problem and alsofor a variant where we include a terminal wealth objective as well as the consumption objective.2.3.
Deflators and the budget constraint.
With the one-jump process N in (2.1) we associatethe non-negative c`adl`ag ( P , F )-martingale M , defined by M t := N t + η (cid:90) t N s d s, t ∈ [0 , T ] . Let Z denote the set of (local) martingale deflators, such that deflated discounted self-financingwealth is a local martingale. The set Z is composed of positive martingales Z given by(2.10) Z := E ( − λ · W − ψ · M ) , where E ( · ) denotes the stochastic exponential, for c`agl`ad adapted processes ψ satisfying ψ > − , ψ < + ∞ almost surely. We note that as long as ψ < + ∞ , each Z ∈ Z is indeed a martingale.The multiplicity of processes Z ∈ Z , equivalently of integrands ψ in (2.10), is the manifestation ofthe market incompleteness induced by the presence of the randomly terminating income. In the caseof no income (corresponding formally to the limit η → ∞ , so the income terminates immediately)and also in the case of non-terminating income (so f t = a > t ∈ [0 , T ], correspondingformally to the limit η →
0) there is a unique martingale deflator Z (0) := E ( − λ · W ).For each Z ∈ Z and X ∈ X , the process (cid:101) X Z is a local martingale (and also a super-martingale),so Z is equivalently defined by(2.11) Z := (cid:110) Z > , c`adl`ag , Z = 1 : (cid:101) X Z is a local martingale, for all X ∈ X (cid:111) . The set Z is clearly convex.For each Z ∈ Z , we may define an associated supermartingale deflator as the discounted martin-gale deflator (cid:101) Z := exp (cid:0) − (cid:82) · r s d s (cid:1) Z, Z ∈ Z , and we denote the set of such supermartingales withinitial value y > (cid:101) Y ( y ):(2.12) (cid:101) Y ( y ) := (cid:110) Y : Y = y (cid:101) Z = y exp (cid:0) − (cid:82) · r s d s (cid:1) Z , for Z ∈ Z (cid:111) , y > . We write (cid:101)
Y ≡ (cid:101) Y (1), and we have (cid:101) Y ( y ) = y (cid:101) Y for y >
0. The set (cid:101) Y is clearly convex, inheriting thisproperty from Z .Define a further set Y ( y ) of supermartingale deflators with initial value y > Y ( y ) := (cid:8) Y > , c`adl`ag , Y = y : X Y is a supermartingale, for all X ∈ X (cid:9) , y > . As before, we write Y ≡ Y (1) and we have Y ( y ) = y Y for y >
0. Clearly, the set Y is convex,and it includes all the processes in the set (cid:101) Y of discounted local martingale deflators, but may includeother processes. Since X ≡ X (one may choose to hold initial wealth without investingin stocks or the cash account), each Y ∈ Y is a supermartingale. We shall refer to Y as the setof supermartingale deflators. The processes Y ∈ Y correspond to the classical deflators defined Note a small typographical error in Vellekoop and Davis [35], who state at the start of their Section III that theprocess (cid:101) N := n − η (cid:82) · (1 − n s ) d s is a martingale, where they have n t := { t ≤ τ } , t ≥
0. This is not correct, but becomesso if n t , t ≥ t := { t ≥ τ } , t ≥ ANDOMLY TERMINATING INCOME 7 by Kramkov and Schachermayer [22, 23] in their seminal treatment of the terminal wealth utilitymaximisation problem. We thus have the inclusion(2.14) Y ⊇ (cid:101) Y . Because the set of supermartingale deflators is non-empty, the no-arbitrage assumption implicitin our model,(2.15) Y ( y ) (cid:54) = ∅ . is tantamount to the no unbounded profit with bounded risk (NUPBR) condition (see Karatzas andKardaras [20]). In fact, given the inclusion (2.14), the one-to-one correspondence between (cid:101) Y and Z , the fact that each Z ∈ Z is a martingale, and the finite horizon, we actually have the existenceof equivalent local martingale measures (ELMMs), so we have the stronger no-arbitrage conditionof no free lunch with vanishing risk (NFLVR), in accordance with Delbaen and Schachermayer[8]. However, we emphasise the existence of deflators, as in (2.15), since the tools we shall use inestablishing a duality for the problem in (2.7) will include optional decomposition results based ondeflators, due to Stricker and Yan [34]. It is now well-known, following Karatzas and Kardaras[20], that the minimal requirement for well-posed utility maximisation problems is the existenceof a suitable class of deflators which act on primal variables to create supermartingales, and ourapproach is in this spirit.2.3.1. The budget constraint.
The key to identifying the dual problem is the following budget con-straint involving admissible consumption plans and deflators. We shall derive this constraint fordeflators Y ∈ (cid:101) Y , and this will motivate the definition of our ultimate dual domain further below,which will be comprised of all processes such that a key supermartingale property holds. Lemma 2.1 (Budget constraint) . For x, y > , and for all admissible consumption plans c ∈ A ( x ) and deflators Y ∈ (cid:101) Y ( y ) we have the budget constraint (2.16) E (cid:20)(cid:90) T ( c t − f t ) Y t d t (cid:21) ≤ xy, x, y > . Proof.
Recalling the wealth dynamics (2.3), the Itˆo product rule applied to XY yields(2.17) XY + (cid:90) · ( c s − f s ) Y s d s = xy + (cid:90) · X s Y s ( π (cid:62) s σ s − λ s ) d W s − (cid:90) · X s − Y s − ψ s d M s . For any t ∈ [0 , T ] the random variable (cid:82) t f s Y s d s is P -integrable, since we have E (cid:20)(cid:90) t f s Y s d s (cid:21) ≤ a (cid:90) t E [ Y s ] d s < ∞ , t ∈ [0 , T ] . Hence, the integrand on the left-hand-side of (2.17) is bounded below by an integrable randomvariable. Since XY is non-negative, the right-hand side of (2.17) is a local martingale boundedbelow by an integrable random variable, so the Fatou lemma gives that(2.18) XY + (cid:82) · ( c s − f s ) Y s d s is a supermartingale, ∀ c ∈ A ( x ) and Y ∈ (cid:101) Y ( y ) . Using (2.18) and noting that XY is non-negative, we immediately obtain (2.16). (cid:3) Consumption deflators.
Motivated by the supermartingale property in (2.18), we now definethe space which will form the dual domain for the consumption problem (2.7). Define(2.19) Y ( y ) := (cid:8) Y > , c`adl`ag , Y = y : XY + (cid:82) ( c s − f s ) Y s d s is a supermartingale, ∀ c ∈ A ( x ) (cid:9) . As usual, we write
Y ≡ Y (1) and we have Y ( y ) = y Y for y >
0. The set Y is easily seen to beconvex. In (2.19), the wealth process X is the one on the left-hand-side of (2.2), so incorporatingconsumption and income. Since ( X, c ) ≡ (1 , f ) is an admissible consumption-investment pair, each ASHLEY DAVEY, MICHAEL MONOYIOS, AND HARRY ZHENG Y ∈ Y ( y ) is a supermartingale, so includes the wealth deflators in the set Y ( y ) of (2.13). Notingthe inclusion in (2.14), we thus have(2.20) Y ⊇ Y ⊇ (cid:101) Y . The dual domain Y ( y ) , y > consumption deflators (or, simply, as deflators, when no confusion arises) todistinguish them from the corresponding deflators when the consumption and income processes areabsent or equal, so cancel out.The budget constraint as derived in Lemma 2.1 then extends trivially from (cid:101) Y to Y , and we have:(2.21) E (cid:20)(cid:90) T ( c t − f t ) Y t d t (cid:21) ≤ xy, ∀ c ∈ A ( x ) , Y ∈ Y ( y ) , x, y > . The budget constraint in (2.21) thus constitutes a necessary condition for admissible consumptionplans. We show later (in Lemma 7.2) that it is also a sufficient condition for such consumptionplans.2.4.
The dual problem.
Let V : R + → R denote the convex conjugate of the utility function U ( · ),defined by V ( y ) := sup x> [ U ( x ) − xy ] , y > . The map y (cid:55)→ V ( y ) , y >
0, is strictly convex, strictly decreasing, continuously differentiable on R + , − V ( · ) satisfies the Inada conditions, and we have the bi-dual relation U ( x ) := inf x> [ V ( y ) + xy ] , x > , as well as V (cid:48) ( · ) = − I ( · ) = − ( U (cid:48) ) − ( · ), where I ( · ) denotes the inverse of marginal utility. Inparticular, we have the inequality(2.22) V ( y ) ≥ U ( x ) − xy, ∀ x, y > , with equality iff U (cid:48) ( x ) = y. The dual to the primal problem (2.7) is motivated in the usual manner with the aid of the budgetconstraint (2.21). We have, on recalling the process ζ of (2.9), E (cid:20)(cid:90) T U ( c t ) d κ t (cid:21) ≤ E (cid:20)(cid:90) T U ( c t ) d κ t (cid:21) + xy − E (cid:20)(cid:90) T ( c t − f t ) Y t d t (cid:21) (2.23) = E (cid:20)(cid:90) T ( U ( c t ) − c t ζ t Y t + f t ζ t Y t ) d κ t (cid:21) + xy ≤ E (cid:20)(cid:90) T ( V ( ζ t Y t ) + f t ζ t Y t ) d κ t (cid:21) + xy, the last inequality a consequence of (2.22). We therefore define the dual value function by(2.24) v ( y ) := inf Y ∈Y ( y ) E (cid:20)(cid:90) T ( V ( ζ t Y t ) + f t ζ t Y t ) d κ t (cid:21) , y > . We shall assume throughout that the dual problem is finitely valued:(2.25) v ( y ) < ∞ , ∀ y > . It is well known that the assumption (2.25) acts an alternative mild feasibility condition to the rea-sonable asymptotic elasticity condition of Kramkov and Schachernayer [22] (the latter accompaniedby the assumption that the primal value function is finitely valued for some x >
0) that ensures theusual tenets of a duality theory can hold, as detailed by Kramkov and Schachermayer [23].
ANDOMLY TERMINATING INCOME 9
Elementary properties of the value function.
We end this section with a simple lemmaon some elementary properties of the primal value function, that we shall take as given at the outset.Let u : R + → R denote the value function in the case where there is no income. In the case wherethe income does not terminate, so f t = a for all t ∈ [0 , T ], let ϕ ( f ) denote the risk-adjusted valueof the non-terminating income stream. With Z (0) := E ( − λ · W ) denoting the unique martingaledeflator of the complete market in the presence of the non-terminating income stream, ϕ ( f ) is givenby ϕ ( f ) = E (cid:104)(cid:82) T a exp (cid:16) − (cid:82) t r s d s (cid:17) Z (0) t d t (cid:105) < ∞ . For example, in the case of a one-dimensionalBlack-Scholes model, with λ ∈ R and r > ϕ ( f ) = a (1 − exp( − rT )) /r . Lemma 2.2.
The primal value function u ( · ) in (2.7) is increasing and concave on R + . It satisfiesthe bounds (2.26) u ( x ) ≤ u ( x ) ≤ u ( x + ϕ ( f )) , x > , and thus the maximal expected utility is finite, u ( x ) ∈ ( −∞ , ∞ ) for all x > .The derivative of the primal value function at infinity is given by (2.27) u (cid:48) ( ∞ ) := lim x →∞ u (cid:48) ( x ) = 0 . Proof.
Monotonicity and concavity of u ( · ) is a standard argument (see Rogers [32, Proposition1.1]), by considering convex combinations of initial wealths and of control processes, and using themonotonicity and concavity of the utility function.It is well known (see Vellekoop and Davis [35]) that the value function for the terminating incomeproblem is bounded above (respectively, below) by the value function for the problem with non-terminating income (respectively, no income). It is also well known (see [35] again) that, denotingthe value function for the non-terminating income problem by u a ( · ), we have u a ( x ) = u ( x + ϕ ( f )) , x + ϕ ( f ) >
0, from which the bounds in (2.26) follow.It is well known that the no-income value function satisfies the Inada condition u (cid:48) ( ∞ ) = 0. Since ϕ ( f ) < ∞ we see from (2.26) that the value functions with non-terminating income, terminatingincome and no income all coalesce at large values of wealth, as do their derivatives, and the Inadacondition for u (cid:48) ( · ) at infinity yields (2.27). (cid:3) Remark . Naturally, the value function for the non-terminating income problem is defined fornegative values of initial wealth. This is the concrete manifestation of the fact that the agent canborrow against the (known in advance) present value of future income, and implement the no-incomeoptimal strategy with the increased initial capital x + ϕ ( f ). (It is precisely this strategy that is notavailable to the agent when the income is randomly terminating.) Remark . We shall also prove (2.27) during the course of proving our abstract duality theorem,using an entirely distinct method (see Lemma 7.14).3.
The duality theorem
Here is the main duality result of the paper (Theorem 3.1 below), a dual characterisation of thesolution to the terminating income problem (2.7). This shows that many of the usual tenets ofduality theory hold, with consumption over income replacing consumption in the budget constraintand optimal wealth process representation. A yet more noteworthy feature is that the dual valuefunction’s effective domain is no longer infinite, but only extends up to the finite value of y > finite . This is a result of the presence of the income stream which, thoughit can terminate randomly, will not do so until a time strictly greater than zero. So the agent isbound to have some income, no matter how small, and so is not infinitely penalised for having zero initial wealth. One conjectures that as η → ∞ we expect to recover the Inada condition at zero, asthis would correspond to the problem without income (though proving continuity in the intensityof the exponential time is far beyond the scope of this paper). The conclusion that u (cid:48) (0) < + ∞ ,and the closing of the duality gap found by Davis and Vellekoop [35] (in the infinite horizon versionof the problem), are major contributions of the theorem which follows. We conjecture that thesefeatures would extend to the infinite horizon version of the problem, but that is also a topic for futureresearch. Note also that our conclusion of a finite primal derivative at zero is in agreement with thenumerical results we present in the second half of the paper. It is not in line with the assertion in[35, Theorem 2.1], which used some heuristic differential equation arguments to conjecture that theprimal derivative at zero was infinite. Theorem 3.1 (Consumption with randomly terminating income duality) . Define the primal valuefunction u ( · ) by (2.7) and the dual value function v ( · ) by (2.24) . Assume (2.6) , (2.15) and (2.25) .Define the variable y ∗ > as the smallest value of y > at which the derivative of the dual valuefunction reaches zero: (3.1) y ∗ := inf { y > v (cid:48) ( y ) = 0 } . Then: (i) u ( · ) and v ( · ) are conjugate: v ( y ) = sup x> [ u ( x ) − xy ] , u ( x ) = inf y ∈ (0 ,y ∗ ) [ v ( y ) + xy ] , x > , < y < y ∗ . (ii) The primal and dual optimisers (cid:98) c ( x ) ∈ A ( x ) and (cid:98) Y ( y ) ∈ Y ( y ) exist and are unique, so that u ( x ) = E (cid:20)(cid:90) T U ( (cid:98) c t ( x )) d κ t (cid:21) , v ( y ) = E (cid:20)(cid:90) T (cid:16) V ( ζ t (cid:98) Y t ( y )) + f t ζ t (cid:98) Y t ( y ) (cid:17) d κ t (cid:21) , x > , y ∈ (0 , y ∗ ) . (iii) With y = u (cid:48) ( x ) (equivalently, x = − v (cid:48) ( y ) ), the primal and dual optimisers are related by U (cid:48) ( (cid:98) c t ( x )) = ζ t (cid:98) Y t ( y ) , equivalently , (cid:98) c t ( x ) = − V (cid:48) ( ζ t (cid:98) Y t ( y )) , t ∈ [0 , T ] , and satisfy (3.2) E (cid:20)(cid:90) T ( (cid:98) c t ( x ) − f t ) (cid:98) Y t ( y ) d t (cid:21) = xy. Moreover, the associated optimal wealth process (cid:98) X ( x ) is given by (3.3) (cid:98) X t ( x ) (cid:98) Y t ( y ) = E (cid:20) (cid:90) Tt ( (cid:98) c s ( x ) − f s ) (cid:98) Y s ( y ) d s (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) , t ∈ [0 , T ] , and the process (cid:98) X t ( x ) (cid:98) Y t ( y ) + (cid:82) · ( (cid:98) c s ( x ) − f s ) (cid:98) Y s ( y ) d s is a uniformly integrable martingale. (iv) The functions u ( · ) and − v ( · ) are strictly increasing, strictly concave and differentiable ontheir respective domains, and the variable y ∗ of (3.1) satisfies y ∗ < + ∞ , so that the primalInada condition is violated at zero: u (cid:48) (0) := lim x ↓ u (cid:48) ( x ) < + ∞ . The primal Inada condition at infinity holds true, so that u (cid:48) ( ∞ ) := lim x →∞ u (cid:48) ( x ) = 0 , − v (cid:48) (0) := lim y ↓ ( − v (cid:48) ( y )) = + ∞ . Moreover, the derivatives of the value functions satisfy xu (cid:48) ( x ) = E (cid:20)(cid:90) T U (cid:48) ( (cid:98) c t ( x ))( (cid:98) c t ( x ) − f t ) d κ t (cid:21) , x > ,yv (cid:48) ( y ) = E (cid:20)(cid:90) T V (cid:48) ( ζ t (cid:98) Y t ( y ))( (cid:98) Y t ( y ) + f t ) d t (cid:21) , y ∈ (0 , y ∗ ) . ANDOMLY TERMINATING INCOME 11
The proof of Theorem 3.1 will be given in Section 7, and will proceed by proving an abstractversion of the theorem, which is stated in the next section.We have used the set Y ( y ) of consumption deflators in the definition (2.24) of the dual valuefunction, and we have the inclusion (2.20). An important question is whether one can also definethe dual value function as an infimum over the set (cid:101) Y ( y ) of discounted martingale deflators. Thenext theorem shows that this is the case if we take the closure of (cid:101) Y ( y ) (with respect the topology ofconvergence in measure κ × P ) as the dual domain. We shall see in due course that this is becausethe set (cid:101) Y is dense (in the appropriate topology) in Y (see the proof of Proposition 4.3). Theorem 3.2.
The dual value function (2.24) also has the representation v ( y ) := inf Y ∈ cl( (cid:101) Y ( y )) E (cid:20)(cid:90) T ( V ( ζ t Y t ) + f t ζ t Y t ) d κ t (cid:21) , y > , where (cid:101) Y ( y ) is the set of discounted local martingale deflators defined in (2.12) , and cl( · ) denotes theclosure with respect to convergence in measure κ × P . The proof of Theorem 3.2 will rest on Proposition 4.3 which connects two dual domains in theabstract formulations of our optimisation problems in Section 4.4.
The abstract duality
In this section we state an abstract duality theorem, from which Theorem 3.1 will follow. Proofsof the results here will follow in Section 7.Set Ω := [0 , T ] × Ω. Let G denote the optional σ -algebra on Ω , that is, the sub- σ -algebra of B ([0 , T ]) ⊗ F generated by evanescent sets and stochastic intervals of the form (cid:74) T , T (cid:74) for arbitrarystopping times 0 ≤ T < T ≤ T . Define the measure µ := κ × P on ( Ω , G ). On the resultingfinite measure space ( Ω , G , µ ), denote by L ( µ ) the space of non-negative µ -measurable functions,corresponding to non-negative processes over [0 , T ].The primal and dual domains for our optimisation problems (2.7) and (2.24) are now considered assubsets of L ( µ ). The abstract primal domain C ( x ) is identical to the set of admissible consumptionplans, now considered as a subset of L ( µ ):(4.1) C ( x ) := { g ∈ L ( µ ) : g = c, µ -a.e., for some c ∈ A ( x ) } , x > . As always we write
C ≡ C (1), and the set C is convex. (Since C = A we do not really need to introducethe new notation, and do so only for some notational symmetry in the abstract formulation.) In theabstract notation, the primal value function (2.7) is written as(4.2) u ( x ) := sup g ∈C ( x ) (cid:90) Ω U ( g ) d µ, x > . For the dual problem, the abstract dual domain D ( y ) , y > h = ζY appearing in the dual value function (2.24). We thus define(4.3) D ( y ) := { h ∈ L ( µ ) : h = ζY, µ -a.e., for some Y ∈ Y ( y ) } , y > . As usual, we write
D ≡ D (1), we have D ( y ) = y D for y >
0, and the set D is convex. With thisnotation, the dual problem (2.24) takes the form(4.4) v ( y ) := inf h ∈D ( y ) (cid:90) Ω ( V ( h ) + f h ) d µ, y > . We note that we have not enlarged the original primal or dual domains in the classical mannerakin to Kramkov and Schachermayer [22, 23], to encompass processes dominated by elements of therespective domains. We shall see that such an enlargement is not needed here, because we shall notbe basing our duality proof on an application of the bipolar theorem of Brannath and Schachermayer [5]. The reason for this is that in the budget constraint (2.21), the variable c − f is not necessarilynon-negative, which precludes the use of the bipolar theorem, as that applies to elements in L ( µ ).For later use, and in particular to state a result further below (Proposition 4.3) that will ultimatelyfurnish us with the proof of Theorem 3.2, we define the abstract domain (cid:101) D ( y ) , y > (cid:101) Y ( y ) of discounted martingale deflators:(4.5) (cid:101) D ( y ) := { h ∈ L ( µ ) : h = ζY, µ -a.e., for some Y ∈ (cid:101) Y ( y ) } , y > . As usual, we write (cid:101)
D ≡ (cid:101) D (1), we have (cid:101) D ( y ) = y (cid:101) D for y >
0, and the set (cid:101) D is convex.The abstract duality theorem will rely on certain basic properties of the sets C and D , whichwe state in Proposition 4.1 below. Further properties of these sets will be derived in the courseof proving the abstract duality theorem. In what follows we shall sometimes employ the notation (cid:104) g, h (cid:105) := (cid:82) Ω gh d µ, g, h ∈ L ( µ ). Proposition 4.1 (Properties of the primal and dual domains) . The abstract primal domain C isbounded in L ( µ ) . Moreover, it satisfies (4.6) g ∈ C ⇐⇒ (cid:104) g − f, h (cid:105) ≤ , ∀ h ∈ D . and so is a closed, convex subset of L ( µ ) .The abstract dual domain D is bounded in L ( µ ) and hence is also bounded in L ( µ ) . The proof of Proposition 4.1 will be given in Section 7.
Theorem 4.2 (Abstract duality theorem) . Define the primal value function u ( · ) by (4.2) and thedual value function v ( · ) by (4.4) . Assume that the utility function satisfies the Inada conditions (2.6) and that u ( x ) > −∞ , ∀ x > , v ( y ) < ∞ , ∀ y > . Set (4.7) y ∗ := inf { y > v (cid:48) ( y ) = 0 } . Then, with Proposition 4.1 in place, we have: (i) u ( · ) and v ( · ) are conjugate: (4.8) v ( y ) = sup x> [ u ( x ) − xy ] , u ( x ) = inf y ∈ (0 ,y ∗ ) [ v ( y ) + xy ] , x > , y ∈ (0 , y ∗ ) . (ii) The primal and dual optimisers (cid:98) g ( x ) ∈ C ( x ) and (cid:98) h ( y ) ∈ D ( y ) exist and are unique, so that u ( x ) = (cid:90) Ω U ( (cid:98) g ( x )) d µ, v ( y ) = (cid:90) Ω (cid:16) V ( (cid:98) h ( y )) + f (cid:98) h ( y ) (cid:17) d µ, x > , y ∈ (0 , y ∗ ) . (iii) With y = u (cid:48) ( x ) (equivalently, x = − v (cid:48) ( y ) ), the primal and dual optimisers are related by U (cid:48) ( (cid:98) g ( x )) = (cid:98) h ( y ) , equivalently , (cid:98) g ( x ) = − V (cid:48) ( (cid:98) h ( y )) , and satisfy (cid:104) (cid:98) g ( x ) − f, (cid:98) h ( y ) (cid:105) = xy. (iv) u ( · ) and − v ( · ) are strictly increasing, strictly concave and differentiable on their respectivedomains. The constant y ∗ in (4.7) is finite: y ∗ < + ∞ , so that the primal value function hasfinite derivative at zero: u (cid:48) (0) := lim x ↓ u (cid:48) ( x ) < + ∞ , while the derivative of the primal value function at infinity and of the dual value function atzero satisfy u (cid:48) ( ∞ ) := lim x →∞ u (cid:48) ( x ) = 0 , − v (cid:48) (0) := lim y ↓ ( − v (cid:48) ( y )) = + ∞ . ANDOMLY TERMINATING INCOME 13
Furthermore, the derivatives of the value functions satisfy xu (cid:48) ( x ) = (cid:90) Ω U (cid:48) ( (cid:98) g ( x ))( (cid:98) g ( x ) − f ) d µ, yv (cid:48) ( y ) = (cid:90) Ω (cid:16) V (cid:48) ( (cid:98) h ( y )) + f (cid:17) (cid:98) h ( y ) d µ, x > , y ∈ (0 , y ∗ ) . The proof of Theorem 4.2 will follow in Section 7.Theorem 3.2 will rest on the following proposition, which connects the sets D and (cid:101) D . Proposition 4.3.
With respect to the topology of convergence in measure µ , the set (cid:101) D ≡ (cid:101) D (1) of (4.5) is dense in the abstract dual domain D ≡ D (1) of (4.3) . That is, we have D = cl( (cid:101) D ) , where cl( · ) denotes the closure with respect to the topology of convergence in measure µ . The proof of Proposition 4.3 will be given in Section 7, following the proofs of the abstract andconcrete duality theorems. 5.
The stochastic control approach
In this section, we prepare for a numerical solution to the terminating income problem, incor-porating in addition a terminal wealth objective, by adopting a stochastic control and dynamicprogramming perspective. We shall, in essence, integrate over the distribution of the random time τ , and convert the problem to one with an objective which combines cumulative consumption andcumulative running wealth, as well as the terminal wealth.For a starting time t ∈ [0 , T ], suppose τ > t , so the income has not yet terminated. Define thevalue function for the terminating income problem starting at t ∈ [0 , T ] by u ( t, x ) := sup ( π,c ) ∈H ( t,x ) E (cid:104)(cid:90) Tt e − δ ( s − t ) U ( c s ) d s + e − δ ( T − t ) U ( X T ) (cid:12)(cid:12)(cid:12) X t = x (cid:105) , t ∈ [0 , T ] , x > , where X has dynamics as in (2.3) and H ( t, x ) denotes the set of x -admissible investment-consumptionpairs ( π, c ) given X t = x , with U ( · ) and U ( · ) utility functions satisfying the Inada conditions. Weassume the portfolio proportion lies in some convex set K ⊆ R d , such that the wealth remainsnon-negative.We suppose that we know how to solve the problem when there is no income, so can compute(analytically, or numerically using [7]) the value function u ( · , · ) defined by u ( t, x ) := sup ( π,c ) ∈H ( t,x ) E (cid:104)(cid:90) Tt e − δ ( s − t ) U ( c s ) d s + e − δ ( T − t ) U ( X (0) T ) (cid:12)(cid:12)(cid:12) X t = x (cid:105) , t ∈ [0 , T ] , x > , where X (0) denotes the wealth process in the absence of the income stream, with dynamics given by(2.3) with f ≡
0, and H ( t, x ) denotes the corresponding set of admissible strategies in the absenceof income, such that X (0) remains almost surely non-negative. Note that u ( T, x ) = U ( x ).We can connect u ( · , · ) and u ( · , · ). Fix t ∈ [0 , T ] and τ > t , define θ := T ∧ τ and α := η + δ .Simple calculus re-casts the objective functional of the terminating income problem according tosup ( π,c ) ∈H ( t,x ) E (cid:104)(cid:90) θt e − δ ( s − t ) U ( c s ) d s + (cid:90) Tθ e − δ ( s − t ) U ( c s ) d s + e − δ ( T − t ) U ( X T ) (cid:12)(cid:12)(cid:12) X t = x (cid:105) = sup ( π,c ) ∈H ( t,x ) E (cid:104)(cid:90) θt e − δ ( s − t ) U ( c s ) d s + e − δ ( θ − t ) u ( θ, X θ ) (cid:12)(cid:12)(cid:12) X t = x (cid:105) = sup ( π,c ) ∈H ( t,x ) E (cid:104)(cid:90) ∞ t η e − η ( z − t ) (cid:16)(cid:90) z ∧ Tt e − δ ( s − t ) U ( c s ) d s + e − δ ( z ∧ T − t ) u ( z ∧ T, X z ∧ T ) (cid:17) d z (cid:12)(cid:12)(cid:12) X t = x (cid:105) = sup ( π a ,c a ) ∈H a ( t,x ) E (cid:104)(cid:90) Tt e − α ( s − t ) f ( s, X as , π as , c as ) d s + e − α ( T − t ) g ( X aT ) (cid:12)(cid:12)(cid:12) X at = x (cid:105) , where, for 0 ≤ t ≤ T , X a is a modified wealth process with non-terminating income (so f t ≡ a forall t ∈ [0 , T ]), which follows(5.1) d X as = (cid:16) X as (cid:16) r s + ( π a ) (cid:62) s σ s λ s (cid:17) − c as + a (cid:17) d s + X as ( π a ) (cid:62) s σ s d W s , with H a ( t.x ) denoting the set of modified controls ( π a , c a ), and with f ( t, x, π, c ) := U ( c ) + ηu ( t, x ) , g ( x ) := U ( x ) . The structure of the problem has changed, as the wealth process X a now has non-terminating incomerate a > ∂u∂t + sup ( π,c ) (cid:16) − αu + U ( c ) + ηu ( t, x ) + ( xr + xπ (cid:62) σλ − c + a ) u x + 12 x | π (cid:62) σ | u xx (cid:17) = 0 , with the terminal condition u ( T, x ) = U ( x ). From (5.2) we can derive the optimal controls in termsof the value function and its derivatives, so ( (cid:98) π t , (cid:98) c t ) ≡ ( (cid:98) π ( t, (cid:98) X at ) , (cid:98) c ( t, (cid:98) X at )) , t ∈ [0 , T ], with (cid:98) π ( t, x ) = arg max π (cid:16) xu x π (cid:62) σλ + 12 u xx x | π (cid:62) σ | (cid:17) , (cid:98) c ( t, x ) = arg max c ( U ( c ) − u x c ) = I ( u x ) , where I ( · ) := ( U (cid:48) ) − ( · ), (cid:98) X a is the optimal wealth process, and the arguments of the value functionhave beeen suppressed for brevity.We can apply results of convex duality [4] to this transformed problem. First, note that theconstrained problem is equivalent to the unconstrained maximisation problem with f ( t, x, π, c ) = U ( c ) + ηu ( t, x ) − Φ (0 , ∞ ) ( c ) − Φ K ( π ) , where Φ A ( z ) is the infinite penalty function that equals zero if z ∈ A and ∞ otherwise. We candefine, in addition to V ( · ), the conjugate functions V ( y ) := sup x> ( U ( x ) − xy ) , v ( t, y ) := sup x> ( u ( t, x ) − xy ) , t ∈ [0 , T ] , y > . The dual running cost is not immediately obvious a priori. In particular, we will see that it is notjust a sum of dual functions. Define a strictly positive dual process ¯ Y byd ¯ Y t = ¯ Y t ( ¯ α t d t + ¯ β (cid:62) t d W t ) , ¯ Y = y , for some processes ¯ α ∈ R , ¯ β ∈ R d to be determined. By Itˆo’s Lemma, the drift of the process(e − αt X at ¯ Y t ) t ∈ [0 ,T ] is given by e − αt X at ¯ Y t ( ¯ α t − α + r t + π (cid:62) t σ t ( λ t + ¯ β t )) + e − αt ¯ Y t ( a − c t ) , t ∈ [0 , T ] . Define L t = e − αt X at ¯ Y t − (cid:90) t e − αs ¯ Y s (cid:16) a − c s + X as ( ¯ α s − α + r s + π (cid:62) s σ s ( λ s + ¯ β s )) (cid:17) d s, t ∈ [0 , T ] . By construction L has zero drift so is a local martingale. Define ¯ v := − σ ( λ + ¯ β ) and ξ := ¯ Y ( ¯ α + r − α ) /η . Then the process ¯ Y has dynamics given byd ¯ Y t = ( ¯ Y t ( α − r t ) + ηξ t ) d t − ¯ Y t ( λ t + σ − t ¯ v t ) (cid:62) d W t . The diffusion coefficient of L is X a ¯ Y ( σ (cid:62) π − λ + ( σ − ) (cid:62) ¯ v ). One can show that L is a true martingaleby using Doob’s L -inequality to show that this diffusion coefficient is square integrable. To do ANDOMLY TERMINATING INCOME 15 this we need to assume that the control processes are themselves square integrable [25]. Thus, E [ L T ] = L = x y , and by the definition of the conjugate of U ( · ) we have, for any X a , π, c, ¯ Y , ¯ v, ξ , E (cid:104)(cid:90) T e − αs f ( s, X as , π s , c s ) d s + e − αT U ( X aT ) (cid:105) ≤ E (cid:104)(cid:90) T e − αs (cid:16) f ( s, X as , π s , c s ) − X s ¯ Y s π (cid:62) s ¯ v s + ηX as ξ s + ( a − c s ) ¯ Y s (cid:17) d s + e − αT V ( ¯ Y T ) (cid:105) + x y . Maximising over X a , c, π yields u (0 , x ) ≤ E (cid:104)(cid:90) T e − αt ¯ f ( t, ¯ Y t , ¯ v t , ξ t ) d s + e − αT V ( ¯ Y T ) (cid:105) + x y , where ¯ f ( t, y, ¯ v, ξ ) = sup x,c,π (cid:16) U ( c ) − Φ (0 , ∞ ) ( c ) − Φ K ( π ) − cy + ηu ( t, x ) + ηxξ − xyπ (cid:62) ¯ v + ay (cid:17) = V ( y ) + ay + Φ (0 , ∞ ) ( y ) + sup x,π (cid:16) ηu ( t, x ) + ηxξ − xyπ (cid:62) ¯ v − Φ K ( π ) (cid:17) = V ( y ) + ay + Φ (0 , ∞ ) ( y ) + sup x ( ηu ( t, x ) + ηxξ + xyδ K (¯ v ))= V ( y ) + ay + Φ (0 , ∞ ) ( y ) + ηv (cid:16) t, − ξ − η yδ K (¯ v ) (cid:17) . Define a new control γ := − ξ − η yδ K (¯ v ). For the running cost to be finite we must have γ > y , γ, ¯ v gives us the following dual problem:inf y,v a ,γ a ˜ J ( y, v a , γ a ) := E (cid:104)(cid:90) T e − αs ˜ f ( s, Y as , γ as ) d s + e − αT ˜ g ( Y aT ) (cid:105) + x y , where Y a is the dual process satisfying the SDE(5.3) d Y at = ( Y at ( α − r t − δ K ( v at )) − ηγ at ) d t − Y at (cid:0) λ t + σ − t v at (cid:1) (cid:62) d W t , Y a = y , and ˜ f ( t, y, γ ) := V ( y ) + ay + ηv ( t, γ ) , ˜ g ( y ) := V ( y ) . The dual value function for starting time t ∈ [0 , T ] is therefore given by v ( t, y ) := inf γ a > ,v a ∈ R d E (cid:104)(cid:90) Tt e − α ( s − t ) ˜ f ( s, Y as , γ as ) d s + e − α ( T − t ) ˜ g ( Y aT ) (cid:12)(cid:12)(cid:12) Y at = y (cid:105) . Here we use the superscript “ a ” once again, since the state variable process Y a is only dual withrespect to the primal non-terminating income process X a . The dual HJB equation is ∂v∂t − αv + V ( y ) + ay + ( α − r ) y ∂v∂y + inf γ> (cid:16) ηv ( t, γ ) − ηγ ∂v∂y (cid:17) (5.4) + inf ¯ v (cid:16) − δ K (¯ v ) y ∂v∂y + 12 | λ + σ − ¯ v | y ∂ v∂y (cid:17) = 0 , with the terminal condition ˜ v ( T, y ) = ˜ g ( y ). If K is a closed convex cone, then (5.4) is reduced to asemilinear PDE:(5.5) ∂v∂t − αv + V ( y ) + ay + ( α − r ) y ∂v∂y + η inf γ> (cid:16) v ( t, γ ) − γ ∂v∂y (cid:17) + 12 | λ + σ − ˆ v | y ∂ v∂y = 0 , with the terminal condition v ( T, y ) = ˜ g ( y ), where ˆ v is the minimum point of | λ + σ − ¯ v | over ¯ v ∈ K and K = { ¯ v : ¯ v (cid:62) π ≥ , ∀ π ∈ K } , the positive polar cone of K . Remark . We have shown in Theorem 3.1 that the primal value function u ( x ) ≡ u (0 , x ) has finitederivative at x = 0 when K is the whole space (no control constraint) and U ( · ) ≡ ∂u/∂x )( t, < ∞ for0 ≤ t < T . Recalling the primal-dual relations y = ( ∂u/∂x )( t, x ) if and only if x = − ( ∂v/∂y )( t, y ),and that x >
0, we must have − ( ∂v/∂y )( t, y ) > y ∈ (0 , y ∗ ( t )), where y ∗ ( t ) < ∞ is thesmallest value of y > ∂v/∂y )( t, y ) of the intermediate dual valuefunction with respect to y reaches zero. In other words, while one can perfectly well define thedual value function as a stand-alone problem in its own right for all y >
0, the conjugacy relationbetween the primal and dual problems holds only for y ∈ (0 , y ∗ ( t )), in accordance with the results inTheorems 3.1 and 4.2. Hence, the dual value function, when viewed as the conjugate of the primalvalue function, has effective domain (0 , y ∗ ( t )), and the dual HJB equation in this case becomes avariational equation(5.6) min (cid:16) inf γ A γ v ( t, y ) , − ∂v∂y ( t, y ) (cid:17) = 0 , where A γ is the operator appearing on the left-hand-side of (5.5) with ˆ v = 0 and v ( T, y ) = 0. Thisvariational structure has also featured in other works in which constraints (such as non-negativity onwealth induced by borrowing restrictions) manifest themselves (see El Karoui and Jeanblanc-Picqu´e[11], He and Pag`es [17], and Bian and Zheng [3, Theorem 2.6]).
Remark . The HJB equations we have derived cover both the pre-stopping and post-stoppingproblems. Indeed, by taking η = 0 in (5.2) and (5.4) we recover the special case of non-terminatingincome, and additionally taking a = 0 recovers the standard no-income HJB equation. Note furtherthat taking η = 0 removes dependence on γ in the dual HJB and dual state process in (5.3). Thisis to be expected, as γ does not appear in the deterministic income problems.The value function u ( · , · ) is the concave conjugate function of the dual value function v ( · , · ), so u ( t, x ) := inf y ∈ (0 ,y ∗ ( t )) ( v ( t, y ) + xy ) , ( t, x ) ∈ [0 , T ] × R + , where y ∗ ( t ) is the free boundary at time t ∈ [0 , T ] of the variational equation (5.6). The optimalwealth process is given by (cid:98) X at = − ∂v∂y ( t, (cid:98) Y at ) , t ∈ [0 , T ] , where (cid:98) Y a is the optimal dual state process starting from (cid:98) Y a = ˆ y , with ˆ y the solution of the equation( ∂v/∂y )(0 , y ) + x = 0, and x > Deep controlled 2BSDE method
In this section we outline the method for using machine learning to solve the pre-income termi-nation problem when the value function is assumed sufficiently regular, using the ideas of [7].Define the Hamiltonian by F ( t, x, π, c, z, Γ) := f ( t, x, π, c ) + b a ( t, x, π, c ) z + 12 Γ | σ ( t, x, π, c ) | , and the generalised Hamiltonian by H ( t, x, π, c, z, q ) = f ( t, x, π, c ) + b a ( t, x, π, c ) z + σ ( t, x, π, c ) q, where b a ( t, x, π, c ) := x (cid:0) r t + π (cid:62) σ t λ t (cid:1) − c + a and σ ( t, x, π, c ) := xπ (cid:62) σ t are the drift and diffusioncoefficients of X a in (5.1) and f ( t, x, π, c ) = U ( c ) + ηu ( t, x ). Assume that u ∈ C , ([0 , T ] × R + ),and that π a , c a are optimal controls. Define the processes V a ,t = u ( t, X at ) , Z a ,t = D x u ( t, X at ) , Γ a ,t = D x u ( t, X at ) , t ∈ [0 , T ] . ANDOMLY TERMINATING INCOME 17
By an application of Itˆo’s formula these processes solve the following 2BSDE:d X at = b a ( t, X at , π at , c at ) d t + σ ( t, X at , π at , c at ) d W t , d V a ,t = (cid:2) αV a ,t − f ( t, X at , π at , c at ) (cid:3) d t + Z a ,t σ ( t, X at , π at , c at ) d W t , d Z a ,t = (cid:2) αZ a ,t − D x H (cid:0) t, X at , π a t, c at , Z a ,t , Γ a ,t σ ( t, X at , π at , c at ) (cid:1)(cid:3) d t + Γ a ,t σ ( t, X at , π at , c at ) d W t , with the terminal conditions V a ,T = U ( X aT ) , Z a ,T = U (cid:48) ( X aT ) and the optimal controls π a , c a satisfying π at , c at ∈ arg max π,c F ( t, X at , π, c, Z a ,t , Γ a ,t ) , t ∈ [0 , T ] . We solve this 2BSDE using the deep controlled 2BSDE method.We also describe the dual problem in this setting. Define the dual Hamiltonian as (cid:101) F ( t, y, v, γ, z, Γ) := ˜ f ( t, y, γ ) + ˜ b ( t, y, v, γ ) z + 12 Γ | ˜ σ ( t, y, v, γ ) | , and the generalised dual Hamiltonian as (cid:101) H ( t, y, v, γ, z, q ) := ˜ f ( t, y, γ ) + ˜ b ( t, y, v, γ ) z + ˜ σ ( t, y, v, γ ) q, where ˜ b ( t, y, v, γ ) := y ( α − r t − δ K ( v )) − ηγ and ˜ σ ( t, y, v, γ ) := − y ( λ t + σ − t v ) (cid:62) are the drift anddiffusion coefficients of Y a in (5.3) and ˜ f ( t, y, γ ) := V ( y ) + ay + ηv ( t, γ ). Then the value processes V a , Z a , Γ a , defined analogously to the primal ones, solve the 2BSDEd V a ,t = (cid:104) αV a ,t − ˜ f ( t, Y at , γ at ) (cid:105) d t + Z a ,t ˜ σ ( t, Y at , v at , γ at ) d W t , d Z a ,t = (cid:104) αZ a ,t − D y (cid:101) H (cid:0) t, Y at , v at , γ at , Z a ,t , Γ a ,t ˜ σ ( t, Y at , v at , γ at ) (cid:1)(cid:105) d t + Γ a ,t ˜ σ ( t, Y at , v at , γ at ) d W t , with the terminal conditions V a ,T = V ( Y aT ) , Z a ,T = V (cid:48) ( Y aT ) and the optimal controls v a , γ a , y satis-fying ( v at , γ at ) ∈ arg min v,γ> (cid:101) F ( t, Y at , v, γ, Z a ,t , Γ a ,t ) , t ∈ [0 , T ] ,y ∈ arg min y> (cid:18) E (cid:20)(cid:90) T e − αs (cid:101) f ( s, Y as , γ as ) d s + e − αT ˜ g ( Y aT ) (cid:21) + x y (cid:19) . By the primal-dual HJB relations, we have the following: X at = − Z a ,t , V a ,t = V a ,t − Z a ,t Y at , Z a ,t = Y at , t ∈ [0 , T ] . Solution by deep learning.
Here we describe the algorithm for solving this problem usingmachine learning. We simulate all processes in the forward direction, where the unknown processesare outputs of neural networks. We use the terminal loss condition and control optimality conditionsas loss functions to train this neural network. Our neural networks are dense feed forward networkswith two hidden layers with ten layers, of the form N ( x ; θ ) = f ,θ ◦ h ◦ f ,θ ◦ h ◦ f ,θ ◦ h ◦ f ,θ ( x ) , where f i,θ ( x ) = A i x + b i for i = 1 , , , h ( x ) = tanh( x ) is the activationfunction. The matrices A i and vectors b i depend on the parameter set θ . The input will be ourone-dimensional wealth process, and the output will be the control process or the second derivativeprocess Γ, so will be either one- or two-dimensional. Using these neural networks, we employ thefollowing Euler-Maruyama scheme. Let ( t i ) Ni =0 be a discretisation of [0 , T ], with step size h := T /N .For the primal problem we use X a = x, V a = v , Z a = z , then for t = 0 , . . . , N − π ai , c ai ) = N ( X ai ; θ i ) , Γ ai = N ( X ai ; λ i ) ,X ai +1 = X ai + b a ( t i , X ai , π ai , c ai ) h + σ ( t i , X ai , π ai , c ai ) √ h d W i ,V ai +1 = V ai + [ αV ai − f ( t i , X ai , π ai , c ai )] h + Z ai σ ( t i , X ai , π ai , c ai ) √ h d W i ,Z ai +1 = Z ai + (cid:2) αZ ai +1 − D x H ( t i , X ai , π ai , c ai , Z ai , Γ ai σ ( t i , X ai , π ai , c ai )) (cid:3) h + Γ ai σ ( t i , X ai , π ai , c ai ) √ h d W i , where d W i is standard normal for each i . The start variables v and z are optimised alongside ( λ i )using the loss function L ( v , z , ( λ i )) = E (cid:104) | V aN − U ( X aN ) | + (cid:12)(cid:12) Z aN − U (cid:48) ( X aN ) (cid:12)(cid:12) (cid:105) , where E [ · ] indicates a sample average over our current batch, when we implement the ADAM gradientdescent algorithm using mini-batches. Simultaneously, we optimise the control variables ( θ i ) usingthe loss functions L ( θ i ) = E [ F ( t i , X ai , π ai , c ai , Z ai , Γ ai )] . The dual BSDE has a similar scheme. In the dual problem we have an additional variable for thestart of the state process. However, we also know by the duality relations that Z a , = − x , so we donot need to optimise this variable. If we denote the state process by ( Y ai ), and the control processby ( γ ai ), then the loss function for y := Y a is L ( y ) = E (cid:34) N − (cid:88) i =0 β i e − αt i ˜ f ( t i , Y ai , γ ai ) + e − αT V ( Y aN ) (cid:35) , where β i = 0 . i = 0 or i = N − ADAM algorithm to update each of the parameter sets in turn, rebuilding the scheme each timewith the new parameters. We repeat this process until we achieve convergence.6.2.
Numerical example.
We take d = 1, K = R and let r, σ, λ be constants. Consider the powerutility case U ( x ) = U ( x ) = x p /p, p < , p (cid:54) = 0. Furthermore, suppose that η = 0, meaning thatthe income is either non-terminating or zero. Then we can solve the HJB equation exactly using asuitable ansatz for the value function that has the form u a ( t, x ) = 1 p g ( t ) − p ( x + h ( t ) a ) p , where g ( t ) = (cid:0) − k (cid:1) e − k ( T − t ) + k and h ( t ) = r (cid:0) − e − r ( T − t ) (cid:1) if k (cid:54) = 0 for t ∈ [0 , T ]. Note that thevalue function for the perpetual optimal consumption problem can be derived by letting T → ∞ ,provided k > δ > rp − qλ , for q = − p/ (1 − p ). We use these functions to buildthe true optimal control processes.We first provide a table of value approximations of the primal and dual algorithms. For theprimal, the approximation is V , , and for the dual algorithm it is V , + X Y . When available,we compare these to analytical solutions. We also consider the duality gap relative to the primalapproximation. For this application we take the values δ = 0 . , r = 0 . , σ = 0 . , λ = 0 . , p = 0 . T = 1 .
0. Tables 1 and 2 show the approximations for varying a , η and number of time steps N . The closed-form solution is 2 . a = 0 . η = 0. For all cases we see a decrease inthe duality gap that is linear with the increasing number of time steps, however in some instancesthe dual value approximation is below the solution. It is also notable that the introduction of apositive η improves the accuracy of the algorithm. This may be due to the extra penalty terms inthe Hamiltonian and process dynamics, which help the training of the neural networks. ANDOMLY TERMINATING INCOME 19 N lower rel-err (%) upper rel-err (%) duality gap (%)5 2.79487E+00 3.72507E-01 2.80100E+00 1.53993E-01 2.19431E-0110 2.79962E+00 2.03185E-01 2.80312E+00 7.84224E-02 1.24986E-0120 2.80202E+00 1.17634E-01 2.80420E+00 3.99241E-02 7.78521E-0250 2.80371E+00 5.76122E-02 2.80501E+00 1.11405E-02 4.64984E-02 Table 1.
Results for a = 0 . , η = 0 for power utility. N lower upper duality gap (%)5 2.72674E+00 2.72798E+00 4.62485E-0210 2.72996E+00 2.73066E+00 2.25402E-0220 2.73167E+00 2.73215E+00 1.51645E-0250 2.73284E+00 2.73305E+00 7.74071E-03 Table 2.
Results for a = 0 . , η = 2 for power utility.Figure 1 shows how the value approximation behaves between these corner cases. Graph (A)shows the relation of the termination frequency η to the income a . We see that the value is moresensitive to changes in the income size than the termination frequency. Graph (B) shows the valueapproximation as the terminal time increases for different termination frequencies. (a) Sensitivity of value to a and η . (b) Initial value approximation against ter-minal time for different values of η . Figure 1.
Sensitivity of initial value for power utility.An interesting property of this problem is that even if one starts with zero initial wealth, theincome allows us to generate wealth, and as such the agent’s value will not be zero. In Figure 2we look at the limit of the initial investment and consumption strategy, value, and derivative of thevalue as the initial wealth x goes to zero. Due to instability issues (and the benchmark investmentgoing to infinity), we cannot simply plug in zero into our algorithms, so we consider sufficiently smallvalues of x instead. We compare the initial terms for various valued of η , while fixing a = 0 .
5. Thelines corresponding to η = 0 and ∞ are calculated using the closed form solutions evaluated at timezero. The two intermediate cases are found using our primal 2BSDE algorithm, using the variables π a , c a , v , and z respectively. We see that the physical amount consumed or invested decreasedlinearly with the initial wealth, but is higher for smaller values of η , which corresponds to expectingto receive income for longer. The value has a roughly square root relationship with initial wealth,the same as the utility. The derivative process increases drastically at zero as η decreases, indicatingthe stark difference between some initial wealth and no wealth when the income terminates rapidly. (a) Initial investment strategy. (b)
Initial consumption strategy. (c)
Initial value. (d)
Initial space derivative of value.
Figure 2.
Initial control strategies and value for different initial wealths for powerutility. 7.
Proofs of the duality theorems
In this section, we prove the abstract duality of Theorem 4.2, and then establish the concreteduality of Theorem 3.1. We proceed via a series of lemmas, establishing first that the budgetconstraint (2.21) is a sufficient as well as necessary condition for admissibility, in Lemma 7.2 below.This allows us to establish Proposition 4.1. We then proceed to establish the abstract duality, byderiving a series of lemmas on further properties of the abstract primal and dual domains.7.1.
Sufficiency of the budget constraint.
Setting x = y = 1, the budget constraint (2.21) givesus the implications(7.1) c ∈ A = ⇒ E (cid:20)(cid:90) T ( c t − f t ) Y t d t (cid:21) ≤ , ∀ Y ∈ Y , and indeed also Y ∈ Y = ⇒ E (cid:104)(cid:82) T ( c t − f t ) Y t d t (cid:105) ≤ , ∀ c ∈ A .We wish to establish the reverse implication to (7.1). This requires some version of the OptionalDecomposition Theorem (ODT), originally due to El Karoui and Quenez [12] in a Brownian setting,generalised to the locally bounded semimartingale case by Kramkov [24], extended to the non-locallybounded case by F¨ollmer and Kabanov [13], and to models with constraints by F¨ollmer and Kramkov[14].The relevant version of the ODT for us is the one due to Stricker and Yan [34], which uses deflators (and in particular local martingale deflators) rather then martingale measures. In the ANDOMLY TERMINATING INCOME 21 proof of Lemma 7.2 we shall apply a part of the Stricker and Yan ODT which applies to the super-hedging of American claims, so is designed to construct a process which can super-replicate a payoffat an arbitrary time. The salient observation is that this result can also be used to dominate aconsumption stream, which is how we shall employ it. For clarity and convenience of the reader,we state here the ODT results we need, and afterwards specify precisely which results from [34] wehave taken.For t ≥
0, let T ( t ) denote the set of F -stopping times with values in [ t, T ]. For t = 0, write T ≡ T (0), and recall the set Z of (local) martingale deflators in (2.11), as well as the set (cid:101) Y ofdiscounted local martingale deflators in (2.12). Theorem 7.1 (Stricker and Yan [34] ODT) . (i) Let W be an adapted non-negative process.The process ZW is a supermartingale for each Z ∈ Z if and only if W admits a decomposi-tion of the form W = W + ( φ · (cid:101) S ) − A, where φ is a predictable S -integrable process such that Z ( φ · (cid:101) S ) is a local martingale for each Z ∈ Z , A is an adapted increasing process with A = 0 , and for all Z ∈ Z and ρ ∈ T , E [ Z ρ A ρ ] < ∞ . In this case, moreover, we have sup Z ∈Z ,ρ ∈T E [ Z ρ A ρ ] ≤ W . (ii) Let b = ( b t ) t ∈ [0 ,T ] be a non-negative c`adl`ag process such that sup Z ∈Z ,ρ ∈T E [ Z ρ b ρ ] < ∞ . Thenthere exists an adapted c`adl`ag process W that dominates b : W t ≥ b t almost surely for all t ∈ [0 , T ] , ZW is a supermartingale for each Z ∈ Z , and the smallest such process W isgiven by (7.2) W t = ess sup Z ∈Z ,ρ ∈T ( t ) Z t E [ Z ρ b ρ |F t ] , t ∈ [0 , T ] . Lemma 7.2 (Sufficiency of the budget constraint) . Suppose c is a non-negative adapted c`adl`agprocess that satisfies, for all Y ∈ Y , (7.3) E (cid:20)(cid:90) T ( c t − f t ) Y t d t (cid:21) ≤ . Then c ∈ A .Proof. Since c is assumed to satisfy (7.3) for all deflators Y ∈ Y , and since (cid:101) Y ⊆ Y , (7.3) is satisfiedfor any Y ∈ (cid:101) Y . For such a deflator, (7.3) thus translates to(7.4) E (cid:20)(cid:90) T ( (cid:101) c t − (cid:101) f t ) Z t d t (cid:21) ≤ , Z ∈ Z . For any stopping time ρ ∈ T , the integration by parts formula gives(7.5) ( C ρ − F ρ ) Z ρ = (cid:90) ρ ( C s − − F s − ) d Z s + (cid:90) ρ ( (cid:101) c s − (cid:101) f s ) Z s d s, ρ ∈ T , where C := (cid:82) · (cid:101) c s d s is the non-decreasing candidate cumulative discounted consumption process, and F denotes cumulative discounted income. We note that C is finite and F is bounded. The processΛ := (cid:82) · ( C s − − F s − ) d Z s is a local martingale which possesses a localising sequence ( T n ) n ∈ N , analmost surely increasing sequence of stopping times with lim n →∞ T n = T a.s. such that the stoppedprocess Λ T n t := Λ t ∧ T n , t ∈ [0 , T ] is a uniformly integrable martingale for each n ∈ N . Therefore, E (cid:104)(cid:82) ρ ∧ T n ( C s − − F s − ) d Z s (cid:105) = 0 for each n ∈ N . Using this along with the finiteness of ρ ∈ T andthe uniform integrability of Λ T n , we have E (cid:20)(cid:90) ρ ( C s − − F s − ) d Z s (cid:21) = E (cid:20) lim n →∞ (cid:90) ρ ∧ T n ( C s − − F s − ) d Z s (cid:21) = lim n →∞ E (cid:20)(cid:90) ρ ∧ T n C s − d Z s (cid:21) = 0 . Using this in (7.5) we obtain E [ Z ρ ( C ρ − F ρ )] = E (cid:20)(cid:90) ρ Z s ( (cid:101) c s − (cid:101) f s ) d s (cid:21) ≤ , the last inequality a consequence of (7.4). Since Z ∈ Z and ρ ∈ T were arbitrary, we havesup Z ∈Z ,ρ ∈T E [ Z ρ ( C ρ − F ρ )] ≤ < ∞ , and therefore, sup Z ∈Z ,ρ ∈T E [ Z ρ C ρ ] ≤ Z ∈Z ,ρ ∈T E [ Z ρ F ρ ] = 1 < ∞ , the last equality following from taking ρ ≡ W that dominates C , so W t ≥ C t , a . s ., ∀ t ∈ [0 , T ], and ZW is a super-martingale for each Z ∈ Z . From (7.2), the smallestsuch W given by W t = ess sup Z ∈Z ,ρ ∈T ( t ) Z t E [ Z ρ C ρ |F t ] , t ∈ [0 , T ] , so that W ≤
1. Further, by part (i) of Theorem 7.1, there exists a predictable S -integrableprocess H and an adapted increasing process A , with A = 0, such that W has decomposition W = W + ( H · (cid:101) S ) − A , with Z ( H · (cid:101) S ) a local martingale for each Z ∈ Z , and E [ Z ρ A ρ ] < ∞ for all Z ∈ Z and ρ ∈ T .Since W dominates C , we can define a process X and its discounted counterpart (cid:101) X by (cid:101) X t := 1 + ( H · (cid:101) S ) t , t ∈ [0 , T ] , which also dominates C , since its initial value is no smaller than W and we have dispensed withthe increasing process A . We can then add the income stream to also conclude that1 + ( H · (cid:101) S ) t + (cid:90) t (cid:101) f s d s ≥ (cid:90) t (cid:101) c s d s, t ∈ [0 , T ] , and we have a wealth process incorporating income, with initial capital 1, which dominates C , sothat c ∈ A . (cid:3) With Lemma 7.2 in place, we can now prove Proposition 4.1.
Proof of Proposition 4.1.
For the L ( µ )-boundedness of C , we shall find a positive element ¯ h ∈ D and show that C is bounded in L (¯ h d µ ) and hence bounded in L ( µ ). From the inclusions in(2.20), we have Y ⊃ (cid:101) Y , so let us choose Z (cid:51) Z ≡
1, so that Y = exp (cid:0) − (cid:82) · r s d s (cid:1) ∈ (cid:101) Y , and then¯ h := ζ exp (cid:0) − (cid:82) · r s d s (cid:1) ∈ D defines a strictly positive element of D . Observe that (cid:90) Ω ¯ h d µ = E (cid:20)(cid:90) T exp (cid:18) − (cid:90) t r s d s (cid:19) ζ t d κ t (cid:21) = E (cid:20)(cid:90) T exp (cid:18) − (cid:90) t r s d s (cid:19) d t (cid:21) ≤ T, By virtue of the budget constraint we have, for any g ∈ C , that (cid:82) Ω ( g − f )¯ h d µ ≤
1, so that (cid:90) Ω g ¯ h d µ ≤ (cid:90) Ω f ¯ h d µ ≤ a (cid:90) Ω ¯ h d µ = 1 + aT < ∞ . Thus, C is bounded in L (¯ h d µ ) and hence bounded in L ( µ ).We now turn to establishing the dual characterisation of C as expressed in (4.6). Lemma 7.2,combined with the implication in (7.1), give the equivalence, invoking the measure κ of (2.8), c ∈ A ⇐⇒ E (cid:20)(cid:90) T ( c t − f t ) ζ t Y t d κ t (cid:21) ≤ , ∀ Y ∈ Y . ANDOMLY TERMINATING INCOME 23
So, with
C (cid:51) g ≡ c and D (cid:51) ζY , in terms of the measure µ we have(7.6) g ∈ C ⇐⇒ (cid:90) Ω ( g − f ) h d µ ≤ , ∀ h ∈ D , which establishes (4.6).The equivalence (7.6) along with Fatou’s lemma yields that the set C is closed with respect to thetopology of convergence in measure µ . To see this, let ( g n ) n ∈ N be a sequence in C which converges µ -a.e. to an element g ∈ L ( µ ). For arbitrary h ∈ D we obtain, via Fatou’s lemma and the factthat g n ∈ C for each n ∈ N , (cid:90) Ω ( g − f ) h d µ ≤ lim inf n →∞ (cid:90) Ω ( g n − f ) h d µ ≤ , so by (7.6), g ∈ C , and thus C is closed. Convexity of C is clear from its definition.Finally, let us show that D is bounded in L ( µ ), and hence also bounded in L ( µ ). Each Y ∈ Y is a supermartingale, so satisfies E [ Y t ] ≤ , t ∈ [0 , T ]. With h = ζY ∈ D , we have (cid:90) Ω h d µ = E (cid:20)(cid:90) T ζ t Y t d κ t (cid:21) = E (cid:20)(cid:90) T Y t d t (cid:21) = (cid:90) T E [ Y t ] d t ≤ T < ∞ . Thus D is bounded in L ( µ ), and hence also bounded in L ( µ ). (cid:3) The abstract duality proof.
We now take Proposition 4.1 as given in the remainder of thissection, and proceed with the proof of the abstract duality of Theorem 4.2 via a series of lemmas.The first step is to establish weak duality.
Lemma 7.3 (Weak duality) . The primal and dual value functions u ( · ) and v ( · ) of (4.2) and (4.4) satisfy the weak duality bounds (7.7) v ( y ) ≥ sup x> [ u ( x ) − xy ] , y > , equivalently u ( x ) ≤ inf y> [ v ( y ) + xy ] , x > . As a result, u ( x ) is finitely valued for all x > . Moreover, we have the limiting relations (7.8) lim sup x →∞ u ( x ) x ≤ , lim inf y →∞ v ( y ) y ≥ . Proof.
For any g ∈ C ( x ) and h ∈ D ( y ), using the budget constraint (cid:82) Ω ( g − f ) h d µ ≤ xy in the samemanner as the arguments leading to (2.23), we may bound the achievable utility according to(7.9) (cid:90) Ω U ( g ) d µ ≤ (cid:90) Ω ( V ( h ) + f h ) d µ + xy, x, y > , Maximising the left-hand-side of (7.9) over g ∈ C ( x ) and minimising the right-hand-side over h ∈D ( y ) gives u ( x ) ≤ v ( y ) + xy , and (7.7) follows.The assumption that v ( y ) < ∞ for all y > u ( x ) is finitely valued forsome x >
0. Since U ( · ) is strictly increasing and strictly concave, and given the convexity of C ,these properties are inherited by u ( · ), which is therefore finitely valued for all x >
0. Finally, therelations in (7.7) easily lead to those in (7.8). (cid:3)
The next step is to give a compactness lemma for the primal domain. The proof is on similar linesto the proof (for the dual domain) in Mostovyi [27, Lemma 3.6], and uses Delbaen and Schachermayer[8, Lemma A1.1] (adapted from a probability space to the finite measure space ( Ω , G , µ )), so forbrevity is omitted. Lemma 7.4 (Compactness lemma for C ) . Let (˜ g n ) n ∈ N be a sequence in C . Then there exists asequence ( g n ) n ∈ N with g n ∈ conv(˜ g n , ˜ g n +1 , . . . ) , which converges µ -a.e. to an element g ∈ C that is µ -a.e. finite. Here is the next step in this chain of results, a uniform integrability property associated withelements of C and the positive part of the utility function. Lemma 7.5 (Uniform integrability of ( U + ( g n )) n ∈ N , g n ∈ C ( x )) . The family ( U + ( g )) g ∈C ( x ) is uni-formly integrable, for any x > . The proof is in the spirit of Kramkov and Schachermayer [23, Lemma 1], but with some adapta-tions due to the presence of the income stream in the budget constraint.
Proof of Lemma 7.5.
Fix x >
0. If U ( ∞ ) ≤ U ( ∞ ) > U + ( g n )) n ∈ N is not uniformly integrable, then, passing if need be to a subsequencestill denoted by ( g n ) n ∈ N , we can find a constant α > A n ) n ∈ N of sets of( Ω , G ) (so A n ∈ G , n ∈ N and A i ∩ A j = ∅ if i (cid:54) = j ) such that (cid:90) Ω U + ( g n ) A n d µ ≥ α, n ∈ N . (See for example Pham [30, Corollary A.1.1].) Define a sequence ( f n ) n ∈ N of elements in L ( µ ) by f n := x + n (cid:88) k =1 g k A k , where x := inf { x > U ( x ) ≥ } .For any h ∈ D (so that (cid:82) Ω h d µ ≤ T , (cid:82) Ω ( g k − f ) h d µ ≤ x, k = 1 , . . . , n , (cid:82) Ω f h d µ ≤ aT ), we have (cid:90) Ω ( f n − f ) h d µ = (cid:90) Ω (cid:32) x + n (cid:88) k =1 g k A k − f (cid:33) h d µ ≤ x T + n (cid:88) k =1 (cid:90) Ω ( g k − f + f ) h d µ − (cid:90) Ω f h d µ = x T + n (cid:88) k =1 (cid:90) Ω ( g k − f ) h d µ + ( n − (cid:90) Ω f h d µ ≤ ( x + ( n − a ) T + nx. Thus, f n ∈ C (( x + ( n − a ) T + nx ) , n ∈ N .On the other hand, since U + ( · ) is non-negative and non-decreasing, (cid:90) Ω U ( f n ) d µ = (cid:90) Ω U + ( f n ) d µ = (cid:90) Ω U + (cid:32) x + n (cid:88) k =1 g k A k (cid:33) d µ ≥ (cid:90) Ω U + (cid:32) n (cid:88) k =1 g k A k (cid:33) d µ = n (cid:88) k =1 (cid:90) Ω U + (cid:16) g k A k (cid:17) d µ ≥ αn. Therefore, lim sup z →∞ u ( z ) z = lim sup n →∞ u (( x + ( n − a ) T + nx )( x + ( n − a ) T + nx ≥ lim sup n →∞ (cid:82) Ω U ( f n ) d µ ( x + ( n − a ) T + nx ≥ lim sup n →∞ (cid:18) αn ( x + ( n − a ) T + nx (cid:19) = αx + aT > , ANDOMLY TERMINATING INCOME 25 which contradicts the limiting weak duality bound in (7.8). This contradiction establishes the result. (cid:3)
We can now prove existence of a unique optimiser in the primal problem.
Lemma 7.6 (Primal existence) . The optimal solution (cid:98) g ( x ) ∈ C ( x ) to the primal problem (4.2) existsand is unique, so that u ( · ) is strictly concave.Proof. Fix x >
0. Let ( g n ) n ∈ N be a maximising sequence in C ( x ) for u ( x ) < ∞ (the finiteness provenin Lemma 7.3). That is(7.10) lim n →∞ (cid:90) Ω U ( g n ) d µ = u ( x ) < ∞ . By the compactness lemma for C (and thus also for C ( x ) = x C ), Lemma 7.4, we can find a sequence( (cid:98) g n ) n ∈ N of convex combinations, so C ( x ) (cid:51) (cid:98) g n ∈ conv( g n , g n +1 , . . . ) , n ∈ N , which converges µ -a.e.to some element (cid:98) g ( x ) ∈ C ( x ). We claim that (cid:98) g ( x ) is the primal optimiser. That is, that we have(7.11) (cid:90) Ω U ( (cid:98) g ( x )) d µ = u ( x ) . By concavity of U ( · ) and (7.10) we havelim n →∞ (cid:90) Ω U ( (cid:98) g n ) d µ ≥ lim n →∞ (cid:90) Ω U ( g n ) d µ = u ( x ) , which, combined with the obvious inequality u ( x ) ≥ lim n →∞ (cid:82) Ω U ( (cid:98) g n ) d µ means that we also have,further to (7.10), lim n →∞ (cid:90) Ω U ( (cid:98) g n ) d µ = u ( x ) . In other words(7.12) lim n →∞ (cid:90) Ω U + ( (cid:98) g n ) d µ − lim n →∞ (cid:90) Ω U − ( (cid:98) g n ) d µ = u ( x ) < ∞ , and note therefore that both integrals in (7.12) are finite.From Fatou’s lemma, we have(7.13) lim n →∞ (cid:90) Ω U − ( (cid:98) g n ) d µ ≥ (cid:90) Ω U − ( (cid:98) g ( x )) d µ. From Lemma 7.5 we have uniform integrability of ( U + ( (cid:98) g n )) n ∈ N , so that(7.14) lim n →∞ (cid:90) Ω U + ( (cid:98) g n ) d µ = (cid:90) Ω U + ( (cid:98) g ( x )) d µ. Thus, using (7.13) and (7.14) in (7.12), we obtain u ( x ) ≤ (cid:90) Ω U ( (cid:98) g ( x )) d µ, which, combined with the obvious inequality u ( x ) ≥ (cid:82) Ω U ( (cid:98) g ( x )) d µ , yields (7.11). The uniquenessof the primal optimiser follows from the strict concavity of U ( · ), as does the strict concavity of u ( · ).For this last claim, fix x < x and λ ∈ (0 , λ (cid:98) g ( x ) + (1 − λ ) (cid:98) g ( x ) ∈ C ( λx + (1 − λ ) x )(yet must be sub-optimal for u ( λx + (1 − λ ) x ) as it is not guaranteed to equal (cid:98) g ( λx + (1 − λ ) x ))and therefore, using the strict concavity of U ( · ), u ( λx + (1 − λ ) x ) ≥ (cid:90) Ω U ( λ (cid:98) g ( x ) + (1 − λ ) (cid:98) g ( x )) d µ > λu ( x ) + (1 − λ ) u ( x ) . (cid:3) We can now move to the dual side of the analysis, which will lead to the demonstration ofconjugacy of the value functions as well as dual existence and uniqueness. In many duality proofsthis is accompanied by an enlargement of the dual domain, a demonstration of closedness of theenlarged domain, and subsequent use of the bipolar theorem of Brannath and Schachermayer [5]on L ( µ ) to confirm that, with the enlargement, we have reached the bipolar of the original dualdomain. Here, because the variable g − f appearing in the budget constraint is not an element of L ( µ ), the use of the bipolar theorem is not available.Our program is to show that our dual domain is closed, use this to directly derive a compactnesslemma, and we proceed from there to show dual existence and conjugacy.The first step is this to show that D is closed. As in some classical proofs (see for example Kramkovand Schachermayer [22, Lemma 4.1]), we shall employ supermartingale convergence results basedon Fatou convergence of processes. For the convenience of the reader, we recall the required conceptfrom F¨ollmer and Kramkov [14] that will be needed. Definition 7.7 (Fatou convergence) . Let ( Y n ) n ∈ N be a sequence of processes on a stochastic basis(Ω , F , F := ( F t ) t ≥ , P ), uniformly bounded from below, and let T be a dense subset of R + . Thesequence ( Y n ) n ∈ N is said to be Fatou convergent on T to a process Y if Y t = lim sup s ↓ t, s ∈ T lim sup n →∞ Y ns = lim inf s ↓ t, s ∈ T lim inf n →∞ Y ns , a.s ∀ t ≥ . If T = R + , the sequence is simply called Fatou convergent .The relevant consequence for our purposes is F¨ollmer and Kramkov [14, Lemma 5.2], that for asequence ( S n ) n ∈ N of supermartingales, uniformly bounded from below, with S n = 0 , n ∈ N , thereis a sequence ( Y n ) n ∈ N of supermartingales, with Y n ∈ conv( S n , S n +1 , . . . ), and a supermartingale Y with Y ≤
0, such that ( Y n ) n ∈ N is Fatou convergent on a dense subset T of R + to Y . Here,conv( S n , S n +1 , . . . ) denotes a convex combination (cid:80) N ( n ) k = n λ k S k for λ k ∈ [0 ,
1] with (cid:80) N ( n ) k = n λ k = 1.The requirement that S n = 0 is of course no restriction, since for a supermartingale with (say) S n = 1 (as we shall have when we apply these results below for supermartingales in Y ), we canalways subtract the initial value 1 to reach a process which starts at zero.Here is the closed property for the abstract dual domain. Lemma 7.8.
The dual domain
D ≡ D (1) of (4.3) is closed with respect to the topology of convergencein measure µ .Proof. Let ( h n ) n ∈ N be a sequence in D , converging µ -a.e. to some h ∈ L ( µ ). We want to showthat h ∈ D .Since h n ∈ D , for each n ∈ N we have h n = ζ (cid:98) Y n , µ -a.e for some supermartingale (cid:98) Y n ∈ Y . FromF¨ollmer and Kramkov [14, Lemma 5.2] there exists a sequence ( Y n ) n ∈ N of supermartingales witheach Y n ∈ conv( (cid:98) Y n , (cid:98) Y n +1 , . . . ), and a supermartingale Y , such that ( Y n ) n ∈ N is Fatou convergenton a dense subset T of R + to Y .Define a supermartingale sequence ( (cid:98) V n ) n ∈ N by (cid:98) V n := X (cid:98) Y n , with X ∈ X . Once again from [14,Lemma 5.2] there exists a sequence ( V n ) n ∈ N of supermartingales with each V n ∈ conv( (cid:98) V n , (cid:98) V n +1 , . . . ) = X conv( (cid:98) Y n , (cid:98) Y n +1 , . . . ), and a supermartingale V , such that ( V n ) n ∈ N is Fatou convergent on T to V . Since V n ∈ X conv( (cid:98) Y n , (cid:98) Y n +1 , . . . ) for each n ∈ N , we have V n = X Y n , for Y n ∈ conv( (cid:98) Y n , (cid:98) Y n +1 , . . . ). Because the sequence ( Y n ) n ∈ N is Fatou convergent on T to the supermartingale Y , the sequence ( V n ) n ∈ N = ( X Y n ) n ∈ N is Fatou convergent on T to the supermartingale V = X Y .Since X Y is a supermartingale and X ∈ X , and because c ≡ f is an admissible consumption plan(equivalently because Y ⊆ Y ) we have Y ∈ Y .Finally, because h n = ζ (cid:98) Y n , µ -a.e.for each n ∈ N , we also have h = ζY, µ -a.e. (as we verify below)and thus h ∈ D , so D is closed. ANDOMLY TERMINATING INCOME 27
To verify that h n = ζ (cid:98) Y n , n ∈ N , µ -a.e. implies h = ζY, µ -a.e., we observe that, since Y n = (cid:80) N ( n ) k = n λ k (cid:98) Y k , we have(7.15) ζY n = N ( n ) (cid:88) k = n λ k ζ (cid:98) Y k ≥ N ( n ) (cid:88) k = n λ k h k , µ -a.e.Because the sequence ( Y n ) n ∈ N Fatou converges on the dense subset T to the supermartingale Y ∈ Y ,the left-hand-side of (7.15) Fatou converges on T to ζY . The right-hand-side of (7.15) converges inmeasure µ to h , so we conclude that h = ζY on a dense subset of R + , and hence that h = ζY, µ -a.e.,and the proof is complete. (cid:3) Lemma 7.8 allows us to establish a compactness lemma for D , as stated below. Lemma 7.9 (Compactness lemma for D ) . Let (˜ h n ) n ∈ N be a sequence in D . Then there exists asequence ( h n ) n ∈ N with h n ∈ conv(˜ h n , ˜ h n +1 , . . . ) , which converges µ -a.e. to an element h ∈ D that is µ -a.e. finite.Proof. Delbaen and Schachermayer [8, Lemma A1.1] (adapted from a probability space to the finitemeasure space ( Ω , G , µ )) implies the existence of a sequence ( h n ) n ∈ N , with h n ∈ conv(˜ h n , ˜ h n +1 , . . . ),which converges µ -a.e. to an element h that is µ -a.e. finite because D is bounded in L ( µ ) (thefiniteness also from [8, Lemma A1.1]). By convexity of D , each h n , n ∈ N lies in D . Finally, wenote that h ∈ D because, according to Lemma 7.8, D is closed with respect to the topology ofconvergence in measure µ . (cid:3) The next step in the chain of results we need is a uniform integrability result for the family( V − ( h )) h ∈D ( y ) . The proof uses the L ( µ )-boundedness of D and is similar to the proof in Kramkovand Schachermayer [22, Lemma 3.2], but the bound on (cid:82) Ω h d µ, h ∈ D ( y ) here is T y (note we haveused the finite horizon at this juncture) as opposed to y in the classical case of [22]. For brevity,therefore, the proof is omitted. Lemma 7.10 (Uniform integrability of ( V − ( h )) h ∈D ( y ) ) . The family ( V − ( h )) h ∈D ( y ) is uniformlyintegrable, for any y > . One can can now proceed to prove existence of a unique optimiser in the dual problem, andconjugacy of the value functions. We proceed first with the former, followed by conjugacy. Theproof of Lemma 7.11 on dual existence is on the same lines as the proof of primal existence (Lemma7.6), with adjustments for minimisation as opposed to maximisation and convexity of V ( · ) (inheritedby V ( f ) ( y ) := V ( y ) + f y, y >
0) replacing concavity of U ( · ), so is omitted for brevity. Lemma 7.11 (Dual existence) . The optimal solution (cid:98) h ( y ) ∈ D ( y ) to the dual problem (4.4) existsand is unique, so that v ( · ) is strictly convex. We now establish conjugacy of the value functions. The proof works by bounding the elementsin the primal domain to create a compact set for the weak ∗ topology σ ( L ∞ , L ) on L ∞ ( µ ), so asto apply the minimax theorem (see Strasser [33, Theorem 45.8]), involving a maximisation over acompact set and a minimisation over a subset of a vector space. This uses the fact that the dualdomain D ( y ) is bounded in L ( µ ).Note that, because u ( · ) and − v ( · ) are strictly concave, they are almost everywhere differentiable.We shall show in Lemma 7.15 that they are in fact differentiable everywhere, so freely use theirderivatives in the statements of some forthcoming lemmas. Recall that a sequence ( g n ) n ∈ N in L ∞ ( µ ) converges to g ∈ L ∞ ( µ ) with respect to the weak ∗ topology σ ( L ∞ , L )if and only if ( (cid:104) g n , h (cid:105) ) n ∈ N converges to (cid:104) g, h (cid:105) for each h ∈ L ( µ ). Lemma 7.12 (Conjugacy) . The dual value function in (4.4) satisfies the conjugacy relation (7.16) v ( y ) = sup x> [ u ( x ) − xy ] , for each y ∈ (0 , y ∗ ) , where u ( · ) is the primal value function in (4.2) and y ∗ is the minimal value of y > at which thedual derivative v (cid:48) ( · ) reaches zero, as in (4.7) .Proof. For n ∈ N denote by B n the set of elements in L ( µ ) lying in a ball of radius n : B n := (cid:8) g ∈ L ( µ ) : g ≤ n, µ − a . e . (cid:9) . The sets ( B n ) n ∈ N are σ ( L ∞ , L )-compact. Because each h ∈ D ( y ) is µ -integrable, D ( y ) is a closed,convex subset of the vector space L ( µ ), so we apply the minimax theorem (Strasser [33, Theorem45.8]) to the compact set B n ( n fixed) and the set D ( y ), with the function w ( g, h ) := (cid:82) Ω ( U ( g ) − gh + f h ) d µ = (cid:82) Ω ( U ( g ) − ( g − f ) h ) d µ , for g ∈ B n , h ∈ D ( y ), to give(7.17) sup g ∈B n inf h ∈D ( y ) (cid:90) Ω ( U ( g ) − ( g − f ) h ) d µ = inf h ∈D ( y ) sup g ∈B n (cid:90) Ω ( U ( g ) − ( g − f ) h ) d µ. By the dual characterisation of C in (7.6) or (4.6), an element g ∈ L ( µ ) lies in C ( x ) if and only ifsup h ∈D ( y ) (cid:82) Ω ( g − f ) h d µ ≤ xy . Thus, the limit as n → ∞ on the left-hand-side of (7.17) is given as(7.18) lim n →∞ sup g ∈B n inf h ∈D ( y ) (cid:90) Ω ( U ( g ) − ( g − f ) h ) d µ = sup x> sup g ∈C ( x ) (cid:18)(cid:90) Ω U ( g ) d µ − xy (cid:19) = sup x> [ u ( x ) − xy ] . Now consider the right-hand-side of (7.17). Define V n ( y ) := sup
0) and (7.22), we obtainlim n →∞ v n ( y ) = lim n →∞ (cid:90) Ω (cid:16) V n (˜ h n ) + f ˜ h n (cid:17) d µ ≥ lim n →∞ (cid:90) Ω ( V n ( h n ) + f h n ) d µ ≥ (cid:90) Ω ( V ( h ) + f h ) d µ ≥ v ( y ) . and the proof is complete. (cid:3) Remark . The range of y > u (cid:48) (ˆ x ) = y which defines the optimal value of x in (7.16). We know that u ( · )is increasing and concave and satisfies u (cid:48) ( ∞ ) = 0 (from Lemma 2.2, and indeed from Lemma 7.14below). If u (cid:48) (0) = + ∞ , then y ∗ = + ∞ and (7.16) is valid for all y >
0. We shall see in Lemma 7.17that y ∗ < ∞ , and this has been anticipated in (7.16).We now proceed to further characterise the derivatives of the value functions, as well as the primaland dual optimisers and the optimal wealth process. Lemma 7.14.
The derivatives of the primal value function in (4.2) at infinity and of the dual valuefunction in (4.4) at zero are given by (7.23) u (cid:48) ( ∞ ) := lim x →∞ u (cid:48) ( x ) = 0 , − v (cid:48) (0) := lim y ↓ ( − v (cid:48) ( y )) = + ∞ . Proof.
By the conjugacy result in Lemma 7.12 between the value functions, the assertions in (7.23)are equivalent. We shall prove the first assertion.The primal value function u ( · ) is strictly concave and strictly increasing, so there is a finitenon-negative limit u (cid:48) ( ∞ ) := lim x →∞ u (cid:48) ( x ). Because U ( · ) is increasing with lim x →∞ U (cid:48) ( x ) = 0,for any (cid:15) > C (cid:15) such that U ( x ) ≤ C (cid:15) + (cid:15)x, ∀ x >
0. Using this and thefact that there exists a positive element ¯ h ∈ D (as in the proof of Proposition 4.1) such that (cid:82) Ω g ¯ h d µ ≤ x (1 + aT ) , ∀ g ∈ C ( x ), and l’Hˆopital’s rule, we have, with (cid:82) Ω d µ = E (cid:104)(cid:82) T d κ t (cid:105) =(1 − e − δT ) /δ =: K > ≤ lim x →∞ u (cid:48) ( x ) = lim x →∞ u ( x ) x = lim x →∞ sup g ∈C ( x ) (cid:90) Ω U ( g ) x d µ ≤ lim x →∞ sup g ∈C ( x ) (cid:90) Ω C (cid:15) + (cid:15)gx d µ = lim x →∞ (cid:32) C (cid:15) Kx + 1¯ h sup g ∈C ( x ) (cid:90) Ω (cid:15)g ¯ hx d µ (cid:33) ≤ lim x →∞ (cid:18) C (cid:15) Kx + (cid:15) (1 + aT )¯ h (cid:19) = (cid:15) (1 + aT )¯ h , µ -a.e. , and taking the limit as (cid:15) ↓ (cid:3) The next lemma shows that the dual value function is differentiable.
Lemma 7.15.
The dual value function v ( · ) in (4.4) is differentiable on (0 , ∞ ) . Proof.
Since (cid:82) Ω h d µ ≤ T y for any h ∈ D ( y ), we have (cid:82) Ω ( (cid:98) h ( y ) /y ) d µ ≤ T, y >
0, which defines aunique integrable element L ( µ ) ⊇ D (cid:51) (cid:98) h y := (cid:98) h ( y ) /y , for any y > y >
0. Then, for any δ >
0, using convexity of V ( · ) and the fact that ( y + δ ) (cid:98) h y will besuboptimal for v ( y + δ ), we have1 δ ( v ( y + δ ) − v ( y )) ≤ (cid:90) Ω (cid:16) V (( y + δ ) (cid:98) h y ) + f ( y + δ ) (cid:98) h y − V ( y (cid:98) h y ) − f y (cid:98) h y (cid:17) d µ = (cid:90) Ω (cid:18) δ (cid:16) V (( y + δ ) (cid:98) h y ) − V ( y (cid:98) h y ) (cid:17) + f (cid:98) h y (cid:19) d µ ≤ (cid:90) Ω (cid:16) V (cid:48) (( y + δ ) (cid:98) h y ) + + f (cid:98) h y (cid:17) d µ. The element ( y + δ ) (cid:98) h y ∈ L ( µ ) is strictly positive, and thus | V (cid:48) (( y + δ ) (cid:98) h y ) | is bounded µ -a.e. (onrecalling the Inada conditions satisfied by − V ( · )), while the non-negative element f (cid:98) h y is boundedabove by the integrable function a (cid:98) h y , so we may apply dominated convergence to obtain that(7.24) v (cid:48) ( y ) ≤ (cid:90) Ω (cid:16) V (cid:48) ( y (cid:98) h y ) + f (cid:98) h y (cid:17) d µ. An identical argument, this time applied to ( v ( y ) − v ( y − δ )) /δ , yields the reverse inequality(7.25) v (cid:48) ( y ) ≥ (cid:90) Ω (cid:16) V (cid:48) ( y (cid:98) h y ) + f (cid:98) h y (cid:17) d µ. Then, (7.24) and (7.25) yield that v ( · ) is differentiable on (0 , ∞ ) with(7.26) v (cid:48) ( y ) = (cid:90) Ω (cid:16) V (cid:48) ( y (cid:98) h y ) + f (cid:98) h y (cid:17) d µ. (cid:3) Remark . We shall see the formula (7.26) for the dual derivative reproduced in the course ofproving Lemma 7.17 (see (7.31)).The conjugacy between the primal and dual value functions, combined with Lemma 7.15, yieldsthat the primal value function u ( · ) is also differentiable on (0 , ∞ ).The final step in the series of lemmas that will furnish us with the proof of Theorem 4.2 is tocharacterise the derivative of the primal value value function u ( · ) at zero, along with a dualitycharacterisation of the primal and dual optimisers. Lemma 7.17. (1)
The primal value function in (4.2) has finite derivative at zero: (7.27) u (cid:48) (0) := lim x ↓ u (cid:48) ( x ) < + ∞ . Equivalently, the value y ∗ := inf { y > v (cid:48) ( y ) = 0 } of y > at which the derivative of dualvalue function reaches zero, is finite. (2) For any fixed x > , with y = u (cid:48) ( x ) ∈ (0 , y ∗ ) (equivalently x = − v (cid:48) ( y ) ), the primal and dualoptimisers (cid:98) g ( x ) , (cid:98) h ( y ) are related by (7.28) U (cid:48) ( (cid:98) g ( x )) = (cid:98) h ( y ) = (cid:98) h ( u (cid:48) ( x )) , µ -a.e. , and satisfy (7.29) (cid:90) Ω ( (cid:98) g ( x ) − f ) (cid:98) h ( y ) d µ = xy = xu (cid:48) ( x ) . ANDOMLY TERMINATING INCOME 31 (3)
The derivatives of the value functions satisfy the relations xu (cid:48) ( x ) = (cid:90) Ω U (cid:48) ( (cid:98) g ( x )) ( (cid:98) g ( x ) − f ) d µ, x > , (7.30) yv (cid:48) ( y ) = (cid:90) Ω (cid:16) V (cid:48) ( (cid:98) h ( y )) + f (cid:17) (cid:98) h ( y ) d µ, y ∈ (0 , y ∗ ) . (7.31) Proof.
Recall the inequality (2.22), which also applies to the value functions because they are alsoconjugate by Lemma 7.12. We thus have, in addition to (2.22),(7.32) v ( y ) ≥ u ( x ) − xy, ∀ x > , y ∈ (0 , y ∗ ) , with equality iff y = u (cid:48) ( x ) . With (cid:98) g ( x ) ∈ C ( x ) , x > (cid:98) h ( y ) ∈ D ( y ) , y ∈ (0 , y ∗ ) denoting the primal and dual optimisers, wehave, because (cid:82) Ω ( g − f ) h d µ ≤ xy for all g ∈ C ( x ) , h ∈ D ( y ), (cid:90) Ω ( (cid:98) g ( x ) − f ) (cid:98) h ( y ) d µ ≤ xy, x > , y ∈ (0 , y ∗ ) . Using this as well as (2.22) and (7.32) we have0 ≤ (cid:90) Ω (cid:16) V ( (cid:98) h ( y )) − U ( (cid:98) g ( x )) + (cid:98) g ( x ) (cid:98) h ( y ) (cid:17) d µ (7.33) = (cid:90) Ω (cid:16) V ( (cid:98) h ( y )) + f (cid:98) h ( y ) − U ( (cid:98) g ( x )) + ( (cid:98) g ( x ) − f ) (cid:98) h ( y ) (cid:17) d µ ≤ v ( y ) − u ( x ) + xy, x > , y ∈ (0 , y ∗ ) . The right-hand-side of (7.33) is zero if and only if y = u (cid:48) ( x ), due to (7.32), and the non-negativeintegrand must then be µ -a.e. zero, which by (2.22) can only happen if (7.28) holds, which establishesthat primal-dual relation.Thus, for any fixed x > y = u (cid:48) ( x ), and hence equality in (7.33), we have0 = (cid:90) Ω (cid:16) V ( (cid:98) h ( y )) + f (cid:98) h ( y ) − U ( (cid:98) g ( x )) + ( (cid:98) g ( x ) − f ) (cid:98) h ( y ) (cid:17) d µ = v ( y ) − u ( x ) + (cid:90) Ω ( (cid:98) g ( x ) − f ) (cid:98) h ( y ) d µ = v ( y ) − u ( x ) + xy, y = u (cid:48) ( x ) , which implies that (7.29) must hold. Inserting the explicit form of (cid:98) h ( y ) = U (cid:48) ( (cid:98) g ( x )) into (7.29) yields(7.30). Similarly, setting (cid:98) g ( x ) = I ( (cid:98) h ( y )) = − V (cid:48) ( (cid:98) h ( y )) into (7.29), with x = − v (cid:48) ( y ) (equivalent to y = u (cid:48) ( x )), yields (7.31).It remains to establish (7.27), and the equivalent assertion that y ∗ < ∞ . Now, since − v (cid:48) (0) = + ∞ and − v ( · ) is concave, − v ( · ) must be increasing near y = 0 (otherwise − v (cid:48) ( · ) would become negative,which is precluded by the relation x = − v (cid:48) ( y ) in the optimal primal-dual relations (7.28)). Thequestion is then whether − v (cid:48) ( · ) reaches zero at infinity (in which case u ( · ) and − v ( · ) would satisfythe Inada conditions, which is the case in the problem without income) or whether the value y ∗ > − v (cid:48) ( · ) reaches zero is finite. We shall show that y ∗ < ∞ , so that (7.27) holds, and so thatthe Inada condition for u ( · ) at zero is violated.Because (cid:82) Ω h d µ ≤ T y for any h ∈ D ( y ), we have(7.34) (cid:90) Ω (cid:98) h ( y ) y d µ ≤ T, y ∈ (0 , y ∗ ) . Using Fatou’s lemma in (7.34) we have T ≥ lim inf y ↑ y ∗ (cid:90) Ω (cid:98) h ( y ) y d µ ≥ (cid:90) Ω lim inf y ↑ y ∗ (cid:32) (cid:98) h ( y ) y (cid:33) d µ, which, given that (cid:98) h ( y ) /y is non-negative, gives that lim inf y ↑ y ∗ ( (cid:98) h ( y ) /y ) < ∞ , µ -a.e. Therefore,writing (cid:98) h ( y ) =: y (cid:98) h y , which defines a unique element (cid:98) h y ∈ D , we have (cid:98) h y ∗ := lim inf y ↑ y ∗ (cid:98) h y = lim inf y ↑ y ∗ (cid:98) h ( y ) y < ∞ , µ -a.e.From (7.31) we thus obtain(7.35) − v (cid:48) ( y ) = (cid:90) Ω (cid:16) − V (cid:48) ( y (cid:98) h y ) − f (cid:17) (cid:98) h y d µ, while from the definition of the dual value function we have(7.36) − v ( y ) = (cid:90) Ω (cid:16) − V ( y (cid:98) h y ) − yf (cid:98) h y (cid:17) d µ. From (7.35) and (7.36) we see that the former is only consistent with the latter if (cid:98) h y has no explicitdependence on y , and this in turn implies (given the monotonicity of V (cid:48) ( · )) that the integrand in(7.35) is monotone in y . We can thus apply monotone convergence to obtain0 = − v (cid:48) ( y ∗ ) = lim y ↑ y ∗ (cid:90) Ω (cid:16) − V (cid:48) ( y (cid:98) h y ) − f (cid:17) (cid:98) h y d µ = (cid:90) Ω (cid:16) − V (cid:48) ( y ∗ (cid:98) h y ∗ ) − f (cid:17) (cid:98) h y ∗ d µ, which in turn yields, since (cid:98) h y ∗ is strictly positive, − V (cid:48) ( y ∗ (cid:98) h y ∗ ) = f > , implying (since − V ( · ) satisfies the Inada conditions) that y ∗ < ∞ , and the proof is complete. (cid:3) We have now established all results that give the abstract duality in Theorem 4.2, so let us confirmthis.
Proof of Theorem 4.2.
Lemma 7.12 implies the relations (4.8) of item (i). The statements in item(ii) are implied by Lemma 7.6 and Lemma 7.11. Items (iii) and (iv) follow from Lemma 7.14, Lemma7.15, Remark 7.16 and Lemma 7.17. (cid:3)
Proof of the concrete duality.
We are ready to prove the concrete duality in Theorem 3.1,because Theorem 4.2 readily implies nearly all of the assertions of Theorem 3.1. The outstandingassertion is the characterisation of the optimal wealth process in (3.3) and the associated uniformlyintegrable martingale property of the deflated wealth plus cumulative deflated consumption overincome process (cid:98) X ( x ) (cid:98) Y ( y ) + (cid:82) · ( (cid:98) c s ( x ) − f s ) (cid:98) Y s ( y ) d s . So we establish these assertions in the course ofproving the concrete duality below. Proof of Theorem 3.1.
Given the definitions of the sets C ( x ) and D ( y ) in (4.1) and (4.3), respec-tively, and the identification of the abstract value functions in (4.2) and (4.4) with their concretecounterparts in (2.7) and (2.24), Theorem 4.2 implies all the assertions of Theorem 3.1, with the ex-ception of the optimal wealth process formula (3.3) and the uniformly integrable martingale propertyof (cid:98) X ( x ) (cid:98) Y ( y ) + (cid:82) · ( (cid:98) c s ( x ) − f s ) (cid:98) Y s ( y ) d s , so let us proceed to establish these assertions.Recall the saturated budget constraint equality in (3.2). It simplifies notation if we take x = y = 1,and is without loss of generality: although y = u (cid:48) ( x ) in (3.2), one can always multiply the utilityfunction by an arbitrary constant so as to ensure that u (cid:48) (1) = 1. We thus have the optimal budgetconstraint(7.37) E (cid:20)(cid:90) T ( (cid:98) c t − f t ) (cid:98) Y t d t (cid:21) = 1 , for (cid:98) c ≡ (cid:98) c (1) ∈ A and (cid:98) Y ≡ (cid:98) Y (1) ∈ Y . Since (cid:98) c ∈ A , we know there exists an optimal wealthprocess (cid:98) X ≡ (cid:98) X (1) and an associated optimal trading strategy (cid:98) H , such that (cid:98) X ≥ ANDOMLY TERMINATING INCOME 33 (cid:98)
Λ := (cid:98) X (cid:98) Y + (cid:82) · ( (cid:98) c s − f s ) (cid:98) Y s d s is a supermartingale over [0 , T ]. The supermartingale condition, bythe same arguments that led to the derivation of the budget constraint in Lemma 2.1, leads to theinequality E (cid:104)(cid:82) T ( (cid:98) c t − f t ) (cid:98) Y t d t (cid:105) ≤ (cid:98) Λ must be a martingale over[0 , T ] (and is uniformly integrable, as it is closed by the random variable Λ T , that is, E [ (cid:98) Λ T ] = (cid:98) Λ = 1,see Protter [31, Theorem I.13]).The martingale condition gives E (cid:104) (cid:98) X T (cid:98) Y T + (cid:82) T ( (cid:98) c t − f t ) (cid:98) Y t d t (cid:105) = 1, which on utilising (7.37) yields E [ (cid:98) X T (cid:98) Y T ] = 0, and since (cid:98) X (cid:98) Y is non-negative, we obtain (cid:98) X T (cid:98) Y T = 0 almost surely, as claimed.Applying the martingale condition again, this time over [ t, T ] for some t ∈ [0 , T ], we have E (cid:20) (cid:98) X T (cid:98) Y T + (cid:90) T ( (cid:98) c s − f s ) (cid:98) Y s d s (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = (cid:98) X t (cid:98) Y t + (cid:90) t ( (cid:98) c s − f s ) (cid:98) Y s d s, t ∈ [0 , T ] , which, on using (cid:98) X T (cid:98) Y T = 0, re-arranges to (3.3), and the proof of the theorem is complete. (cid:3) It remains to prove Theorem 3.2, which in turn rests upon Proposition 4.3, to which we now turn.
Proof of Proposition 4.3.
With x = y = 1, from Lemma 2.1 we know that the budget constraint(2.16) holds for all Y ∈ (cid:101) Y and all c ∈ A , so we have the implication(7.38) Y ∈ (cid:101) Y = ⇒ E (cid:20)(cid:90) T ( c t − f t ) Y t d t (cid:21) ≤ , ∀ c ∈ A . Then, as in the proof of Lemma 7.2 we can establish the reverse implication. Translating the resultingequivalence into the abstract notation on the measure space ( Ω , G , µ ), we have the analogue of (4.6),with (cid:101) D in place of D : g ∈ C ⇐⇒ (cid:104) g − f, h (cid:105) ≤ , ∀ h ∈ (cid:101) D . An examination of the abstract duality proof shows that it was crucial to establish that theabstract dual domain D was closed, as in Lemma 7.8, and that this was done via supermartingaleconvergence results, where we found a supermartingale Y such that X Y (for a self-financing wealthprocess X ∈ X ) was also a supermartingale, so we could conclude that Y ∈ Y (because of thedefinition of Y and the fact that c ≡ f is an admissible consumption plan) and hence that h = ζY ∈ D . This argument would fail with (cid:101) D in place of D , because the limiting supermartingale inthe Fatou convergence argument is known only to be a supermartingale, and cannot be shown tobe a discounted local martingale deflator Y ∈ (cid:101) Y .However, if we enlarge (cid:101) D to its closure, this is automatically closed, and if we define an abstractdual value function by ˜ v ( y ) := inf h ∈ cl( (cid:101) D ( y )) (cid:90) Ω ( V ( h ) + f h ) d µ, y > , then the rest of the abstract duality proof goes through unaltered, and by conjugacy of the abstractprimal and dual value functions, the dual function ˜ v ( · ) coincides with v ( · ). The dual minimiser liesin cl( (cid:101) D ( y )), and this establishes the proposition. (cid:3) Proof of Theorem 3.2.
Furnished with Proposition 4.3, we have established the result, due to theone-to-one correspondence between (cid:101) D and (cid:101) Y . (cid:3) Conclusions
In this paper we have proven a rigorous duality for a consumption problem with randomly termi-nating income, over a finite horizon, thereby closing the duality gap that arose in the infinite horizonversion of the problem as introduced by Vellekoop and Davis [35]. The key property of the finitenessof marginal utility at zero initial wealth emerged, while all the other main tenets of duality theorywere shown to hold. For the most part our methods could well apply to the infinite horizon versionof the problem, but this remains an open problem as we did use the finite horizon to derive the L -boundedness of the abstract dual domain. Some adapted arguments would therefore be neededfor the infinite horizon problem, which remains an interesting topic for future research.In the second part of the paper we applied a deep learning-based stochastic control problem solverto the optimal investment and consumption problem with randomly terminating income. The solvertackles the 2BSDE associated with the HJB equation of the problem, with optimality conditionsfor the control arising from the Hamiltonian of the system. We transformed the problem into onewithin the remit of this solver, and found accurate approximations of the optimal state and controlfor benchmark cases, where analytical solutions could be found. We found tight bounds for thevalue function at time zero for the terminating income problem, for which there are no analyticalsolutions. There remain many open questions such as deep learning methodology to solve the infinitehorizon problem. We leave these for future research. References [1]
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Ashley Davey, Department of Mathematics, Imperial College London, London SW7 2BZ, UK
Email address : [email protected] Michael Monoyios, Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter,Woodstock Road, Oxford OX2 6GG, UK
Email address : [email protected] Harry Zheng, Department of Mathematics, Imperial College London, London SW7 2BZ, UK
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