An elementary approach to the Merton problem
aa r X i v : . [ q -f i n . M F ] J un An elementary approach to the Merton problem
Martin Herdegen, David Hobson, Joseph Jerome ∗ June 11, 2020
Abstract
In this article we consider the infinite-horizon Merton investment-consumption problemin a constant-parameter Black–Scholes–Merton market for an agent with constant relativerisk aversion R . The classical primal approach is to write down a candidate value functionand to use a verification argument to prove that this is the solution to the problem. However,features of the problem take it outside the standard settings of stochastic control, andthe existing primal verification proofs rely on parameter restrictions (especially, but notonly, R < ), restrictions on the space of admissible strategies, or intricate approximationarguments.The purpose of this paper is to show that these complications can be overcome using asimple and elegant argument involving a stochastic perturbation of the utility function. In the Merton investment-consumption problem (Merton [10, 11]) an agent seeks to maximisethe expected integrated discounted utility of consumption over the infinite horizon in a modelwith a risky asset and a riskless bond. When parameters are constant and the utility function isgiven by a power law, it is straightforward to write down the candidate value function. However,it is more difficult to give a complete verification argument. For general strategies the wealthprocess may hit zero at which point the application of Itô’s formula to the candidate valuefunction breaks down; the local martingale which arises from the application of Itô’s formulamay fail to be a martingale; even for constant proportional strategies transversality may fail.For all these reasons, it is difficult to give a concise, rigorous verification proof via analysisof the value function, and many textbooks either finesse the issues or restrict attention to asubclass of admissible strategies, and/or restrict attention to a subset of parameter combinations(especially
R < , but even then there can be substantive points which are often overlooked).The need for such a verification argument has been obviated by the development of proofs usingthe dual method, which provides a powerful and intuitive alternative approach, see Biagini [1]for a survey (and also [5, 7, 8, 13]). Nonetheless, it would be nice to provide a short proofbased on the primal approach. The goal of this paper is didactic – to give a simple, brief proofthat the candidate value function is the value function via the primal approach, and moreover, ∗ We thank Steve Shreve and Ioannis Karatzas for sharing their recollections of their motivations behind [9]and [6]. Our original motivation for returning to the Merton problem arose from consideration of a problem involv-ing stochastic differential utility. There, dual approaches are more involved and do not cover all parametercombinations, so that the primal method is not redundant, and indeed may provide a more direct approach.
1o give a proof which is valid for all parameter combinations for which the Merton problem iswell-posed.The first full verification of the solution to the Merton problem of which we are aware (underan assumption of positive discounting and strictly positive interest rates) is Karatzas et al [6],which built on the previous work of Lehoczky et al [9]. There, the idea is to solve a perturbationof the original problem in which the agent may go bankrupt, at which point they receive aresidual value P . (Part of their motivation was to better understand the results of Merton [11]on HARA utilities, see also Sethi and Taskar [14].) The solution to the perturbed problem isvery clever, and is developed in the case of a general utility function, but it is also very intricateand takes many pages of calculation. Moreover, when specialised to the case of CRRA utilities,it places some assumptions on the parameter values beyond the necessary assumption of well-posedness of the Merton problem. The problem with bankruptcy is of independent interest,but more important for our purposes is the fact that, given the solution to the problem withbankruptcy for a CRRA utility, by letting P ↓ ( R < ) or P ↓ −∞ ( R > ) Karatzas et al [6]recover the solution to the original Merton problem.In their seminal paper on transaction costs, Davis and Norman [2, Section 2] briefly considerthe Merton problem without transaction costs. They assume that the proportion of wealth in-vested in the risky asset is bounded, and for R < they go on to prove a verification theorem forstrategies restricted to this class. Further, in the case R > they propose a different perturba-tion, this time of the candidate value function. The key point is that in the perturbed problemthe candidate value function has a finite lower bound, and this allows Davis and Norman [2]to re-apply arguments from the R < case, although the restriction to ‘regular’ investmentstrategies remains. The candidate value for the perturbed problem can be used to give an up-per bound on the true value function, which converges to the candidate solution to the Mertonproblem as the perturbation disappears. Unlike the argument in Karatzas et al [6], the proof isquite short, but again it only works for certain parameter combinations, and more importantlyit restricts attention to a subclass of admissible strategies.Our goal is to give a complete, simple verification argument via primal methods. At itsheart, our idea is a modification of the approach in [2]. We perturb the utility function, whichleads to a perturbed value function. Moreover, rather than perturbing by the addition of adeterministic constant, we perturb by adding a multiple of the optimal wealth process. The greatbenefit is that the optimal consumption and the optimal investment are unchanged under theperturbation, which means that mathematical calculations remain strikingly simple. Moreover,these arguments are valid whenever the Merton problem is well-posed.This paper is structured as follows. In the next section we introduce the problem, andin Section 3 we give the candidate value function. In Section 4 we give a proof of the mainresult under a set of clearly-stated assumptions which are precisely designed to make the proofwork. Often, proofs in the stochastic control literature (see, for example, Davis and Norman [2],Fleming and Soner [3, Example 5.2] and Pham [12]) artificially impose restrictions on the set ofadmissible strategies or on the parameter values to ensure that these assumptions are satisfiedby default. In Section 5 we give a small amount of detail on the Karatzas et al [6] and Davis andNorman [2] approaches to the verification problem before in Section 6 we give our argument.Finally, in Section 7 we explain the insights which arise from considering a numéraire changeof the problem, in particular on what are the appropriate parameter restrictions, and what isthe best formulation of the problem. We also show how a change of numéraire can lead to asimplified solution of a slightly modified version of the problem with bankruptcy considered in2aratzas et al [6], and hence to a simplified verification argument.Our proof is an improvement on the existing results in at least three important ways. First,it places no restrictions on the class of admissible strategies: for example, unlike much of thestochastic control literature, it does not require the fraction of wealth invested in the riskyasset to be bounded. (The argument in Karatzas et al [6] also applies to general investmentstrategies.) Second, the proof covers all parameter combinations for which the Merton problemis well-posed (and does not assume that interest rates and discounting are positive – as we shallargue these quantities depend on the choice of currency units, and therefore are not absolutesin themselves). Third, our proof is simple, elegant and concise and not counting the derivationof the candidate solution and candidate value function can be written up in just over one page(Theorem 6.1 and Corollary 6.4). One final contribution of this article is to argue that someformulations of the Merton problem are more natural than others, in the sense that they arerobust to changes of currency unit, and in consequence have simpler dependencies on parametercombinations. Throughout this paper we will work on a filtered probability space (Ω , F , P , F = ( F t ) t> ) sat-isfying the usual conditions and supporting a Brownian motion W = ( W t ) t ≥ . We will assumea Black–Scholes–Merton financial market consisting of a risk-free asset with interest rate r ∈ R whose price process S = ( S t ) t ≥ is given by S t = exp( rt ) and a risky asset whose price process S = ( S t ) t ≥ follows a geometric Brownian motion with drift µ ∈ R and volatility σ > : dS t S t = µ d t + σ d W t , S = s > . An agent operating with this investment opportunity set and initial wealth x > chooses an admissible investment-consumption strategy ( ϑ , ϑ, C ) = ( ϑ t , ϑ t , C t ) t ≥ , where ϑ t ∈ R denotesthe number of riskless assets held at time t , ϑ t ∈ R denotes the number of risky assets held attime t , and C t ∈ R + represents the rate of consumption at time t . We require that ϑ , ϑ, C are progressively measurable processes, ϑ is integrable with respect to S , ϑ is integrable withrespect to S , C is integrable with respect to the identity process, the wealth process X = ( X t ) t ≥ defined by X t := ϑ t S t + ϑ t S t (1)is P -a.s. nonnegative and the self-financing condition, X t = x + Z t ϑ s d S s + Z t ϑ s d S s − Z t C s d s, t ≥ , is satisfied. We then denote by Π t := ϑ t S t X t and Π t := ϑ t S t X t the fraction of wealth invested in theriskless and risky asset at time t , respectively. Noting that Π t + Π t = 1 by (1), it follows that X satisfies the SDE d X t = ϑ t d S t + ϑ t d S t − C t d t = X t Π t r d t + X t Π t ( µ d t + σ d W t ) − C t d t = X t Π t σ d W t + ( X t ( r + Π t ( µ − r )) − C t ) d t, X = x. (2) Strictly speaking, Π t and Π t are not defined for X t = 0 , but this does not matter. We can for example set Π t := 0 and Π t := 1 for X t = 0 . x > more succinctly by a pair (Π , C ) = (Π t , C t ) t ≥ of progressively measurableprocesses, where Π is real-valued and C is nonnegative, such that the SDE (2) has a uniquestrong solution X x, Π ,C that is P -a.s. nonnegative. We denote the set of admissible investment-consumption strategies for x > by A ( x ) . A consumption stream C is called attainable forinitial wealth x > if there exists an investment process Π such that (Π , C ) ∈ A ( x ) , and wedenote the set of attainable consumption streams for x > by C ( x ) .The objective of the agent is to maximise the expected discounted utility of consumptionover an infinite time horizon for a given initial wealth x > . To any attainable consumptionstream C ∈ C ( x ) , they associate a value J ( C ) ∈ [ −∞ , ∞ ] , where J ( C ) := E (cid:20)Z ∞ e − δt U ( C t ) d t (cid:21) . Here, δ ∈ R can be seen as a discount or impatience parameter; see Section 7 for a discussion onthe economic interpretation of δ . We assume that the agent has constant relative risk aversion(CRRA) or equivalently that U : [0 , ∞ ) → [ −∞ , ∞ ) takes the form U ( c ) = c − R − R , where R ∈ (0 , ∞ ) \ { } is the coefficient of relative risk aversion. Note that since R = 1 , the signof U ( c ) is uniquely determined. Thus, if R ∞ e − δt U ( C t ) d t is not integrable, we can define J ( C ) := + ∞ when R < and J ( C ) := −∞ when R > .In summary, the problem facing the agent is to determine V ( x ) := sup C ∈ C ( x ) J ( C ) = sup C ∈ C ( x ) E "Z ∞ e − δt C − Rt − R d t . From the homogeneous structure of the problem we expect (see for example, Rogers [13, Propo-sition 1.2]) that V ( κx ) = κ − R V ( x ) and that if ( ˆΠ , ˆ C ) is an optimal strategy in A ( x ) then ( ˆΠ κ = ˆΠ , ˆ C κ = κ ˆ C ) is optimal in A ( κx ) for κ > . For this reason, we may guess that it isoptimal to invest a constant fraction of wealth in the risky asset, and to consume a constant frac-tion of wealth. (Of course, this will be verified later.) So, consider an investment-consumptionstrategy that at each time t , invests a constant proportion of wealth Π t = π into the risky assetand consumes a constant fraction ξ > of wealth per unit time, i.e., C t = ξX t .Then the agent’s wealth process X = X x,π,ξX is given by X t = x exp (cid:18) πσW t + (cid:18) r + π ( µ − r ) − ξ − π σ (cid:19) t (cid:19) , X = x. Denoting the market price of risk or
Sharpe ratio by λ := µ − rσ , we obtain C − Rt − R = ( ξX t ) − R − R = x − R ξ − R − R exp (cid:18) πσ (1 − R ) W t + (1 − R ) (cid:18) r + λσπ − ξ − π σ (cid:19) t (cid:19) . We follow the convention that − R := ∞ for R > . The case of R = 1 is the case of logarithmic utility U ( c ) = ln c . This may be treated by very similarmathematics, but we will not consider it here. e − δt and taking expectations gives E " e − δt C − Rt − R = x − R ξ − R − R e − F ( π,ξ ) t , (3)where F ( π, ξ ) = F ( π, ξ ; R, δ, λ, r, σ ) := δ − (1 − R ) (cid:18) r + λσπ − π σ R − ξ (cid:19) . Provided that F ( π, ξ ) > , we find that J ( ξX ) = E "Z ∞ e − δt ξ − R X − Rt − R d t = x − R − R ξ − R F ( π, ξ ) . (4)In order to maximise this over π , we want to minimise (1 − R ) F ( π, ξ ) , which is equivalent tomaximising λσπ − π σ R . This is achieved at ˆ π = λσR . In this case λσ ˆ π − ˆ π σ R = λ R and theproblem then becomes to maximise x − R − R ξ − R F (ˆ π, ξ ) = x − R − R ξ − R (cid:16) δ − (1 − R )( r + λ R − ξ ) (cid:17) over ξ . A simple calculation shows that the maximum is attained at ˆ ξ = η , where η := 1 R (cid:20) δ − (1 − R ) (cid:18) r + λ R (cid:19)(cid:21) , provided that η > . (If η ≤ , no maximum exists.)Therefore, when η > , the agent’s optimal behaviour (at least over constant proportionalstrategies) and corresponding value function are given by ˆ π = µ − rσ R , ˆ ξ = η, ˆ V ( x ) := J ( ˆ ξX ) = η − R x − R − R . (5)When η ≤ , the problem is ill-posed. Indeed, if R < , then F (ˆ π, ξ ) ↓ as ξ ↓ − ηR (1 − R ) andhence J ( ξX ) ↑ ∞ by (4). If R > , then F ( π, ξ ) ≤ F (ˆ π, ξ ) = Rη + (1 − R ) ξ ≤ Rη ≤ for every π ∈ R and ξ ≥ . Hence, at least for constant proportional strategies J ( ξX ) = −∞ . We willsee in Corollary 6.5 that J ( C ) = −∞ for every admissible consumption stream C ∈ C ( x ) . In this section, we prove that our candidate optimal strategy (ˆ π, ˆ ξX ) from (5) is optimal in asubset of the class of all admissible strategies. Since the conditions defining that class are chosenprecisely in such a way that the proof works, we call them fiat conditions. Definition 4.1.
Fix x > . An investment-consumption strategy (Π , C ) ∈ A ( x ) is called fiatadmissible if the following three conditions are satisfied:(P) The wealth process X x, Π ,C is P -a.s. positive.(M) The local martingale R · e − δt σ Π t ( X x, Π ,Ct ) − R d W t is a supermartingale.5T) The transversality condition lim inf t →∞ E [ e − δt ( X x, Π ,Ct ) − R − R ] ≥ is satisfied.We denote the set of all fiat admissible investment-consumption strategies for x > by A ∗ ( x ) .A consumption stream C ∈ C ( x ) is called fiat attainable for x > if there is an investmentprocess Π such that (Π , C ) ∈ A ∗ ( x ) . We denote the set of fiat attainable consumption streamsby C ∗ ( x ) . Remark . As far as we are aware, the above notion of fiat admissible strategies has notbeen explicitly used in the literature before. However, the conditions (P), (M) and (T) orstronger versions thereof have been used explicitly or implicitly throughout the stochastic controlliterature on the Merton problem:1. Condition (P) is (implicitly) assumed throughout most of the stochastic control literaturedealing with the Merton problem; a notable exception is [6]. However, for
R > , (P)can be assumed without loss of generality because any admissible strategy (Π , C ) ∈ A ( x ) violating (P) has J ( C ) = −∞ .2. Condition (M) is implied by the stronger condition(M1) The local martingale R · σ Π t ( X x, Π ,Ct ) − R e − δt d W t is a martingale.It is not difficult to check that for R < , (M1) is implied by the even stronger condition(B) Π is uniformly bounded.A common approach in the stochastic control literature is to assume (B), see e.g. Davisand Norman [2, Equation (2.1)(B)], Fleming and Soner [3, Equation IV.5.2], or Pham [12,Equation (3.2)], and then prove (M1) for R < .
3. Condition (T) is implied by the stronger standard transversality condition (T1) lim t →∞ E [ e − δt ( X x, Π ,Ct ) − R − R ] = 0 .When R < , Davis and Norman [2, page 682] prove that (T1) is satisfied for any admissiblestrategy satisfying (B). Pham [12, Equation (3.39)] and Fleming and Soner [3, EquationIV.5.11] require (T1), and prove that the candidate optimal strategy has this property.It is clear that C ∗ ( x ) ⊂ C ( x ) . The following result shows that the candidate optimal strategy (ˆ π, ˆ ξX ) from (5) is optimal in the class of fiat admissible strategies. Theorem 4.3.
Suppose η := R [ δ − (1 − R )( r + λ R )] > . Let the function ˆ V : (0 , ∞ ) → R begiven by ˆ V ( x ) = x − R − R η − R . Then for x > , V ∗ ( x ) := sup C ∈ C ∗ ( x ) J ( C ) = J ( ˆ C ) = ˆ V ( x ) , where the corresponding optimal investment-consumption strategy is given by (Π , C ) = ( ˆΠ , ˆ C ) where ˆΠ = λσR , ˆ C = ηX x, ˆΠ , ˆ C . Davis and Norman [2, Proof of Theorem 2.1] argue that (B) implies (M1) also in the case
R > but this isnot the case. See Example 4.8. Note, however, that if
R > , (T) and (T1) are equivalent. roof. First, we show that V ∗ ( x ) ≥ ˆ V ( x ) = J ( ˆ C ) . By the arguments in Section 3, it onlyremains to show that ˆ C is fiat attainable. It follows from the construction of ˆ C , that the wealthprocess X x, ˆΠ , ˆ C is P -a.s. positive. Next, a similar calculation as in (3) shows that for each T > , E (cid:20)Z T e − δt σ ˆ π (cid:16) X x, ˆΠ , ˆ Ct (cid:17) − R d t (cid:21) = σ ˆ π Z T exp (cid:18)(cid:18) λ (1 − R ) R − η (cid:19) t (cid:19) d t < ∞ . This implies that the local martingale R · exp( − δt ) σ ˆΠ t ( X x, ˆΠ ,Ct ) − R d W t is a (square-integrable)martingale and hence a supermartingale. Finally, (3) together with the fact that F (ˆ π, η ) = η > ,implies that ( ˆΠ , ˆ C ) satisfies the transversality condition (T1).Next, we show that V ∗ ( x ) ≤ ˆ V ( x ) . Let (Π , C ) ∈ A ∗ ( x ) be arbitrary. If R > , we may inaddition assume without loss of generality that C − R is integrable with respect to the identityprocess; for otherwise J ( C ) = −∞ . It suffices to argue that J ( C ) ≤ ˆ V ( x ) .Set X := X x, Π ,C for brevity and define the process M = ( M t ) t ≥ by M t = Z t e − δs U ( C s ) d s + e − δt ˆ V ( X t ) . We want to apply Itô’s formula to M . This is indeed possible as ˆ V is in C (0 , ∞ ) and X ispositive by fiat admissibility of (Π , C ) . Note that ˆ V x ( X t ) is positive and ˆ V xx ( X t ) is negative.Then, noting that the argument of ˆ V and its derivatives is X t throughout, and adding andsubtracting R − R ˆ V − /Rx and λ V x ˆ V xx for the second equality, dM t = σ Π t X t e − δt ˆ V x d W t + e − δt " C − Rt − R − δ ˆ V + ( X t ( r + σλ Π t ) − C t ) ˆ V x + σ t X t ˆ V xx d t = η − R σ Π t X − Rt e − δt d W t + e − δt " C − Rt − R − C t ˆ V x − R − R ( ˆ V x ) − /R d t + e − δt " σλ Π t X t ˆ V x + σ t X t ˆ V xx + λ V x ˆ V xx d t + e − δt " − δ ˆ V + rX t ˆ V x + R − R ( ˆ V x ) − /R − λ V x ˆ V xx d t =: d N t + A t d t + A t d t + A t d t. Here, N given by N t = R t η − R σ Π s X − Rs e − δs d W s is a local martingale. Next, A ≤ and A ≤ . The first inequality follows from the elementary inequality a − R − R − a − R − R ≤ for a ≥ and R ∈ (0 , ∞ ) \ { } , when setting a := C t ( V x ) R , and the second inequality follows from theelementary inequality ab − a − b ≤ for a, b ≥ , for a := σ Π t X t p − ˆ V xx and b := λ ˆ V x ( X t ) √ − ˆ V xx .Finally, a simple calculation using the definition of ˆ V and η shows that A = 0 .It follows that M t ≤ ˆ V ( x ) + N t , t ≥ . (7)7aking expectations and using fiat admissibility of (Π , C ) to ensure that N is a supermartingale,we find for each t ≥ , E [ M t ] ≤ E h ˆ V ( x ) + N t i ≤ ˆ V ( x ) . Taking the limit as t goes to infinity, and using the monotone convergence theorem as well asthe transversality condition, we obtain J ( C ) = lim t →∞ E (cid:20)Z t e − δs C − Rs − R ds (cid:21) = lim t →∞ E h M t − e − δt ˆ V ( X x, Π ,Ct ) i ≤ lim sup t →∞ E [ M t ] − lim inf t →∞ E h e − δt ˆ V ( X x, Π ,Ct ) i ≤ lim sup t →∞ E [ M t ] ≤ ˆ V ( x ) . (8)This establishes the claim. Remark . A close inspection of the proof of Theorem 4.3 shows that for the optimal strategy ( ˆΠ , ˆ C ) , the process ˆ M = ( ˆ M t ) t ≥ given by ˆ M t := R t e − δs U ( ˆ C s ) d s + e − δt ˆ V ( X x, ˆΠ , ˆ C ) is a uniformlyintegrable martingale. Indeed, in this case ˆ N is a martingale and ˆ M = ˆ V ( x ) + ˆ N . Hence, ˆ M is a martingale. It is uniformly integrable because, by the transversality condition (T1) andmonotone convergence, equation (8) implies that ˆ M t converges in L to ˆ M ∞ := R ∞ e − δs U ( ˆ C s ) d s .For R < , the above fiat verification theorem can be easily generalised to a general verifica-tion theorem. Corollary 4.5.
Suppose
R < and η > . Then V ( x ) = ˆ V ( x ) .Proof. It is sufficient to show that (P), (M) and (T) are satisfied for general strategies, or tofind a way of bypassing the relevant part of the argument. First, (T) is automatically satisfiedby the fact that X − R / (1 − R ) is nonnegative. Next, M is nonnegative and hence N is boundedbelow by − ˆ V ( x ) by (7). Therefore, N is always a supermartingale and (M) is automaticallysatisfied.Finally, to avoid imposing (P), one has to refine the argument in Theorem 4.3 by a stoppingargument. To wit, fix an admissible strategy (Π , C ) ∈ A ( x ) . Then for n ∈ N , set τ n :=inf { t ≥ X x, Π ,C ≤ n } and let τ ∞ := lim n →∞ τ n . Then it is not difficult to check that X t = X x, Π ,Ct ≥ /n > if t ≤ τ n and X t = 0 = C t if t ≥ τ ∞ . Moreover, for each n , we get E [ M τ n t ] ≤ E h ˆ V ( x ) + N τ n t i ≤ ˆ V ( x ) . Now first taking the limit t → ∞ , we obtain E (cid:20)Z τ n e − δs C − Rs − R ds (cid:21) ≤ lim sup t →∞ E [ M τ n t ] ≤ ˆ V ( x ) . Next, taking the limit n → ∞ , the result follows from the monotone convergence theorem andthe fact that R ∞ τ ∞ C s d s = 0 P -a.s. Remark . The above approach of avoiding (P) is taken in [6, Theorem 4.1]. Note, however,that there the stopping argument is slightly more involved as it also requires stopping when thewealth process X x, Π ,C or the quadratic variation of R · σ Π d W gets too large. But this additionalstopping rather obfuscates the argument. More precisely, we have R ∞ τ ∞ C s d s = 0 P -a.s. emark . If R > , extending Theorem 4.3 to general admissible strategies is far moreinvolved. While condition (P) can be assumed without loss of generality (recall Part 1 ofRemark 4.2), condition (M) is in general not satisfied as there are investment strategies Π andconsumption strategies C such that N fails to be a supermartingale, see Example 4.8 below.Note that these strategies are suboptimal because A and A are (very) negative. Finally, wehave no reason to expect that the transversality condition (T) is satisfied. Indeed, (T) even failsfor constant proportional strategies: If ξ > ηRR − , then F (ˆ π, ξ ) < , and it follows from (3) that lim t →∞ E h e − δt − R X x, ˆ π,ξXt i = −∞ . Example . For
R > , the process N in the proof Theorem 4.3 can fail to be supermartingale.We first give an abstract version of an example and then two concrete specifications.Let (Π , C ) ∈ A ( x ) be such that X = X x, Π ,C has P -a.s. positive paths. Define the stoppingtime τ := inf (cid:26) t ≥ Z t η − R σ Π s X − Rs e − δs d W s = 1 (cid:27) . If τ is bounded, then N fails to be a supermartingale because E [ N τ ] = 1 > E [ N ] .The above abstract situation can be achieved either by “wild” investment or by “too fast“consumption, or a combination of the two.For an example of a “wild” investment strategy Π , assume that µ ≥ r > and define thestopping time ˜ τ := inf (cid:26) t ≥ Z t η − R σe − δs − s d W s = 1 (cid:27) . Note that ˜ τ < P -a.s. since R (cid:16) η − R σe − δs − s (cid:17) d s = ∞ . Then define (Π , C ) ∈ A ( x ) by Π t = 11 − t X R − t { t ≤ ˜ τ } , C t := rX t + Π t X t ( µ − r ) . Then the corresponding wealth process X is a stopped and time changed CEV process: d X t = X Rt σ − t { t ≤ ˜ τ } d W t , X = x. Since
R > , X remains positive. Since τ = ˜ τ P -a.s. we have τ < P -a.s. and N fails to be asupermartingale.For an example of a “too fast” consumption strategy C (with bounded investment strategy Π ), assume that µ ≥ r > and define the stopping time ¯ τ := inf ( t ≥ Z t x − R η − R σe σW s − ( δ + σ ) s − s d W s = 1 ) . Note that ¯ τ < P -a.s. since R ( x − R η − R σe σWs − ( δ + 12 σ s − s ) d s = ∞ P -a.s. Then define (Π , C ) ∈ A ( x ) by Π t = { t ≤ ¯ τ } , C t := 1 R − X t − t { t ≤ ¯ τ } + rX t + Π t X t ( µ − r ) . Then the corresponding wealth process satisfies the SDE d X t = σX t { t ≤ ¯ τ } d W t − R − X t − t { t ≤ ¯ τ } d t, X = x.
9t is not difficult to check that this has the solution X t = x (1 − t ∧ ¯ τ ) R − e σW t ∧ ¯ τ − σ ( t ∧ ¯ τ ) which is well-defined and positive by the fact that ¯ τ < P -a.s.. Since τ = ¯ τ P -a.s., we have τ < P -a.s. and N fails to be a supermartingale. R > As we have explained in Remark 4.7, a verification argument for general admissible strategiesrequires some new ideas for the case
R > . In this section, we discuss the two most generalapproaches in the extant stochastic control literature. Both approaches first consider a pertur-bation of the problem (or the candidate solution) and then let the perturbation disappear. The first perturbation approach is by Karatzas et al [6] who study an optimal investment-consumption problem with bankruptcy for a general utility function which is of interest in itsown right, building on earlier work [9] by a subset of the authors. In the following, we onlydescribe their contribution towards the solution of the Merton problem for CRRA utilities. Weassume
R > , and we use our notation.Assume that δ > and r > . For an admissible strategy (Π , C ) ∈ A ( x ) , denote the bankruptcy time τ = τ x, Π ,C = inf { t : X x, Π ,Ct = 0 } . Then choose a finite bankruptcy value P ∈ ( −∞ , and consider the problem with bankruptcy: V P ( x ) := sup C ∈ C ( x ) J P ( C ) = sup C ∈ C ( x ) E "Z τ x, Π ,C e − δt U ( C t ) d t + e − δτ x, Π ,C P . (9)Note that the classical Merton problem corresponds to the limiting case P = −∞ .Karatzas et al [6] show the following:(A) Suppose that a C -function ˆ V P : (0 , ∞ ) → ( P, solves the HJB equation correspondingto the optimisation problem (9) given by δ ˜ V ( x ) = sup c ≥ ,π (cid:20) ˜ V ′ ( x )(( µ − r ) πx + ( rx − c )) + 12 π σ x ˜ V ′′ ( x ) + U ( c ) (cid:21) , x > . (10)subject to lim x ↓ ˜ V ( x ) = P .Then ˆ V P ( x ) = V P ( x ) for all x ∈ [0 , ∞ ) .(B) For each P ∈ ( −∞ , , there exists a C -function ˆ V P : (0 , ∞ ) → ( P, ∞ ) that solves theHJB equation (10) with lim x ↓ ˆ V P ( x ) = P .(C) V ( x ) ≤ lim P ↓−∞ ˆ V P ( x ) = ˆ V ( x ) , which together with ˆ V ( x ) ≤ V ( x ) establishes the claim.Here, the argument for (A) is relatively straightforward; see [6, Theorem 4.1] and not moredifficult than the proof of our Theorem 4.3. Similarly, the argument for (C) is easy: the firstinequality follows from the fact that V ( x ) ≤ V P ( x ) ≤ ˆ V P ( x ) for each x > and P ∈ R − by10he definition of V P and (A); the second inequality is straightforward using the explicit form for ˆ V P .But the main difficulty – and great ingenuity – of the argument in [6] is (B). Indeed, a directcalculation for r > case takes at least two pages and yields the answer: ˆ V P ( x ) = νη ( R − ν ) (cid:18) ηR R − ν − ν (1 − R ) P (cid:19) − ν − R ( ˆ C P ( x )) ν − R + η − ( ˆ C P ( x )) − R − R , (11)where the function ˆ C P ( x ) describing the optimal consumption is the inverse of the function I P ( c ) = − η − (cid:18) ηR ν − Rν − − R ) P (cid:19) − ν − R c ν + cη , and ν is the negative root of the equation λ ζ + ( r − δ − λ ) Rζ − rR = 0 . We return to thisapproach in Section 7.2. The second perturbation approach is by Davis and Norman [2] who study the Merton problemwith transaction costs; the perturbation argument for
R > in the frictionless case is a fortunateby-product, and not the main contribution of the paper. Again we will use our notation todescribe their approach.Assume that δ > , r > and that condition (B) of Remark 4.2 is satisfied. Moreover, denoteby A b ( x ) all admissible strategies (Π , C ) for which Π is uniformly bounded, write C b ( x ) for thecorresponding set of attainable consumption strategys and set V b ( x ) := sup C ∈ C b ( x ) J ( C ) . For ζ > , consider the perturbed value function ˆ V ζ ( x ) = ˆ V ( x + ζ ) and for (Π , C ) ∈ A b ( X ) (suchthat C − R is integrable with respect to the identity process), consider the process M ζ definedby M ζ = Z t e − δt U ( C s ) d s + e − δt ˆ V ζ ( X t ) . Then the same argument as in the proof of Theorem 4.3 but with ˆ V replaced by ˆ V ζ yields d M ζt = d N ζt + A ,ζt d t + A ,ζt d t + A ,ζt d t, where the only difference is that ˆ V and its derivatives are replaced by ˆ V ζ and its derivatives.Then as in the proof of Theorem 4.3 it follows that A ,ζ , A ,ζ ≤ . Moreover, using that ˆ V ζ ( x ) = ˆ V ( x + ζ ) , it is straightforward to check that A ,ζt = − rζ ˆ V ζx ( X t ) e − δt ≤ , whichcrucially uses that r ≥ . Finally, using that Π and ˆ V ζx are bounded, it is not difficult to checkthat N ζ is a square integrable martingale. Now following the proof of Theorem 4.3, and usingthat | ˆ V ζ | is bounded and δ > it follows that J ( C ) ≤ lim sup E h M ζt i − lim inf E h e − δt ˆ V ζ ( X x, Π ,Ct ) i ≤ lim sup E h M ζt i ≤ ˆ V ζ ( x ) . We may conclude that V b ( x ) ≤ ˆ V ζ ( x ) and taking the limit as ζ ↓ , it follows that V b ( x ) = ˆ V ( x ) .11 The general verification argument
In this section, we present our general verification argument. It is inspired by the perturbationargument of Davis and Norman. The key idea is to use the candidate optimal consumptionstrategy as a stochastic perturbation of the utility function. This yields a very elegant and simpleargument that has the trio of advantages that it is no more difficult than the fiat verificationargument in Theorem 4.3, it does not need to distinguish between the case
R > and R < and it does not involve any stopping argument. The following theorem contains the solution to the stochastically perturbed Merton prob-lem. The subsequent corollary then lets this perturbation disappear. Recall the notations ofTheorem 4.3: η = R [ δ − (1 − R )( r + λ R )] , ˆΠ = λσR and ˆ V ( x ) = x − R − R η − R . Theorem 6.1.
Suppose η > . Denote by Y = ( Y t ) t ≥ the candidate optimal wealth processstarted from unit initial wealth , i.e., Y t := X , ˆΠ ,ηXt , and by G = ( G t ) t ≥ , the correspondingoptimal consumption stream, i.e., G t = ηY t . Fix ε > , define the function U ε : [0 , ∞ ) × (0 , ∞ ) → ( −∞ , ∞ ) by U ε ( c, g ) = ( c + εg ) − R − R , and for an attainable consumption stream C consider J ε ( C ) := E (cid:20)Z ∞ e − δt U ε ( C t , G t ) d t (cid:21) = J ( C + εG ) . Then for x > , V ε ( x ) := sup C ∈ C ( x ) J ε ( C ) = ˆ V ( x + ε ) . Moreover, the supremum is attained when
Π = ˆΠ and C = ˆ C where ˆ C = ηX x, ˆΠ , ˆ C .Proof. First, from the SDE for the wealth process (2) we have that X x, ˆΠ ,ηX + εY = X x + ε, ˆΠ ,ηX .It follows that ˆ C + εG = ηX x + ε, ˆΠ ,ηX ∈ C ( x + ε ) , which together with Theorem 4.3 implies that J ε ( ˆ C ) = J ( ˆ C + εG ) = ˆ V ( x + ε ) .It remains to show that V ε ( x ) ≤ ˆ V ( x + ε ) . The argument is very similar to the one in theproof of Theorem 4.3. Let (Π , C ) ∈ A ( x ) be arbitrary and set X := X x, Π ,C for brevity. Thedynamics of X + εY are given by d( X t + εY t ) = (cid:18) σ Π t X t + λR εY t (cid:19) d W t + (cid:18) X t ( r + Π t σλ ) − C t + (cid:18) r + λ R − η (cid:19) εY t (cid:19) d t. Define the process M ε = ( M εt ) t ≥ by M εt = Z t e − δs U ε ( C s , G s ) d s + e − δt ˆ V ( X t + εY t ) . We proceed to apply Itô’s formula to M ε . Adding and subtracting R − R ( ˆ V x ) − /R + εηY t ˆ V x and λ V x V xx + λ R εY t ˆ V x and noting that the argument of ˆ V and its derivatives is ( X t + εY t ) throughout, Mutatis mutandis the same argument gives also a full verification argument for the log case, i.e., R = 1 .
12e obtain d M εt = e − δt ( C t + εηY t ) − R − R d t + e − δt (cid:20) − δ ˆ V d t + e − δt ˆ V x d( X t + εY t ) + 12 e − δt ˆ V xx d[ X + εY ] t (cid:21) = e − δt ˆ V x (cid:18) σ Π t X t + λR εY t (cid:19) d W t + e − δt (cid:20) ( C t + εηY t ) − R − R − ( C t + εηY t ) ˆ V x − R − R ( ˆ V x ) − /R (cid:21) d t + e − δt " λ (cid:18) σ Π t X t + λR εY t (cid:19) ˆ V x + 12 (cid:18) σ Π t X t + λR εY t (cid:19) ˆ V xx + λ V x ˆ V xx dt + e − δt " − δ ˆ V + r ( X t + εY t ) ˆ V x + R − R ( ˆ V x ) − /R − λ V x ˆ V xx + (cid:18) λ R − η (cid:19) εY t ˆ V x + εηY t ˆ V x − λ R εY t ˆ V x (cid:21) d t =: d N εt + A ,εt d t + A ,εt d t + A ,εt d t. By the same arguments as in the proof of Theorem 4.3, it follows that A ,ε ≤ , A ,ε ≤ and A ,ε = 0 . This gives M εt ≤ ˆ V ( x + ε ) + N εt , t ≥ . (19)Next, define the process Λ ε = (Λ εt ) t ≥ by Λ εt := Z t e − δs U ε (0 , G s ) d s + e − δt ˆ V (0 + εY t ) = Z t e − δs U ( εG s ) + e − δt ˆ V ( εY t ) . Then Λ ε ≤ M ε by monotonicity of U and ˆ V . Using that Λ ε is a (UI) martingale by Remark4.4, it follows that N ε is bounded below by the (UI) martingale − ˆ V ( x + ε ) − Λ ε and hence asupermartingale.Taking expectation in (19), we find for each t ≥ , E [ M εt ] ≤ E h ˆ V ( x + ε ) + N εt i ≤ ˆ V ( x + ε ) . (20)Next, note that X + εY satisfies the transversality condition (T) since lim inf t →∞ E (cid:20) e − δt ( X t + εY t ) − R − R (cid:21) ≥ ε − R lim inf t →∞ E " e − δt Y − Rt − R = 0 . (21)Taking the limit in (20) as t goes to infinity and using (21), we may conclude that for any C ∈ C ( x ) , J ε ( C ) = lim t →∞ E (cid:20)Z t e − δs ( C s + εG s ) − R − R ds (cid:21) = lim t →∞ E h M εt − e − δt ˆ V ( X t + εY t ) i ≤ lim sup t →∞ E [ M εt ] − lim inf t →∞ E (cid:20) e − δt η − R ( X t + εY t ) − R − R (cid:21) ≤ lim sup t →∞ E [ M εt ] ≤ ˆ V ( x + ε ) . emark . The perturbation of the problem by the additional consumption of εG elegantlyand simply transforms the problem to one in which the fiat conditions (P), (M) and (T) aresatisfied. Since Y is positive P -a.s., the same is trivially true for X + εY . Moreover, J ( εG ) = ε − R J ( G ) > −∞ and this allows us to easily find an integrable lower bound on N ε and henceconclude it is a supermartingale. Again Y satisfies a transversality condition (T) and so thesame is trivially true for X + εY . Remark . One interpretation of the theorem is that a financially-savvy benefactor gives theagent an additional consumption stream based on an initial wealth ε which is invested opti-mally by the benefactor. Then, if the agent behaves optimally with their own wealth, the twoconsumption streams and investment strategies remain perfectly aligned to each other, and thederivation and valuation of the candidate optimal strategy is simple and immediate. Corollary 6.4.
Suppose η > . Then for x > , V ( x ) := sup C ∈ C ( x ) J ( C ) = J ( ˆ C ) = ˆ V ( x ) . Proof.
The equality J ( ˆ C ) = ˆ V ( x ) follows from Theorem 4.3. It remains to establish that V ( x ) ≤ ˆ V ( x ) . Using the notation of Theorem 6.1, for any C ∈ C ( x ) , we get J ( C ) ≤ J ε ( C ) ≤ V ε ( x ) = ˆ V ( x + ε ) . Letting ε ↓ , we conclude that V ( x ) ≤ ˆ V ( x ) .We finish this section by showing that in the case R > if η ≤ , every C ∈ C ( x ) has J ( C ) = −∞ . Corollary 6.5.
Suppose that
R > and η ≤ . Then V ( x ) = sup C ∈ C ( x ) J ( C ) = −∞ . Proof.
Fix C ∈ C ( x ) . It suffices to show that J ( C ) = −∞ . We use an approximation argument.For n ∈ N set δ n := δ + R ( n − η ) . Then δ n > δ and η n := R [ δ n − (1 − R )( r + λ R )] = n > .Then using that U ( c ) < for c ≥ , it follows from Theorem 6.1 J ( C ) = E (cid:20)Z ∞ e − δs U ( C s ) d s (cid:21) ≤ E (cid:20)Z ∞ e − δ n s U ( C s ) d s (cid:21) ≤ x − R − R ( η n ) − R , n ∈ N . Taking the limit on the right hand side as n goes to ∞ , it follows that J ( C ) = −∞ . We close the paper with some remarks on change of numéraire ideas. As we have seen inSection 5, using the perturbation arguments of Karatzas et al [6] or Davis and Norman [2], weget verification arguments for the case
R > under the parameter restrictions δ > and r > .In this section we show by using a change of numéraire that this parameter restriction can beweakened, although not to the extent that it covers all the parameter combinations for which η > . We then apply these ideas to present another new verification argument that is based onKaratzas et al [6] but far simpler. 14 .1 The Merton problem under a change of numéraire We say that a pair ( ˜ S , ˜ S ) = ( ˜ S t , ˜ S t ) t ≥ of semimartingales is economically equivalent to ( S , S ) if there exists a positive continuous semimartingale D = ( D t ) t ≥ such that ˜ S = DS and ˜ S = DS . Here, the interpretation of D is an exchange rate process and ( ˜ S , ˜ S ) describes thefinancial market in a different currency unit; see [4, Section 2.1] for more details.Next, recall that if ( ϑ , ϑ, C ) is a admissible investment-consumption strategy for initialwealth x > , (where ϑ and ϑ denote the number of shares held in the riskless and risky asset,respectively), then the corresponding wealth process X = ϑ t S t + ϑ t S satisfies the SDE d X t = ϑ t d S t + ϑ t d S t − C t d t. Now if ( ˜ S , ˜ S ) is economically equivalent to ( S , S ) with corresponding exchange rate process D , it is not difficult to check that the corresponding wealth process ˜ X := ϑ ˜ S + ϑ ˜ S = DX satisfies the SDE d ˜ X t = ϑ t d ˜ S + ϑ t d ˜ S − ˜ C t d t, where ˜ C = DC . This means that if C describes an attainable consumption strategy in unitscorresponding to ( S , S ) , then ˜ C = DC describes the same consumption strategy in unitscorresponding to ( ˜ S , ˜ S ) (which is also attainable for those units).Consider now the case that D t = e γt for some γ ∈ R . Then ( ˜ S , ˜ S ) is again a Black-Scholes-Merton model with interest rate ˜ r = r + γ , drift ˜ µ = µ + γ and volatility ˜ σ = σ .Let C be an attainable consumption strategy in units corresponding to ( S , S ) and ˜ C = DC the corresponding attainable consumption strategy in units corresponding to ( ˜ S , ˜ S ) . Then ˜ C/ ˜ S = DC/DS = C/S and J ( C ; δ ) := E (cid:20)Z ∞ e − δt − R C − Rt d t (cid:21) = E "Z ∞ e − ( δ + r ( R − t − R (cid:18) C t S t (cid:19) − R d t = J ( C/S ; δ + r ( R − J ( ˜ C/ ˜ S ; δ + (˜ r − γ )( R − E Z ∞ e − ( δ − ( R − γ +˜ r ( R − t − R ˜ C t ˜ S t ! − R d t = E "Z ∞ e − ( δ − ( R − γ ) t − R ˜ C − Rt d t = J ( ˜ C ; δ − ( R − γ ) It follows from the above calculation that the Merton problem for
R, r, µ, σ, δ is equivalentto the Merton problem for
R, r + γ, µ + γ, σ, δ − ( R − γ for each γ ∈ R . This means that ifwe have a verification argument for the parameters R, r + γ, µ + γ, σ, δ − ( R − γ , we also haveverification argument for the parameters R, r, µ, σ, δ . Hence, if δ + r ( R − > we can choose γ = δ − r ( R − R − so that ˜ δ = ˜ r = δ + r ( R − > and then we can extend the verification argumentsof Karatzas et al [6] or Davis and Norman [2] to this case. It follows that instead of needing toassume δ > and r > as in [6] and [2] it is sufficient to assume only that δ + r ( R − > .Nonetheless, the condition δ + r ( R − > is stronger than the condition for a well-posedproblem (namely η > ) and there are parameter values which we would like to consider (andwhich are covered by Theorem 6.1) for which the verification arguments of [6] and [2] do notapply, even after the change of numéraire arguments of this section.15 emark . An alternative formulation of the Merton problem is to associate to an attainableconsumption stream C the expected utility K ( C ; φ ) = E "Z ∞ e − φt − R (cid:18) C t S t (cid:19) − R d t , (24)where φ := δ + r ( R − . Then K ( C ; φ ) = J ( C, φ − r ( R − . In order to emphasise thedependence of the problem on the currency units which are being used we might expand thenotation to write J ( C ; S , S ; δ ) and K ( C ; S , S ; φ ) and then (23) becomes J ( C ; S , S ; δ ) = J ( ˜ C ; ˜ S , ˜ S ; δ − ( R − γ ) , whilst, for K ( C, φ ) = K ( C ; S , S, φ ) we find K ( ˜ C ; ˜ S , ˜ S, φ ) = E "Z ∞ e − φt − R (cid:18) D t C t D t S t (cid:19) − R d t = K ( C ; S , S, φ ) In particular, K defined via (24) has the advantage that (unlike J ) it is numéraire-independentin the sense that a change of currency unit leaves the problem value unchanged. With this inmind it makes sense to call φ (rather than δ ) the impatience rate . Note that η = R φ + ( R − R λ R so that the optimal consumption rate is a linear (convex if R > ) combination of the impatiencerate and (half of) the squared Sharpe ratio per unit of risk aversion, with the weights dependingon the risk aversion. Using the ideas of this section we can revisit the argument of Karatzas et al [6] to give a muchsimpler proof for V ( x ) = ˆ V ( x ) in the case that δ + (1 − R ) r > .The idea is to consider the case r = 0 , which is not studied in Karatzas et al [6]. For r = 0 ,it is not difficult to check that the HJB equation (10) has the solution ˆ V P,r =0 η = ( ηx + ( η (1 − R ) P ) / (1 − R ) ) − R η (1 − R ) , which is substantially simpler than the solution (11) for r > . Since the argument in (A) and(C) of Section 5.1 carry verbatim over to r = 0 , we have a verification argument for the MertonProblem in the case that δ > and r = 0 . Now if δ + r ( R − > , we choose γ = − r , so that ˜ δ = δ + r ( R − > and ˜ r = 0 , and the above change of numéraire argument in Section 7.1give a verification argument also in this case. Remark . Motivated by the above change of numéraire ideas, one might want to consider ageneralisation of (9) to allow for different discount rates on the utility of consumption and thebankruptcy payout: For δ, χ > , let J P ( C ; δ, χ ) = E "Z τ x, Π ,C e − δt C − Rt − R d t + e − χτ x, Π ,C P (25)and set V P ( x ; δ, χ ) := sup C ∈ C ( x ) J P ( C ; δ, χ ) . D t = e γt for some γ ∈ R be an exchange rate process, ( ˜ S , ˜ S ) = ( DS , DS ) the corresponding economically equivalent Black-Scholes-Merton model with interest rate ˜ r = r + γ , drift ˜ µ = µ + γ and volatility ˜ σ = σ , C an attainable consumption strategy in unitscorresponding to ( S , S ) and ˜ C = DC the corresponding attainable consumption strategy inunits corresponding to ( ˜ S , ˜ S ) . Then, noting that τ is numéraire independent and recallingthat C/S = ˜ C/ ˜ S , we have J P ( C ; δ, χ ) = E "Z τ e − ( δ + r (1 − R )) t − R (cid:18) C t S t (cid:19) − R d t + e − χτ P = J P ( C/S ; δ + r ( R − , χ ) = J P ( ˜ C/ ˜ S ; δ + (˜ r − γ )( R − , χ )= E Z τ e − ( δ − ( R − γ +˜ r (1 − R )) t − R ˜ C t ˜ S t ! − R d t + e − χτ P = J P ( ˜ C ; δ − ( R − γ, χ ) . It follows from the above calculation that the problem with two different discount rates ismathematically equivalent to the problem with the same discount rates; indeed, choose γ = χ − δ − R ,then J P ( C ; δ, χ ) = J P ( ˜ C ; χ, χ ) . However, the problem (25) with the same discount rates doesnot have a simpler solution than the one with different discount rates—unless r = 0 .This suggests that it might be useful to consider the numéraire invariant analogue to (25)and to set K P ( C ; φ, χ ) := E "Z τ e − φt − R (cid:18) C t S t (cid:19) − R d t + e − χτ P . (26)Then K P has the property that it is invariant under a change of currency units: K P ( C ; φ, χ ) = K P ( ˜ C ; φ, χ ) . Note that by construction J P ( C ; δ, χ ) = K P ( C ; δ + r ( R − , χ ) . In particular, J P ( C, δ, δ ) = K P ( C, δ + r (1 − R ) , δ ) so that the numéraire-dependent problem (25) with the same discount rates corresponds to the numéraire-independent problem with different discountrates—unless r = 0 .Unlike the numéraire-dependent problem (25), the numéraire-independent problem (26) hasa much simpler solution for the same discount rates than for different discount rates. Indeed, ˆ V P,K ( x ; φ, φ ) := sup C ∈ C ( x ) K P ( C ; φ, φ ) = ( ηx + ( η (1 − R ) P ) / − R ) − R η (1 − R ) . where η = R φ + ( R − R λ R . This gives a further justification for considering the numéraire-independent formulation of the Merton problem (cf. Remark 7.1) or its bankruptcy variant. References [1] S. Biagini. Expected utility maximization: Duality methods. In R. Cont, editor,
Encyclo-pedia of Quantitative Finance . Wiley Online Library, 2010.[2] M. H. A. Davis and A. R. Norman. Portfolio selection with transaction costs.
Math. Oper.Res. , 15(4):676–713, 1990. 173] W. H. Fleming and H. M. Soner.
Controlled Markov processes and viscosity solutions ,volume 25. Springer, New York, second edition, 2006.[4] M. Herdegen. No-arbitrage in a numéraire-independent modeling framework.
Math. Fi-nance , 27(2):568–603, 2017.[5] I. Karatzas. Optimization problems in the theory of continuous trading.
SIAM J. ControlOptim. , 27(6):1221–1259, 1989.[6] I. Karatzas, J. P. Lehoczky, S. P. Sethi, and S. E. Shreve. Explicit solution of a generalconsumption/investment problem.
Math. Oper. Res. , 11(2):261–294, 1986.[7] I. Karatzas, J. P. Lehoczky, S. E. Shreve, and G. L. Xu. Martingale and duality methodsfor utility maximization in an incomplete market.
SIAM J. Control. Optim. , 29(3):702–730,1991.[8] I. Karatzas and S. E. Shreve.
Methods of mathematical finance , volume 39 of
Applicationsof Mathematics (New York) . Springer-Verlag, New York, 1998.[9] J. Lehoczky, S. P. Sethi, and S. Shreve. Optimal consumption and investment policiesallowing consumption constraints and bankruptcy.
Math. Oper. Res. , 8(4):613–636, 1983.[10] R. C. Merton. Lifetime portfolio selection under uncertainty: The continuous-time case.
Rev. Econom. Statist. , 51(3):247–257, 1969.[11] R. C. Merton. Optimum consumption and portfolio rules in a continuous-time model.
J.Econom. Theory , 3(4):373–413, 1971.[12] H. Pham.
Continuous-time stochastic control and optimization with financial applications ,volume 61 of
Stochastic Modelling and Applied Probability . Springer-Verlag, Berlin, 2009.[13] L. C. G. Rogers.
Optimal investment . SpringerBriefs in Quantitative Finance. Springer,Heidelberg, 2013.[14] S. P. Sethi and M. Taksar. A note on Merton’s ‘Optimum consumption and portfolio rulesin a continuous-time model’.