Investment/consumption problem in illiquid markets with regime-switching
aa r X i v : . [ q -f i n . P M ] A p r Investment/consumption problem in illiquid markets withregime-switching
Paul Gassiat , Fausto Gozzi , Huyˆen Pham , January 8, 2018 Laboratoire de Probabilit´es et Dipartimento di Scienze EconomicheMod`eles Al´eatoires, CNRS, UMR 7599 ed Aziendali - Facolt`a di Economia,Universit´e Paris 7 Diderot, Universit`a LUISS Guido Carli,pgassiat, pham at math.jussieu.fr fgozzi at luiss.it CREST-ENSAEand Institut Universitaire de France
Abstract
We consider an illiquid financial market with different regimes modeled by a conti-nuous-time finite-state Markov chain. The investor can trade a stock only at the discretearrival times of a Cox process with intensity depending on the market regime. Moreover,the risky asset price is subject to liquidity shocks, which change its rate of return andvolatility, and induce jumps on its dynamics. In this setting, we study the problemof an economic agent optimizing her expected utility from consumption under a non-bankruptcy constraint. By using the dynamic programming method, we provide thecharacterization of the value function of this stochastic control problem in terms ofthe unique viscosity solution to a system of integro-partial differential equations. Wenext focus on the popular case of CRRA utility functions, for which we can provesmoothness C results for the value function. As an important byproduct, this allowsus to get the existence of optimal investment/consumption strategies characterized infeedback forms. We analyze a convergent numerical scheme for the resolution to ourstochastic control problem, and we illustrate finally with some numerical experimentsthe effects of liquidity regimes in the investor’s optimal decision. Key words :
Optimal consumption, liquidity effects, regime-switching models, viscositysolutions, integro-differential system.
MSC Classification (2000) :
Introduction
A classical assumption in the theory of optimal portfolio/consumption choice as in Merton[16] is that assets are continuously tradable by agents. This is not always realistic inpractice, and illiquid markets provide a prime example. Indeed, an important aspect ofmarket liquidity is the time restriction on assets trading: investors cannot buy and sell themimmediately, and have to wait some time before being able to unwind a position in somefinancial assets. In the past years, there was a significant strand of literature addressingthese liquidity constraints. In [19], [15], the price process is observed continuously but thetrades succeed only at the jump times of a Poisson process. Recently, the papers [17], [4], [8]relax the continuous-time price observation by considering that asset is observed only at therandom trading times. In all these cited papers, the intensity of trading times is constant ordeterministic. However, the market liquidity is also affected by long-term macroeconomicconditions, for example by financial crisis or political turmoil, and so the level of tradingactivity measured by its intensity should vary randomly over time. Moreover, liquiditybreakdowns would typically induce drops on the stock price in addition to changes in itsrate of return and volatility.In this paper, we investigate the effects of such liquidity features on the optimal portfoliochoice. We model the index of market liquidity as an observable continuous-time Markovchain with finite-state regimes, which is consistent with some cyclicality observed in finan-cial markets. The modelisation of financial stock prices by regime-switching processes wasoriginally proposed and justified in [9], and since then this approach has been extensivelypursued in the financial litterature, see e.g. [1], [20] and the references therein.The economic agent can trade only at the discrete arrival times of a Cox process withintensity depending on the market regimes. Moreover, the risky asset price is subject toliquidity shocks, which switch its rate of return and volatility, while inducing jumps onits dynamics. In this hybrid jump-diffusion setting with regime switching, we study theoptimal investment/consumption problem over an infinite horizon under a nonbankruptcystate constraint. We first prove carefully that dynamic programming principle (DPP) holdsin our framework. Due to the state constraints in two dimensions, we have to slightly weakenthe standard continuity assumption, see Remark 3.1. Then, using DPP, we characterizethe value function of this stochastic control problem as the unique constrained viscositysolution to a system of integro-partial differential equations. In the particular case ofCRRA utility function, we can go beyond the viscosity properties, and prove C regularityresults for the value function in the interior of the domain. As a consequence, we showthe existence of optimal strategies expressed in feedback form in terms of the derivativesof the value function. Due to the presence of state constraints, the value function is notsmooth at the boundary, and so the verification theorem cannot be proved with the classicalarguments of Dynkin’s formula. To overcome this technical problem, we use an ad hocapproximation procedure (see Proposition 5.2). We also provide a convergent numericalscheme for solving the system of equations characterizing our control problem, and weillustrate with some numerical results the effect of liquidity regimes in the agent’s optimalinvestment/consumption. We also measure the impact of continuous time observation with2espect to a discrete time observation of the stock prices. Our paper contributes andextends the existing literature in several ways. First, we extend the papers [19] and [15] byconsidering stochastic intensity trading times and regime switching in the asset prices. Fora two-state Markov chain modulating the market liquidity, and in the limiting case wherethe intensity in one regime goes to infinity, while the other one goes to zero, we recover thesetup of [5] and [14] where an investor can trade continuously in the perfectly liquid regimebut faces a threat of trading interruptions during a period of market freeze. On the otherhand, regime switching models in optimal investment problems was already used in [23],[20] or [21] for continuous-time trading.The rest of the paper is structured as follows. Section 2 describes our continuous-timemarket model with regime-switching liquidity, and formulates the optimization problemfor the investor. In Section 3 we state some useful properties of the value function of ourstochastic control problem. Section 4 is devoted to the analytic characterization of thevalue function as the unique viscosity solution to the dynamic programming equation. Thespecial case of CRRA utility functions is studied in Section 5: we show smoothness resultsfor the value functions, and obtain the existence of optimal strategies via a verificationtheorem. Some numerical illustrations complete this last section. Finally two appendicesare devoted to the proof of two technical results: the dynamic programming principle, andthe existence and uniqueness of viscosity solutions. Let us fix a probability space (Ω , F , P ) equipped with a filtration F = ( F t ) t ≥ satisfyingthe usual conditions. It is assumed that all random variables and stochastic processes aredefined on the stochastic basis (Ω , F , F , P ).Let I be a continuous-time Markov chain valued in the finite state space I d = { , . . . , d } ,with intensity matrix Q = ( q ij ). For i = j in I d , we can associate to the jump process I ,a Poisson process N ij with intensity rate q ij ≥
0, such that a switch from state i to j corresponds to a jump of N ij when I is in state i . We interpret the process I as a proxyfor market liquidity with states (or regimes) representing the level of liquidity activity, inthe sense that the intensity of trading times varies with the regime value. This is modeledthrough a Cox process ( N t ) t ≥ with intensity ( λ I t ) t ≥ , where λ i > i ∈ I d . Forexample, if λ i < λ j , this means that trading times occur more often in regime j than inregime i . The increasing sequence of jump times ( τ n ) n ≥ , τ = 0, associated to the countingprocess N represents the random times when an investor can trade a risky asset of priceprocess S . Note that under these assumptions the jumps of I and N are a.s. disjoint.In the liquidity regime I t = i , the stock price follows the dynamics dS t = S t ( b i dt + σ i dW t ) , where W is a standard Brownian motion independent of ( I, N ), and b i ∈ R , σ i ≥
0, for i ∈ I d . Moreover, at the times of transition from I t − = i to I t = j , the stock changes as follows: S t = S t − (1 − γ ij )3or a given γ ij ∈ ( −∞ , γ ij > γ ij ≤ γ ij > λ j < λ i . Overall, the risky asset is governed bya regime-switching jump-diffusion model: dS t = S t − (cid:16) b I t − dt + σ I t − dW t − γ It − ,It dN I t − ,I t t (cid:17) . (2.1) Portfolio dynamics under liquidity constraint.
We consider an agent investing and con-suming in this regime-switching market. We denote by ( Y t ) the total amount investedin the stock, and by ( c t ) the consumption rate per unit of time, which is a nonnegativeadapted process. Since the number of shares Y t /S t in the stock held by the investor has tobe kept constant between two trading dates τ n and τ n +1 , then between such trading times,the process Y follows the dynamics: dY t = Y t − dS t S t − , τ n ≤ t < τ n +1 , n ≥ , The trading strategy is represented by a predictable process ( ζ t ) such that at a tradingtime t = τ n +1 , the rebalancing on the number of shares induces a jump ζ t in the amountinvested in the stock : ∆ Y t = ζ t . Overall, the c`adl`ag process Y is governed by the hybrid controlled jump-diffusion process dY t = Y t − (cid:16) b I t − dt + σ I t − dW t − γ It − ,It dN I t − ,I t t (cid:17) + ζ t dN t . (2.2)Assuming for simplicity a constant savings account (see Remark 2.2), i.e. zero interest rate,the amount ( X t ) invested in cash then follows dX t = − c t dt − ζ t dN t . (2.3)The total wealth is defined at any time t ≥
0, by R t = X t + Y t , and we shall require thenon-bankruptcy constraint at any trading time: R τ n ≥ , a.s. ∀ n ≥ . (2.4)Actually since the asset price may become arbitrarily large or small between two tradingdates, this non-bankruptcy constraint means a no-short sale constraint on both the stockand savings account, as showed by the following Lemma. Lemma 2.1
The nonbankruptcy constraint (2.4) is formulated equivalently in the no-shortsale constraint: X t ≥ , and Y t ≥ , ∀ t ≥ . (2.5) This is also written equivalently in terms of the controls as: − Y t − ≤ ζ t ≤ X t − , t ≥ , (2.6) Z τ n +1 t c s ds ≤ X t , τ n ≤ t < τ n +1 , n ≥ . (2.7)4 roof. By writing by induction the wealth at any trading time as R τ n +1 = R τ n + Y τ n (cid:18) S τ n +1 S τ n − (cid:19) − Z τ n +1 τ n c t dt, n ≥ , and since (conditionally on F τ n ) the stock price S τ n +1 has support in (0 , ∞ ), we see thatthe nonbankruptcy condition R τ n +1 ≥ ≤ Y τ n ≤ R τ n , n ≥ , (2.8)together with the condition on the nonnegative consumption rate Z τ n +1 τ n c t dt ≤ R τ n − Y τ n = X τ n , n ≥ . (2.9)Since Y τ n = Y τ n − + ζ τ n , and since R τ n = R τ n − a.s., the no-short sale constraint (2.8) meansequivalently that (2.6) is satisfied for t = τ n . Since ζ is predictable, this is equivalent to(2.6) being satisfied d P ⊗ dt almost everywhere. Indeed, letting H t = { ζ t < − Y t − or ζ t >X t − } , H is predictable, so that ∀ t ≥
0, 0 = E hP τ n ≤ t H τ n i = E hR t H s λ I s ds i , and we deduce that H t = 0 d P ⊗ dt a.e. since λ I t > X t = X τ n − R tτ n c s ds for τ n ≤ t < τ n +1 , the condition (2.9) is equivalentto (2.7). By rewriting the conditions (2.8)-(2.9) as Y τ n ≥ , X τ n ≥ , X ( τ n +1 ) − ≥ , ∀ n ≥ , and observing that for τ n ≤ t < τ n +1 , Y t = S t S τ n Y τ n , X τ n ≥ X t ≥ X ( τ n +1 ) − , we see that they are equivalent to (2.5). ✷ Remark 2.1
Under the nonbankruptcy (or no-short sale constraint), the wealth ( R t ) t ≥ is nonnegative, and follows the dynamics: dR t = R t − Z t − (cid:16) b I t − dt + σ I t − dW t − γ It − ,It dN I t − ,I t t (cid:17) − c t dt, (2.10)where Z t := Y t R t valued in [0 ,
1] is the proportion of wealth invested in the risky asset; andevolves according to the dynamics: dZ t = Z t − (1 − Z t − ) h(cid:0) b I t − − Z t − σ I t − (cid:1) dt + σ I t − dW t − γ It − ,It − Z t − γ It − ,It dN I t − ,I t t i + ζ t R t − dN t + Z t − c t R t − dt, (2.11)for t < τ = inf { t ≥ R t = 0 } . 5iven an initial state ( i, x, y ) ∈ I d × R + × R + , we shall denote by A i ( x, y ) the set ofinvestment/consumption control process ( ζ, c ) such that the corresponding process ( X, Y )solution to (2.2)-(2.3) with a liquidity regime I , and starting from ( I − , X − , Y − ) = ( i, x, y ),satisfy the non-bankruptcy constraint (2.5) (or equivalently (2.6)-(2.7)). Optimal investment/consumption problem.
The preferences of the agent are describedby a utility function U which is increasing, concave, C on (0 , ∞ ) with U (0) = 0, andsatisfies the usual Inada conditions: U ′ (0) = ∞ , U ′ ( ∞ ) = 0. We assume the followinggrowth condition on U : there exist some positive constant K , and p ∈ (0 ,
1) s.t. U ( x ) ≤ Kx p , x ≥ . (2.12)We denote by ˜ U the convex conjugate of U , defined from R into [0 , ∞ ] by:˜ U ( ℓ ) = sup x ≥ [ U ( x ) − xℓ ] , which satisfies under (2.12) the dual growth condition on R + :˜ U ( ℓ ) ≤ ˜ Kℓ − ˜ p , ∀ ℓ ≥ , with ˜ p = p − p > , (2.13)for some positive constant ˜ K .The agent’s objective is to maximize over portfolio/consumption strategies in the aboveilliquid market model the expected utility from consumption rate over an infinite horizon.We then consider, for each i ∈ I d , the value function v i ( x, y ) = sup ( ζ,c ) ∈A i ( x,y ) E (cid:20)Z ∞ e − ρt U ( c t ) dt (cid:21) , ( x, y ) ∈ R , (2.14)where ρ is a discount factor. We also introduce, for i ∈ I d , the functionˆ v i ( r ) = sup x ∈ [0 ,r ] v i ( x, r − x ) , r ≥ , (2.15)which represents the maximal utility performance that the agent can achieve starting froman initial nonnegative wealth r and from the regime i . More generally, for any locallybounded function w i on R , we associate the function ˆ w i defined on R + by: ˆ w i ( r ) =sup x ∈ [0 ,r ] w i ( x, y ), so that:ˆ w i ( x + y ) = sup e ∈ [ − y,x ] w i ( x − e, y + e ) , ( x, y ) ∈ R . In the sequel, we shall often identify a d -tuple function ( w i ) i ∈ I d defined on R with thefunction w defined on R × I d by w ( x, y, i ) = w i ( x, y ).In this paper, we focus on the analytic characterization of the value functions v i (andso ˆ v i ), i ∈ I d , and on their numerical approximation. Remark 2.2
For simplicity we have assumed zero interest rate for the riskless asset. Thecase of constant r = 0 can actually be reduced to this case, at the cost of allowing time-dependent utility of consumption. This can be seen from the identity E Z ∞ e − ρs U ( c s ) ds = E Z ∞ e − ¯ ρs ¯ U ( s, ¯ c s ) ds, ρ = ρ − pr , ¯ c s = e − rs c s and ¯ U ( s, ¯ c ) = e − prs U ( e rs ¯ c ). Note that ¯ U ( s, · ) still satisfies(2.12), and in the special case of power utility U ( c ) = c p , one actually has ¯ U ( s, · ) = U . We state some preliminary properties of the value functions that will be used in the nextsection for the PDE characterization. We first need to check that the value functions arewell-defined and finite. Let us consider for any p >
0, the positive constant: k ( p ) := max i ∈ I d ,z ∈ [0 , h pb i z − σ i p (1 − p ) z + X j = i q ij ((1 − zγ ij ) p − i < ∞ . We then have the following lemma.
Lemma 3.1
Fix some initial conditions ( i, x, y ) ∈ I d × R + × R + , and some p > . Then: (1) For any admissible control ( ζ, c ) ∈ A i ( x, y ) associated with wealth process R , theprocess ( e − k ( p ) t R pt ) t ≥ is a supermartingale. So, in particular, for ρ > k ( p ) , lim t →∞ e − ρt E [ R pt ] = 0 . (3.1) (2) For fixed T ∈ (0 , ∞ ) , the family ( R pT ∧ τ ) τ,ζ,c is uniformly integrable, when τ rangesover all stopping times, and ( ζ, c ) runs over A i ( x, y ) . Proof. (1)
By Itˆo’s formula and (2.10), we have d ( e − k ( p ) t R pt ) = − k ( p ) e − k ( p ) t R pt dt + e − k ( p ) t d ( R pt )= e − k ( p ) t h − k ( p ) R pt + pR p − t − (cid:0) − c t + b I t − R t − Z t − (cid:1) + p ( p − R p − t − (cid:0) σ I t − R t − Z t − (cid:1) + X j = I t − q I t − ,j ( R pt − (1 − γ I t − j Z t − ) p − R pt − ) i dt + dM t , where M is a local martingale. Now, by definition of k ( p ), we have pR p − t − (cid:0) − c t + b I t − R t − Z t − (cid:1) + p ( p − R p − t − (cid:0) σ I t − R t − Z t − (cid:1) + X j = I t − q I t − ,j ( R pt − (1 − γ I t − j Z t − ) p − R pt − ) ≤ − pc t R p − t − + k ( p ) R pt − ≤ k ( p ) R pt − . Since R has countable jumps, R t = R t − , d P ⊗ dt a.e., and so the drift term in d ( e − k ( p ) t R pt )is nonpositive. Hence ( e − k ( p ) t R pt ) t ≥ is a local supermartingale, and since it is nonnegative,it is a true supermartingale by Fatou’s lemma. In particular, we have0 ≤ e − ρt E [ R pt ] ≤ e − ( ρ − k ( p )) t ( x + y ) p (3.2)which shows (3.1). 7 For any q >
1, we get by the supermartingale property of the process ( e − k ( pq ) t R pqt ) t ≥ and the optional sampling theorem: E (cid:2)(cid:0) R pT ∧ τ (cid:1) q (cid:3) ≤ e k ( pq ) T ( x + y ) pq < ∞ , ∀ ( ζ, c ) ∈ A i ( x, y ) , τ stopping time , which proves the required uniform integrability. ✷ The next proposition states a comparison result, and, as a byproduct, a growth conditionfor the value function.
Proposition 3.1(1)
Let w = ( w i ) i ∈ I d be a d -tuple of nonnegative functions on R , twice differentiable on R \ { (0 , } such that ρw i − b i y ∂w i ∂y − σ i y ∂ w i ∂y − X j = i q ij [ w j ( x, y (1 − γ ij )) − w i ( x, y )] − λ i [ ˆ w i ( x + y ) − w i ( x, y )] − ˜ U (cid:18) ∂w i ∂x (cid:19) ≥ , (3.3) for all i ∈ I d , ( x, y ) ∈ R \ { (0 , } . Then, for all i ∈ I d , v i ≤ w i , on R . (2) Under (2.12) , suppose that ρ > k ( p ) . Then, there exists some positive constant C s.t. v i ( x, y ) ≤ C ( x + y ) p , ∀ ( i, x, y ) ∈ I d × R . (3.4) Proof. (1)
First notice that for ( x, y ) = (0 , A i ( x, y )is the zero control ζ = 0, c = 0, so that v i (0 ,
0) = 0. Now, fix ( x, y ) ∈ R \ { (0 , } , i ∈ I d , and consider an arbitrary admissible control ( ζ, c ) ∈ A i ( x, y ). By Itˆo’s formula to e − ρt w ( X t , Y t , I t ), we get: d [ e − ρt w ( X t , Y t , I t )] = e − ρt h − ρw − c t ∂w∂x + b I t − Y t − ∂w∂y + 12 σ I t − Y t − ∂ w∂y + X j = I t − q It − j [ w ( X t − , Y t − (1 − γ It − j ) , j ) − w ( X t − , Y t − , I t − )]+ λ It − (cid:2) w ( X t − − ζ t , Y t − + ζ t , I t − ) − w ( X t − , Y t − , I t − ) (cid:3)i dt + e − ρt σ I t − Y t − ∂w∂y ( X t − , Y t − , I t − ) dW t + e − ρt X j = I t − [ w ( X t − , Y t − (1 − γ It − j ) , j ) − w ( X t − , Y t − , I t − )] (cid:0) dN I t − j − q It − j dt (cid:1) + e − ρt (cid:2) w ( X t − − ζ t , Y t − + ζ t , I t − ) − w ( X t − , Y t − , I t − ) (cid:3)i(cid:0) dN t − λ It − dt (cid:1) . (3.5)Denote by τ = inf { t ≥ X t , Y t ) = (0 , } , and consider the sequence of bounded stoppingtimes τ n = inf { t ≥ X t + Y t ≥ n or X t + Y t ≤ /n } ∧ n , n ≥
1. Then, τ n ր τ a.s. when n goes to infinity, and c t = 0, X t = Y t = 0 for t ≥ τ , and so E h Z ∞ e − ρt U ( c t ) dt i = E h Z τ e − ρt U ( c t ) dt i . (3.6)8rom Itˆo’s formula (3.5) between time t = 0 and t = τ n , and observing that the integrandsof the local martingale parts are bounded for t ≤ τ n , we obtain after taking expectation: w ( x, y, i ) = E h e − ρτ n w ( X τ n , Y τ n , I τ n )+ Z τ n e − ρt (cid:16) ρw + c t ∂w∂x − b I t − Y t − ∂w∂y − σ I t − Y t − ∂ w∂y − X j = I t − q It − j [ w ( X t − , Y t − (1 − γ It − j ) , j ) − w ( X t − , Y t − , I t − )] − λ It − (cid:2) w ( X t − − ζ t , Y t − + ζ t , I t − ) − w ( X t − , Y t − , I t − ) (cid:3)(cid:17) dt i ≥ E h e − ρτ n w ( X τ n , Y τ n , I τ n ) + Z τ n e − ρt U ( c t ) dt i ≥ E h Z τ n e − ρt U ( c t ) dt i , where we used (3.3), and the nonnegativity of w . By sending n to infinity with Fatou’slemma, and (3.6), we obtain the required inequality: w i ≥ v i since ( c, ζ ) are arbitrary. (2) Consider the function w i ( x, y ) = C ( x + y ) p . Then, for ( x, y ) ∈ R \ { (0 , } , anddenoting by z = y/ ( x + y ) ∈ [0 , ρw i − b i y ∂w i ∂y − σ i y ∂ w i ∂y − X j = i q ij [ w j ( x, y (1 − γ ij )) − w i ( x, y )] − λ i [ ˆ w i ( x + y ) − w i ( x, y )] − ˜ U ( ∂w i ∂x )= C ( x + y ) p h ρ − pb i z + σ i p (1 − p ) z − X j = i q ij ((1 − zγ ij ) p − i − ˜ U (( x + y ) p − pC ) ≥ ( x + y ) p (cid:16) C ( ρ − k ( p )) − ˜ K ( pC ) − p − p (cid:17) (3.7)by (2.13). Hence, for ρ > k ( p ), and for C sufficiently large, the r.h.s. of (3.7) is nonnegative,and we conclude by using the comparison result in assertion 1). ✷ In the sequel, we shall assume the standing condition that ρ > k ( p ) so that the valuefunctions are well-defined and satisfy the growth condition (3.4). We now prove continuityproperties of the value functions. Proposition 3.2
The value functions v i , i ∈ I d , are concave, nondecreasing in both vari-ables, and continuous on R . This implies also that ˆ v i , i ∈ I d , are nondecreasing, concaveand continuous on R + . Moreover, we have the boundary conditions for v i , i ∈ I d , on { } × R + : v i (0 , y ) = ( , if y = 0 E h e − ρτ ˆ v Iiτ (cid:0) y S τ S (cid:1)i , if y > . (3.8) Here I i denotes the continuous-time Markov chain I starting from i at time . Proof.
Fix some ( x, y, i ) ∈ R × I d , δ ≥ δ ≥
0, and take an admissible control ( ζ, c ) ∈ A i ( x, y ). Denote by R and R ′ the wealth processes associated to ( ζ, c ), starting from9nitial state ( x, y, i ) and ( x + δ , y + δ , i ). We thus have R ′ = R + δ + δ S/S . Thisimplies that ( ζ, c ) is also an admissible control for ( x + δ , y + δ , i ), which shows clearly thenondecreasing monotonicity of v i in x and y , and thus also the nondecreasing monotonicityof ˆ v i by its very definition.The concavity of v i in ( x, y ) follows from the linearity of the admissibility constraints in X, Y, ζ, c , and the concavity of U . This also implies the concavity of ˆ v i ( r ) by its definition.Since v i is concave, it is continuous on the interior of its domain R . From (3.4), andsince v i is nonnegative, we see that v i is continuous on ( x , y ) = (0 ,
0) with v i (0 ,
0) = 0.Then, ˆ v i is continuous on R + with ˆ v i (0) = 0. It remains to prove the continuity of v i at( x , y ) when x = 0 or y = 0. We shall rely on the following implication of the dynamicprogramming principle v i ( x, y ) = sup c ∈C ( x ) E h Z τ e − ρt U ( c t ) dt + e − ρτ ˆ v Iiτ ( R τ ) i (3.9)= sup c ∈C ( x ) E h Z τ e − ρt U ( c t ) dt + e − ρτ ˆ v Iiτ (cid:0) x − Z τ c t dt + y S τ S (cid:1)i , ∀ ( x, y ) ∈ R , where C ( x ) denotes the set of nonnegative adapted processes ( c t ) s.t. R τ c t dt ≤ x a.s.(i) We first consider the case x = 0 (and y > c in C ( x ) means that c t = 0, t ≤ τ , sothat (3.9) implies (3.8). Now, since v i is nondecreasing in x , we have: v i ( x, y ) ≥ v i (0 , y ).Moreover, by concavity and thus continuity of v i (0 , . ), we have: lim y → y v i (0 , y ) = v i (0 , y ).This implies that lim inf ( x,y ) → (0 ,y ) v i ( x, y ) ≥ v i (0 , y ). The proof of the converse inequalityrequires more technical arguments. For any x, y ≥
0, we have: v i ( x, y ) = sup c ∈C ( x ) E h Z τ e − ρs U ( c s ) ds + e − ρτ ˆ v Iiτ (cid:0) x − Z τ c s ds + y S τ S (cid:1)i ≤ sup c ∈C ( x ) E h Z τ e − ρs U ( c s ) ds i + E h e − ρτ ˆ v I τ (cid:0) x + y S τ S (cid:1)i =: E ( x ) + E ( x, y ) . (3.10)Now, by Jensen’s inequality, and since U is concave, we have: Z ∞ U (cid:0) c s { s ≤ τ } (cid:1) ρe − ρs ds ≤ U (cid:18)Z ∞ c s { s ≤ τ } ρe − ρs ds (cid:19) , and thus: Z τ e − ρs U ( c s ) ds ≤ U ( ρx ) ρ , a.s. ∀ c ∈ C ( x ) , (3.11)by using the fact that R τ c t dt ≤ x a.s. By continuity of U in 0 with U (0) = 0, this showsthat E ( x ) converges to zero when x goes to x = 0. Next, by continuity of ˆ v i , we have:ˆ v Iiτ (cid:0) x + y S τ S (cid:1) → ˆ v Iiτ (cid:0) y S τ S (cid:1) a.s. when ( x, y ) → (0 , y ). Let us check that this convergenceis dominated. Indeed from (3.4), there is some positive constant C s.t.ˆ v Iiτ (cid:0) x + y S τ S (cid:1) ≤ C (cid:0) x + y S τ S (cid:1) p ≤ C ( x + y ) p (cid:16) ∨ (cid:16) S τ S (cid:17) p (cid:17) . E h e − ρτ (cid:16) S τ S (cid:17) p (cid:12)(cid:12)(cid:12) I, W i = Z ∞ λ I t e − R t λ Is e − ρt (cid:16) S t S (cid:17) p dt ≤ max i ∈ I d λ i Z ∞ e − ρt (cid:18) S t S (cid:19) p dt, and so E h e − ρτ (cid:16) S τ S (cid:17) p i ≤ max i ∈ I d λ i Z ∞ E h e − ρt (cid:16) S t S (cid:17) p i dt ≤ max i ∈ I d λ i Z ∞ e − ( ρ − k ( p )) t dt < ∞ , where we used in the second inequality the supermartingale property in Lemma 3.1 (and,more precisely, equation (3.2)) for x = 0 , y = 1 , c ≡ ζ ≡
0. One can then applythe dominated convergence theorem to E ( x, y ), to deduce that E ( x, y ) converges to E h e − ρτ ˆ v Iiτ (cid:0) y S τ S (cid:1)i when ( x, y ) → (0 , y ). This, together with (3.8), (3.10), proves thatlim sup ( x,y ) → (0 ,y ) v i ( x, y ) ≤ v i (0 , y ), and thus the continuity of v i at (0 , y ).(ii) We consider the case y = 0 (and x > v i ( ., ( x,y ) → ( x , v i ( x, y ) ≥ v i ( x , x ≥
0, and c ∈ C ( x ),let us consider the stopping time τ c = inf (cid:8) t ∈≥ R t c s ds = x (cid:9) . Then, the nonnegativeadapted process c ′ defined by: c ′ t = c t (cid:8) t ≤ τ c ∧ τ (cid:9) , lies obviously in C ( x ). Furthermore, Z τ e − ρs U ( c s ) ds = Z τ c ∧ τ e − ρs U ( c ′ s ) ds + Z τ τ c ∧ τ e − ρs U ( c s ) ds ≤ Z τ e − ρs U ( c ′ s ) ds + U ( ρ ( x − x ) + ) ρ , (3.12)by the same Jensen’s arguments as in (3.11), and for all y ≥ v Iiτ (cid:16) x − Z τ c t dt + y S τ S (cid:17) ≤ ˆ v Iiτ (cid:16) x − Z τ c ′ t dt + ( x − x ) + + y S τ S (cid:17) ≤ ˆ v Iiτ (cid:16) x − Z τ c ′ t dt (cid:17) + ˆ v Iiτ (cid:16) ( x − x ) + + y S τ S (cid:17) , (3.13)where we have used the fact that ˆ v i is nondecreasing, and subadditive (as a concave functionwith ˆ v i (0) ≥ v i ( x, y ) ≤ v i ( x ,
0) + U ( ρ ( x − x ) + ) ρ + E h e − ρτ ˆ v Iiτ (cid:16) ( x − x ) + + y S τ S (cid:17)i , and by the same domination arguments as in the first case, this shows thatlim sup ( x,y ) → ( x , v i ( x, y ) ≤ v i ( x , , which ends the proof. ✷ emark 3.1 The above proof of continuity of the value functions at the boundary bymeans of the dynamic programming principle is somehow different from other similar proofsthat one can find e.g. in [6, 17, 23]. Indeed in such problems the proof of dynamic pro-gramming principle is done (or referred to) in two parts: the “easy” one ( ≤ ) which doesnot require continuity of the value function, and the ‘difficult” one ( ≥ ) which requires thecontinuity of the value function up to the boundary. The proof of continuity at the bound-ary in such cases uses only the “easy” inequality. In our case, due to the specific boundarycondition of our problem, the “easy” inequality is not enough to prove the continuity atthe boundary. We need also the “hard” inequality. For this reason we give, in AppendixA, a proof of the dynamic programming principle in our case that, in the “hard” inequalitypart, uses the continuity of v i in the interior and the continuity of its restriction to theboundary (which are both implied by the concavity and by the growth condition (3.4)). Remark 3.2
For simplicity we have restricted our study to the case where U is definedon the positive half-line R + . With some work, our results can be extended to the case U (0) = −∞ , assuming U ( c ) ≥ − Kc q , for some K ≥ q <
0. In that case (assuming ρ > v i ( x, y ) > −∞ whenever x > y ≥
0, while v i (0 + , y ) = −∞ for all y .We shall also need in Section 5 the following technical lemma. Lemma 3.2
There exists some positive constant
C > s.t. ∂v i ∂x ( x + , y ) := lim δ ↓ v i ( x + δ, y ) − v i ( x, y ) δ ≥ C U ′ (2 x ) , ∀ x, y ∈ R + , i ∈ I d . (3.14) Proof.
Fix some x, y ≥
0, and set x = x + δ for δ >
0. For any ( ζ, c ) ∈ A i ( x, y ) withassociated cash/amount in shares ( X, Y ), notice that (˜ ζ, ˜ c ) := ( ζ, c + δ [0 , ∧ τ ] ) is admissiblefor ( x , y ). Indeed, the associated cash amount satisfies˜ X t = X t + ( x − x ) − Z t δ [0 , ∧ τ ] ( s ) ds ≥ X t ≥ , while the amount in cash ˜ Y t = Y t ≥ ζ is unchanged. Thus, (˜ ζ, ˜ c ) ∈ A i ( x , y ), andwe have v i ( x , y ) ≥ E (cid:20)Z ∞ e − ρt U ( ˜ c t ) dt (cid:21) = E (cid:20)Z ∞ e − ρt U ( c t ) dt (cid:21) + E (cid:20)Z ∧ τ e − ρt ( U ( c t + δ ) − U ( c t )) dt (cid:21) . (3.15)Now, by concavity of U : U ( c t + δ ) − U ( c t ) ≥ δU ′ ( c t + δ ), and Z ∧ τ e − ρt ( U ( c t + δ ) − U ( c t )) dt ≥ Z ∧ τ e − ρt δU ′ ( c t + δ ) dt ≥ δe − ρ (1 ∧ τ ) Z ∧ τ U ′ ( c t + δ ) dt ≥ δe − ρ (1 ∧ τ ) U ′ (2 x + δ ) Z ∧ τ { c t < x } dt. (3.16)12oreover, 2 x Z ∧ τ { c t ≥ x } dt ≤ Z ∧ τ c t dt ≤ x, since ( ζ, c ) is admissible for ( x, y ), so that Z ∧ τ { c t < x } dt ≥ (1 ∧ τ ) − (cid:18) ∧ τ (cid:19) ≥ { τ ≥ } . (3.17)By combining (3.16) and (3.17), and taking the expectation, we get E (cid:20)Z ∧ τ e − ρt ( U ( c t + δ ) − U ( c t )) dt (cid:21) ≥ δU ′ (2 x + δ ) E h e − ρ (1 ∧ τ ) { τ ≥ } i . By taking the supremum over ( ζ, c ) in (3.15), we thus obtain with the above inequality v i ( x + δ, y ) ≥ v i ( x, y ) + δU ′ (2 x + δ ) E (cid:20) e − ρ (1 ∧ τ ) { τ ≥ } (cid:21) . Finally, by choosing C = E (cid:2) e − ρ (1 ∧ τ ) 12 { τ ≥ } (cid:3) >
0, and letting δ go to 0, we obtain therequired inequality (3.14). ✷ In this section, we provide an analytic characterization of the value functions v i , i ∈ I d ,to our control problem (2.14), by relying on the dynamic programming principle, which isshown to hold and formulated as: Proposition 4.1 (Dynamic programming principle) For all ( x, y, i ) ∈ R × I d , and anystopping time τ , we have v i ( x, y ) = sup ( ζ,c ) ∈A i ( x,y ) E h Z τ e − ρt U ( c t ) dt + e − ρτ v Iτ ( X τ , Y τ ) i . (4.1) Proof.
See Appendix A. ✷ The associated dynamic programming system (also called Hamilton-Jacobi-Bellman orHJB system) for v i , i ∈ I d , is written as ρv i − b i y ∂v i ∂y − σ i y ∂ v i ∂y − ˜ U (cid:18) ∂v i ∂x (cid:19) (4.2) − X j = i q ij h v j (cid:0) x, y (1 − γ ij ) (cid:1) − v i ( x, y ) i − λ i (cid:2) ˆ v i ( x + y ) − v i ( x, y ) (cid:3) = 0 , ( x, y ) ∈ (0 , ∞ ) × R + , i ∈ I d , together with the boundary condition (3.8) on { } × R + for v i , i ∈ I d . Notice that, arguingas one does for the deduction of the HJB system above, the boundary condition (3.8) may13lso be written as: ρv i (0 , . ) − b i y ∂v i ∂y (0 , . ) − σ i y ∂ v i ∂y (0 , . ) − X j = i q ij h v j (cid:0) , y (1 − γ ij ) (cid:1) − v i (0 , y ) i − λ i (cid:2) ˆ v i ( y ) − v i (0 , y ) (cid:3) = 0 , y > , i ∈ I d . (4.3)Notice that in this boundary condition the term ˜ U (cid:18) ∂v i ∂x (cid:19) has disappeared. This implicitlycomes from the fact that, on the boundary x = 0 the only admissible consumption rate is c = 0. We will say more on this in studying the case of CRRA utility function in Section5.1.In our context, the notion of viscosity solution to the non local second-order system ( E )is defined as follows. Definition 4.1 (i) A d-tuple w = ( w i ) i ∈ I d of continuous functions on R is a viscositysupersolution (resp. subsolution) to (4.2) if ρϕ i (¯ x, ¯ y ) − b i ¯ y ∂ϕ i ∂y (¯ x, ¯ y ) − σ i ¯ y ∂ ϕ i ∂y (¯ x, ¯ y ) − ˜ U (cid:18) ∂ϕ i ∂x (¯ x, ¯ y ) (cid:19) − X j = i q ij h ϕ j (cid:0) ¯ x, ¯ y (1 − γ ij ) (cid:1) − ϕ i (¯ x, ¯ y ) i − λ i (cid:2) ˆ ϕ i (¯ x + ¯ y ) − ϕ i (¯ x, ¯ y ) (cid:3) ≥ ( resp. ≤ ) 0 , for all d-tuple ϕ = ( ϕ i ) i ∈ I d of C functions on R , and any (¯ x, ¯ y, i ) ∈ (0 , ∞ ) × R + × I d ,such that w i (¯ x, ¯ y ) = ϕ i (¯ x, ¯ y ) , and w ≥ (resp. ≤ ) ϕ on R × I d .(ii) A d-tuple w = ( w i ) i ∈ I d of continuous functions on R is a viscosity solution to (4.2) ifit is both a viscosity supersolution and subsolution to (4.2) . The main result of this section is to provide an analytic characterization of the valuefunctions in terms of viscosity solutions to the dynamic programming system.
Theorem 4.1
The value function v = ( v i ) i ∈ I d is the unique viscosity solution to (4.2) satisfying the boundary condition (3.8) , and the growth condition (3.4) . Proof.
The proof of viscosity property follows as usual from the dynamic programmingprinciple. The uniqueness and comparison result for viscosity solutions is proved by ratherstandard arguments, up to some specificities related to the non local terms and state con-straints induced by our hybrid jump-diffusion control problem. We postponed the detailsin Appendix B. ✷ In this section, we consider the case where the utility function is of CRRA type in the form: U ( x ) = x p p , x > , for some p ∈ (0 , . (5.1)14e shall exploit the homogeneity property of the CRRA utility function, and go beyond theviscosity characterization of the value function in order to prove some regularity results, andprovide an explicit characterization of the optimal control through a verification theorem.We next give a numerical analysis for computing the value functions and optimal strategies,and illustrate with some tests for measuring the impact of our illiquidity features. For any ( i, x, y ) ∈ I d × R , ( ζ, c ) ∈ A ( x, y ) with associated state process ( X, Y ), we noticefrom the dynamics (2.3)-(2.2) that for any k ≥
0, the state ( kX, kY ) is associated to thecontrol ( kζ, kc ). Thus, for k > ζ, c ) ∈ A i ( x, y ) iff ( kζ, kc ) ∈ A ( kx, kc ), and sofrom the homogeneity property of the power utility function U in (5.1), we have: v i ( kx, ky ) = k p v i ( x, y ) , ∀ ( i, x, y ) ∈ I d × R , k ∈ R + . (5.2)Let us now consider the change of variables:( x, y ) ∈ R \ { (0 , } −→ (cid:0) r = x + y, z = yx + y (cid:1) ∈ (0 , ∞ ) × [0 , . Then, from (5.2), we have v i ( x, y ) = v i ( r (1 − z ) , rz ) = r p v i (1 − z, z ), and we can separatethe value function v i into: v i ( x, y ) = U ( x + y ) ϕ i (cid:16) yx + y (cid:17) , ∀ ( i, x, y ) ∈ I d × ( R \ { (0 , } ) (5.3)where ϕ i ( z ) = p v i (1 − z, z ) is a continuous function on [0 , v i into the dynamic programming equation (4.2) and the boundary condition(4.3), and after some straightforward calculations, we see that ϕ = ( ϕ i ) i ∈ I d should solvethe system of (nonlocal) ordinary differential equations (ODEs):( ρ − pb i z + 12 p (1 − p ) σ i z ) ϕ i − (1 − p ) (cid:16) ϕ i − zp ϕ ′ i (cid:17) − p − p (5.4) − z (1 − z )( b i − z (1 − p ) σ i ) ϕ ′ i − z (1 − z ) σ i ϕ ′′ i − X j = i q ij h (1 − zγ ij ) p ϕ j (cid:16) z (1 − γ ij )1 − zγ ij (cid:17) − ϕ i ( z ) i − λ i sup π ∈ [0 , (cid:2) ϕ i ( π ) − ϕ i ( z ) (cid:3) = 0 , z ∈ [0 , , i ∈ I d , together with the boundary condition for z = 1:( ρ − pb i + 12 p (1 − p ) σ i ) ϕ i (1) − X j = i q ij (cid:2) (1 − γ ij ) p ϕ j (1) − ϕ i (1) (cid:3) − λ i sup π ∈ [0 , (cid:2) ϕ i ( π ) − ϕ i (1) (cid:3) = 0 , i ∈ I d . (5.5)The following boundary condition for z = 0, obtained formally by taking z = 0 in (5.4), ρϕ i (0) − (1 − p ) (cid:0) ϕ i (0) (cid:1) − p − p − X j = i q ij (cid:2) ϕ j (0) − ϕ i (0) (cid:3) − λ i sup π ∈ [0 , (cid:2) ϕ i ( π ) − ϕ i (0) (cid:3) = 0 , i ∈ I d , (5.6)is proved rigorously in the below Proposition.15 roposition 5.1 The d -tuple ϕ = ( ϕ i ) i ∈ I d is concave on [0 , , C on (0 , . We furtherhave lim z → zϕ ′ i ( z ) = 0 , (5.7)lim z → z ϕ ′′ i ( z ) = 0 , (5.8)lim z → (1 − z ) ϕ ′ i ( z ) = 0 , (5.9)lim z → (1 − z ) ϕ ′′ i ( z ) = 0 , (5.10)lim z → ϕ ′ i ( z ) = −∞ , (5.11) and ϕ is the unique bounded classical solution of (5.4) on (0 , , with boundary conditions (5.5) - (5.6) . Proof.
Since ϕ i ( z ) = p v i (1 − z, z ), and by concavity of v i ( ., . ) in both variables, it is clearthat ϕ i is concave on [0 , v i in Theorem 4.1, and thechange of variables (5.3), this implies that ϕ is the unique bounded viscosity solution to(5.4) on [0 , q ii = − P j = i q ij ,we observe that the system (5.4) can be written as:( ρ − q ii + λ i − pb i z + 12 p (1 − p ) σ i z ) ϕ i − z (1 − z )( b i − z (1 − p ) σ i ) ϕ ′ i − z (1 − z ) σ i ϕ ′′ i − (1 − p ) (cid:0) ϕ i − zp ϕ ′ i (cid:1) − p − p = X j = i q ij (cid:20) (1 − zγ ij ) p ϕ j (cid:16) z (1 − γ ij )1 − zγ ij (cid:17)(cid:21) + λ i sup π ∈ [0 , ϕ i ( π ) , z ∈ (0 , , i ∈ I d . (5.12)Let us fix some i ∈ I d , and an arbitrary compact [ a, b ] ⊂ (0 , ρ − q ii + λ i − pb i z + 12 p (1 − p ) σ i z ) w i − z (1 − z )( b i − z (1 − p ) σ i ) w ′ i − z (1 − z ) σ i w ′′ i − (1 − p ) (cid:0) w i − zp w ′ i (cid:1) − p − p = X j = i q ij (cid:20) (1 − zγ ij ) p ϕ j (cid:16) z (1 − γ ij )1 − zγ ij (cid:17)(cid:21) + λ i sup π ∈ [0 , ϕ i ( π ) (5.13)has a unique viscosity solution w i satisfying w i ( a ) = ϕ i ( a ), w i ( b ) = ϕ i ( b ), and that thissolution w i is twice differentiable on [ a, b ] since the second term z (1 − z ) σ i is uniformlyelliptic on [ a, b ], see [13]. Since ϕ i is a viscosity solution to (5.13) by (5.12), we deduce byuniqueness that ϕ i = w i on [ a, b ]. Since a, b are arbitrary, this means that ϕ is C on (0 , ϕ i , we have for all z ∈ (0 , ϕ i (1) − ϕ i ( z )1 − z ≤ ϕ ′ i ( z ) ≤ ϕ i ( z ) − ϕ i (0) z . Letting z → z →
1, and by continuity of ϕ i , we obtain (5.7) and (5.9).16ow letting z go to 0 in (5.4), we obtain lim z → z ϕ ′′ i ( z ) = l for some finite l ≤
0. If l < z ϕ ′′ i ( z ) ≤ l whenever z ≤ η , for some η >
0. By writing that z ( ϕ ′ i ( z ) − ϕ ′ i ( η )) = z Z zη ϕ ′′ i ( u ) du ≥ − l z Z ηz duu = l z (cid:18) η − z (cid:19) , and sending z →
0, we get lim inf z → zϕ ′ i ( z ) ≥ − l/
2, which contradicts (5.7). Thus l = 0,and the boundary condition (5.6) follows by letting z → z → z →
12 (1 − z ) ϕ ′′ i ( z ) = (cid:0) ϕ i (1) − ϕ ′ i (1 − ) (cid:1) − p − p ∈ [0 , ∞ ] . (5.9) implies that this limit is 0, and we obtain (5.10) and (5.11). ✷ Remark 5.1
From (5.3) and the above Proposition, we deduce that the value functions v i , i ∈ I d , are C on (0 , ∞ ) × (0 , ∞ ), and so are solutions to the dynamic programmingsystem (4.2) on (0 , ∞ ) × (0 , ∞ ) in classical sense.We now provide an explicit construction of the optimal investment/consumption strate-gies in feedback form in terms of the smooth solution ϕ to (5.4)-(5.6)-(5.5). We start withthe following Lemma. Lemma 5.1
For any i ∈ I d , let us define: c ∗ ( i, z ) = (cid:16) ϕ i ( z ) − zp ϕ ′ i ( z ) (cid:17) − − p when < z < ϕ i (0)) − − p when z = 00 when z = 1 ,π ∗ ( i ) ∈ arg max π ∈ [0 , ϕ i ( π ) . Then for each i ∈ I d , c ∗ ( i, . ) is continuous on [0 , , C on (0 , , and given any initialconditions ( r, z ) ∈ I d × R + × [0 , , there exists a solution ( ˆ R t , ˆ Z t ) t ≥ valued in R + × [0 , to the SDE: d ˆ R t = ˆ R t − ˆ Z t − (cid:16) b I t − dt + σ I t − dW t − γ It − ,It dN I t − ,I t t (cid:17) − ˆ R t − c ∗ ( I t − , ˆ Z t − ) dt, (5.14) d ˆ Z t = ˆ Z t − (1 − ˆ Z t − ) h(cid:0) b I t − − ˆ Z t − σ I t − (cid:1) dt + σ I t − dW t − γ It − ,It − ˆ Z t − γ It − ,It dN I t − ,I t t i + ( π ∗ ( I t − ) − ˆ Z t − ) dN t + ˆ Z t c ∗ ( I t − , ˆ Z t − ) dt. (5.15) Moreover, if r > , then ˆ R t > , a.s. for all t ≥ . Proof.
First notice that Lemma 3.2, written in terms of the variables ( r, z ), is formulatedequivalently as ϕ i ( z ) − zp ϕ ′ i ( z ) ≥ C p − (1 − z ) p − , z ∈ (0 , . c ∗ ( i, . ) is well-defined on (0 , C since ϕ is C . The continuity of c ∗ ( i, . ) at 0 and 1 comes from (5.7) and (5.11).Let us show the existence of a solution Z to the SDE (5.15). We start by the existenceof a solution for t < τ (recall that ( τ n ) is the sequence of jump times of N ). In the casewhere z = 1 (resp. z = 0), then Z t ≡ Z t ≡
0) is clearly a solution on [0 , τ ).Consider now the case where z ∈ (0 , z zc ∗ ( i, z ),and recalling that γ ij <
1, we know, adapting e.g. the result of Theorem 38, page 303 of[18], that there exists a solution to d ˆ Z t = ˆ Z t − (1 − ˆ Z t − ) h(cid:0) b I t − − ˆ Z t − σ I t − (cid:1) dt + σ I t − dW t − γ It − ,It − ˆ Z t − γ It − ,It dN I t − ,I t t i + ˆ Z t c ∗ ( I t − , ˆ Z t − ) dt, (5.16)which is valued in [0 ,
1] up to time t < τ ′ := τ ∧ (cid:16) lim ε → inf n t ≥ | ˆ Z t (1 − ˆ Z t ) ≤ ε o(cid:17) . Bynoting that ˆ Z t ≥ Z t , where Z t = z S t S z S t S + (1 − z ) , t ≥ , is the solution to (5.16) without the consumption term, and since S is locally bounded awayfrom 0, we have lim t → τ ′ Z t = 1 on { τ ′ < τ } . By extending ˆ Z t ≡ τ ′ , τ ), we obtainactually a solution on [0 , τ ). Then at τ , by taking ˆ Z τ = π ∗ ( I τ − ), we obtain a solutionto (5.15) valued in [0 ,
1] on [0 , τ ]. Next, we obtain similarly a solution to (5.15) on [ τ , τ ]starting from ˆ Z τ . Finally, since τ n ր ∞ , a.s., by pasting we obtain a solution to (5.15) for t ∈ R + .Given a solution ˆ Z to (5.15), the solution ˆ R to (5.14) starting from r at time 0 isdetermined by the stochastic exponential:ˆ R t = r · E (cid:18)Z · ˆ Z s − (cid:16) b I s − ds + σ I s − dW s − γ Is − ,Is dN I s − ,I s s (cid:17) − c ∗ ( I s − , ˆ Z s − ) dt (cid:19) t . Since − ˆ Z t − γ It − ,It > −
1, we see that R t > t ≥
0, whenever r >
0, while R ≡ r = 0. ✷ Proposition 5.2
Given some initial conditions ( i, x, y ) ∈ I d × ( R \{ (0 , } ) , let us considerthe pair of processes (ˆ ζ, ˆ c ) defined by: ˆ ζ t = ˆ R t − ( π ∗ ( I t − ) − ˆ Z t − ) (5.17)ˆ c t = ˆ R t − c ∗ ( I t − , ˆ Z t − ) , (5.18) where the functions ( c ∗ , π ∗ ) are defined in Lemma 5.1, and ( ˆ R, ˆ Z ) are solutions to (5.14) - (5.15) , starting from r = x + y , z = y/ ( x + y ) , with I starting from i . Then, (ˆ ζ, ˆ c ) is anoptimal investment/consumption strategy in A i ( x, y ) , with associated state process ( ˆ X, ˆ Y )= ( ˆ R (1 − ˆ Z ) , ˆ R ˆ Z ) , for v i ( x, y ) = U ( r ) ϕ i ( z ) . roof. For such choice of (ˆ ζ, ˆ c ), the dynamics of ( ˆ R, ˆ Z ) evolve according to (2.10)-(2.11)with a feedback control (ˆ ζ, ˆ c ), and thus correspond (via Itˆo’s formula) to a state process( ˆ X, ˆ Y ) = ( ˆ R (1 − ˆ Z ) , ˆ R ˆ Z ) governed by (2.2)-(2.3), starting from ( x, y ), and satisfying thenonbankruptcy constraint (2.5). Thus, (ˆ ζ, ˆ c ) ∈ A i ( x, y ). Moreover, since r = x + y > R >
0, and so ( ˆ X, ˆ Y ) lies in R \ { (0 , } .As in the proof of the standard verification theorem, we would like to apply Itˆo’s formulato the function e − ρt v ( ˆ X t , ˆ Y t , I t ) (denoting by v ( x, y, i ) = v i ( x, y ) = U ( x + y ) ϕ i ( y/ ( x + y ))).However this is not immediately possible since the process ( ˆ X t , ˆ Y t ) may reach the boundaryof R where the derivatives of v do not have classical sense. To overcome this problem,we approximate the function ϕ i (and so v ( x, y, i )) as follows. We define, for every ε > ϕ ε = ( ϕ ε ) i ∈ I d ∈ C ([0 , , R d ) as in the proof of Theorem 4.24 in [6], such that • ϕ εi = ϕ i on [ ε, − ε ], • ϕ εi → ϕ i uniformly on [0 ,
1] as ε → • z (1 − z )( ϕ εi ) ′ → z (1 − z ) ϕ ′ i uniformly on [0 ,
1] as ε → • z (1 − z ) ( ϕ εi ) ′′ → z (1 − z ) ϕ ′′ i uniformly on [0 ,
1] as ε → v ε ( x, y, i ) = U ( x + y ) ϕ εi ( y/ ( x + y ))calculated on the process ( ˆ X, ˆ Y , I ) between time 0 and τ n ∧ T , where τ n = inf { t ≥ X t + ˆ Y t ≥ n } : v ε ( x, y, i ) = E h e − ρ ( τ n ∧ T ) v ε ( ˆ X τ n ∧ T , ˆ Y τ n ∧ T , I τ n ∧ T )+ Z τ n ∧ T e − ρt (cid:16) ρv ε + ˆ c t ∂v ε ∂x − b I t − ˆ Y t − ∂v ε ∂y − σ I t − ˆ Y t − ∂ v ε ∂y − X j = I t − q It − j [ v ε ( ˆ X t − , ˆ Y t − (1 − γ It − j ) , j ) − v ε ( ˆ X t − , ˆ Y t − , I t − )] − λ It − (cid:2) v ε ( ˆ X t − − ˆ ζ t , ˆ Y t − + ˆ ζ t , I t − ) − v ε ( ˆ X t − , ˆ Y t − , I t − ) (cid:3)(cid:17) dt i (5.19)We denote by ˆ ζ ( i, r, z ) = r ( π ∗ ( i ) − z ), ˆ c ( i, r, z ) = rc ∗ ( i, z ), and define g ε on ( R \{ (0 , } ) × I d by ρv εi − b i y ∂v εi ∂y − σ i y ∂ v εi ∂y + ˆ c ( i, x + y, yx + y ) ∂v εi ∂x − U (cid:0) ˆ c ( i, x + y, yx + y ) (cid:1) − X j = i q ij h v εj (cid:0) x, y (1 − γ ij ) (cid:1) − v εi ( x, y ) i − λ i h v εi (cid:16) x − ˆ ζ (cid:0) i, x + y, yx + y (cid:1) , y + ˆ ζ (cid:0) i, x + y, yx + y (cid:1)(cid:17) − v εi ( x, y ) i =: g εi ( x, y ) , so that from (5.19): v ε ( i, x, y ) = E h e − ρ ( τ n ∧ T ) v ε ( ˆ X τ n ∧ T , ˆ Y τ n ∧ T , I τ n ∧ T )+ Z τ n ∧ T e − ρt ( U (ˆ c t ) + g ε ( ˆ X t , ˆ Y t , I t )) dt i . (5.20)Notice that the properties of ϕ ε imply : 19 v εi = v i on n ε ≤ yx + y ≤ − ε o , • v εi → v i uniformly on bounded subsets of R , • ˆ c ( i, x + y, yx + y ) ∂v εi ∂x → c ( i, x + y, yx + y ) ∂v i ∂x , x > , x = 0 uniformly on bounded subsetsof R , • y ∂v εi ∂y → y ∂v i ∂y , y > , y = 0 uniformly on bounded subsets of R , • y ∂ v εi ∂y → y ∂ v i ∂y , y > , y = 0 uniformly on bounded subsets of R .The details can be found in [7]. Since v is a classical solution of (4.2) on (0 , ∞ ) × (0 , ∞ ),this implies that g ε converges to 0 uniformly on bounded subsets of R when ε goes to 0.We then obtain by letting ε → v ( x, y, i ) = E h e − ρ ( τ n ∧ T ) v ( ˆ X τ n ∧ T , ˆ Y τ n ∧ T , I τ n ∧ T ) + Z τ n ∧ T e − ρt U (ˆ c t ) dt i , From the growth condition (3.4) we get E h e − ρ ( τ n ∧ T ) v ( ˆ X τ n ∧ T , ˆ Y τ n ∧ T , I τ n ∧ T ) i ≤ C E h e − ρ ( τ n ∧ T ) R pτ n ∧ T i . So, using Lemma 3.1, sending n to infinity, and then T to infinity, we getlim T →∞ lim n →∞ E h e − ρ ( τ n ∧ T ) v ( ˆ X τ n ∧ T , ˆ Y τ n ∧ T , I τ n ∧ T ) i = 0 . Applying monotone convergence theorem to the second term in the r.h.s. of (5.20), we thenobtain v i ( x, y ) = E h Z ∞ e − ρt U (ˆ c t ) dt i , which proves the optimality of (ˆ ζ, ˆ c ). ✷ We focus on the numerical resolution of the system of ODEs (5.4)-(5.6)-(5.5) satisfied by( ϕ i ) i ∈ I d , and rewritten for all i ∈ I d as:( ρ − q ii + λ i − pb i z + 12 p (1 − p ) σ i z ) ϕ i − z (1 − z )( b i − z (1 − p ) σ i ) ϕ ′ i − z (1 − z ) σ i ϕ ′′ i − (1 − p ) (cid:0) ϕ i − zp ϕ ′ i (cid:1) − p − p = X j = i q ij (cid:20) (1 − zγ ij ) p ϕ j (cid:16) z (1 − γ ij )1 − zγ ij (cid:17)(cid:21) + λ i sup π ∈ [0 , ϕ i ( π ) , z ∈ (0 , , ρ − q ii + λ i ) ϕ i (0) − (1 − p ) ϕ i (0) − p − p = X j = i q ij ϕ j (0) + λ i sup π ∈ [0 , ϕ i ( π ) , ( ρ − q ii + λ i − pb i + 12 p (1 − p ) σ i ) ϕ i (1) = X j = i q ij (1 − γ ij ) p ϕ j (1) + λ i sup π ∈ [0 , ϕ i ( π ) . We shall adopt an iterative method to solve this system of integro-ODEs : starting with ϕ = ( ϕ i ) i ∈ I d = 0, we solve ϕ n +1 = ( ϕ n +1 i ) i ∈ I d as the (classical) solution to the local ODEswhere the non local terms are calculated from ( ϕ ni ) :( ρ − q ii + λ i − pb i z + 12 p (1 − p ) σ i z ) ϕ n +1 i − z (1 − z )( b i − z (1 − p ) σ i )( ϕ n +1 i ) ′ − z (1 − z ) σ i ( ϕ n +1 i ) ′′ − (1 − p ) (cid:0) ϕ n +1 i − zp ( ϕ n +1 i ) ′ (cid:1) − p − p = X j = i q ij (cid:20) (1 − zγ ij ) p ϕ nj (cid:16) z (1 − γ ij )1 − zγ ij (cid:17)(cid:21) + λ i sup π ∈ [0 , ϕ ni ( π ) , with boundary conditions( ρ − q ii + λ i ) ϕ n +1 i (0) − (1 − p ) ϕ n +1 i (0) − p − p = X j = i q ij ϕ nj (0) + λ i sup π ∈ [0 , ϕ ni ( π ) , ( ρ − q ii + λ i − pb i + 12 p (1 − p ) σ i ) ϕ n +1 i (1) = X j = i q ij (1 − γ ij ) p ϕ nj (1) + λ i sup π ∈ [0 , ϕ ni ( π ) . Let us denote by: v ni ( x, y ) = ( U ( x + y ) ϕ ni (cid:16) yx + y (cid:17) , for ( i, x, y ) ∈ I d × ( R \ { (0 , } )0 , for i ∈ I d , ( x, y ) = (0 , . A straightforward calculation shows that v n = ( v ni ) i ∈ I d are solutions to the iterative localPDEs: ( ρ − q ii + λ i ) v n +1 i − b i y ∂v n +1 i ∂y − σ i y ∂ v n +1 i ∂y − ˜ U (cid:0) ∂v n +1 i ∂x (cid:1) = X j = i q ij v nj (cid:0) x, y (1 − γ ij ) (cid:1) + λ i ˆ v ni ( x + y ) , ( x, y ) ∈ (0 , ∞ ) × R + , i ∈ I d , (5.21)together with the boundary condition (3.8) on { } × (0 , ∞ ) for v i , i ∈ I d :( ρ − q ii + λ i ) v n +1 i (0 , . ) − b i y ∂v n +1 i ∂y (0 , . ) − σ i y ∂ v n +1 i ∂y (0 , . )= X j = i q ij v nj (cid:0) , y (1 − γ ij ) (cid:1) + λ i ˆ v ni ( y ) , y > , i ∈ I d . (5.22)We then have the stochastic control representation for v n (and so for ϕ n ). Proposition 5.3
For all n ≥ , we have v ni ( x, y ) = sup ( ζ,c ) ∈A i ( x,y ) E h Z θ n e − ρt U ( c t ) dt i , ( i, x, y ) ∈ I d × R , (5.23)21 here the sequence of random times ( θ n ) n ≥ are defined by induction from θ = 0 , and: θ n +1 = inf n t > θ n : ∆ N t = 0 or ∆ N I t − ,I t t = 0 o , i.e. θ n is the n -th time where we have either a change of regime or a trading time. Proof.
Denoting by w ni ( x, y ) the r.h.s. of (5.23), we need to show that w ni = v ni . First(with a similar proof to Proposition 4.1) we have the following Dynamic ProgrammingPrinciple for the w n : for each finite stopping time τ , w n +1 i ( x, y ) = sup ( ζ,c ) ∈A i ( x,y ) E (cid:20)Z τ ∧ θ e − ρt U ( c t ) dt + { τ ≥ θ } e − ρθ w n Iθ ( X θ , Y θ )+ { τ<θ } e − ρτ w n +1 Iτ ( X τ , Y τ ) i (5.24)The only difference with the statement of Proposition 4.1 is the fact that when τ ≥ θ , wesubstitute w n +1 with w n since there are only n stopping times remaining before consump-tion is stopped due to the finiteness of the horizon in the definition of w n .By using (5.24), we can show as in Theorem 4.1 that w n is the unique viscosity solutionto (5.21), satisfying boundary condition (5.22) and growth condition (3.4) (it is actuallyeasier since there are only local terms in this case). Since we already know that v n is sucha solution, it follows that w n = v n . ✷ As a consequence, we obtain the following convergence result for the sequence ( v n ) n . Proposition 5.4
The sequence ( v n ) n converges increasingly to v , and there exists somepositive constants C and δ < s.t. ≤ v i − v ni ≤ Cδ n ( x + y ) p , ∀ ( i, x, y ) ∈ I d × R . (5.25) Proof.
First let us show that δ := sup ( c, ζ ) ∈ A i ( x, y ) { ( x, y ) ∈ R : x + y = 1 } E h e − ρθ R pθ i < . (5.26)By writing that e − ρt R pt = D t L t , where ( L t ) t = ( e − k ( p ) t R pt ) t is a nonnegative supermartingaleby Lemma 3.1, and ( D t ) t = ( e − ( ρ − k ( p )) t ) t is a decreasing process, we see that ( e − ρt R pt ) t isalso a nonnegative supermartingale for all ( ζ, c ) ∈ A i ( x, y ), and so: E h e − ρθ R pθ i ≤ E h e − ρ ( θ ∧ R pθ ∧ i = E h e − ( ρ − k ( p ))( θ ∧ e − k ( p )( θ ∧ R pθ ∧ i . Now, since e − ( ρ − k ( p ))( θ ∧ < E h e − k ( p )( θ ∧ R pθ ∧ i ≤
1, for all ( ζ, c ) ∈ A i ( x, y ) with x + y = 1 (recall the supermartingale property of ( e − k ( p ) t R pt ) t ), and by using also theuniform integrability of the family (cid:16) e − k ( p )( θ ∧ R pθ ∧ (cid:17) c,ζ from Lemma 3.1, we obtain therelation (5.26). 22he nondecreasing property of the sequence ( v ni ) n follows immediately from the rep-resentation (5.23), and we have: v ni ≤ v n +1 i ≤ v for all n ≥
0. Moreover, the dynamicprogramming principle (5.24) applied to τ = θ gives v n +1 i ( x, y ) = sup ( ζ,c ) ∈A i ( x,y ) E (cid:20)Z θ e − ρt U ( c t ) dt + e − ρθ v n Iθ ( X θ , Y θ ) (cid:21) (5.27)Let us show (5.25) by induction on n . The case n = 0 is simply the growth condition (3.4)since v = 0. Assume now that (5.25) holds true at step n . From the dynamic programmingprinciple (4.1) and (5.27) for v and v n +1 , we then have: v n +1 i ( x, y ) ≥ v i ( x, y ) − sup ( ζ,c ) ∈A i ( x,y ) E h e − ρθ ( v I θ − v nI θ ) (cid:0) X θ , Y θ (cid:1)i ≥ v i ( x, y ) − sup ( ζ,c ) ∈A i ( x,y ) E h e − ρθ Cδ n R pθ i = v i ( x, y ) − Cδ n +1 ( x + y ) p , by definition of δ . This proves the required inequality at step n + 1, and ends the proof. ✷ In the next section, we solve the local ODEs for ϕ n with Newton’s method by a finite-difference scheme (see section 3.2 in [12]). In this paragraph, we consider the case where there is only one regime ( d = 1). In this case,our model is similar to the one studied in [17], with the key difference that in their model,the investor only observes the stock price at the trading times, so that the consumptionprocess is piecewise-deterministic. We want to compare our results with [17], and take thesame values for our parameters : p = 0 . ρ = 0 . b = 0 . σ = 1.Let us recall from [17] the reason behind this choice of parameters (which are not veryrealistic for a typical financial asset) : to allow meaningful comparison to the Merton (liquid)problem, the optimal Merton investment proportion should be in [0 , v M should be significantly higher than the value function v correspondingto the consumption problem without trading. These two constraints correspond to a highrisk-return market. In the next subsection (multi-regime case), the choice of parameterswill also follow from this reasoning.Defining the cost of liquidity P ( x ) as the extra amount needed to have the same utilityas in the Merton case : v ( x + P ( x )) = v M ( x ), we compare the results in our model and inthe discrete observation model in [17]. The results in Table 1 indicate that the impact ofthe lack of continuous observation is quite large, and more important than the constraintof only being able to trade at discrete times.In Figure 1 we have plotted the graph of ϕ ( z ) (actually ϕ n ( z ) for n large) and of theoptimal consumption rate c ∗ ( z ) for different values of λ . Notice how the value function,the optimal proportion and the optimal consumption rate converge to the Merton valueswhen λ increases. 23 Discrete observation Continuous observation1 0.275 0.1535 0.121 0.01640 0.054 0.001Table 1: Cost of liquidity P (1) as a function of λ .We observe that the optimal investment proportion is increasing with λ . When z isclose to 1 i.e. the cash proportion in the portfolio is small, the investor faces the riskof “having nothing more to consume” and the further away the next trading date is thesmaller the consumption rate should be, i.e. c ∗ is increasing in λ . When z is far from 1it is the opposite : when λ is smaller the investor will not be able to invest optimally tomaximize future income and should consume more quickly.1.651.71.751.81.851.91.9522.05 ϕ ( z ) z c ∗ ( z ) zλ = 1 λ = 3 λ = 5 λ = 10Merton z ∗ Figure 1:
Value function ϕ ( z ) (left) and optimal consumption rate c ∗ ( z )(right) for different values of λ In this paragraph, we consider the case of d = 2 regimes. We assume that the asset priceis continuous, i.e. γ = γ = 0. In this case, the value functions and optimal strategiesfor the continuous trading (Merton) problem are explicit, see [20]: v i,M ( r ) = r p p ϕ i,M where( ϕ i,M ) i =1 , is the only positive solution to the equations: (cid:16) ρ − q ii − b i p σ i (1 − p ) (cid:17) ϕ i,M − (1 − p ) ϕ − p − p i,M = q ij ϕ j,M , i, j ∈ { , } , i = j. The optimal proportion invested in the asset π ∗ i,M = b i (1 − p ) σ i is the same as in the single-regime case, and the optimal consumption rate is c ∗ i,M = ( ϕ i,M ) − p . We take for values of24he parameters p = 0 . ,q = q = 1 ,b = b = 0 . ,σ = 1 , σ = 2 , i.e. the difference between the two market regimes is the volatility of the asset. In Figure2, we plot the value function and optimal consumption for each of the two regimes in thismarket, for various values of the liquidity parameters ( λ , λ ). As in the single-regime case,when the liquidity increases, ϕ and c ∗ converge to the Merton value.Note that while in the single regime-case the optimal investment proportion is usuallyincreasing with the liquidity parameter λ , in the presence of several regimes there does notappear to be a simple similar effect, as can be seen for instance in the upper-right panel ofFigure 2.To quantify the impact of regime-switching on the investor, it is also interesting tocompare the cost of liquidity with the single-regime case, see Tables 2 and 3. We observethat, for equivalent trading intensity, the cost of liquidity is higher in the regime-switchingcase. This is economically intuitive : in each regime the optimal investment proportion isdifferent, so that the investor needs to rebalance his portfolio more often (at every changeof regime). In this paper we proposed a simple model of an illiquid market with regime-switching, inwhich the investor may only trade at discrete times corresponding to the arrival times ofa Cox process. In this context, we studied an investment/consumption problem over aninfinite horizon. In the general case, we proved that the value function for this problem ischaracterized as the unique viscosity solution to the HJB equation (which is a system ofintegro-PDEs). In the case of power utility, we proved the regularity of our value functionand we were able to characterize the optimal policies. Finally we have presented somenumerical results in this special case.With some straightforward modifications, our viscosity results could be extended tomore general regime-switching diffusions (assuming e.g. Lipschitz coefficients). However,the dimension reduction in the case of power utility which allowed us to prove regularity,and made the numerical resolution easier, is specific to our (regime-switching) Black Scholesdynamics. 25 λ , λ ) P (1) P (1)(1,1) 0.257 0.224(5,5) 0.112 0.103(10,10) 0.069 0.064Table 2: Cost of liquidity P i (1) as afunction of ( λ , λ ). λ P (1) P (1)1 0.153 0.0875 0.015 0.04210 0.004 0.024Table 3: Cost of liquidity P i (1) for thesingle-regime case. ϕ ( z ) z ϕ ( z ) z c ∗ ( z ) z c ∗ ( z ) z ( λ , λ ) = (1 , λ , λ ) = (10 ,
1) ( λ , λ ) = (1 , λ , λ ) = (10 , ϕ i and c ∗ i for different values of ( λ , λ )26 ppendix A: Dynamic Programming Principle We introduce the weak formulation of the control problem.
Definition A.1
Given ( i, x, y ) ∈ I d × R + × R + , a control U is a 9-tuple (Ω , F , P , F = ( F t ) t ≥ , W, I, N, c, ζ ) , where :1. (Ω , F , P , F ) is a filtered probability space satisfying the usual conditions.2. I is a Markov chain with space state I d and generator Q , I = i a.s., N is a Coxprocess with intensity ( λ I t ) , and W is an F -Brownian motion independent of ( I, N ) .3. F t = σ ( W s , I s , N s ; s ≤ t ) ∨ N , where N is the collection of all P -null sets of F .4. ( c t ) is F -progressively measurable, ( ζ t ) is F -predictable.We say that U is admissible , (writing U ∈ A wi ( x, y ) ), if the solution ( X, Y ) to (2.3) - (2.2) with X = x, Y = y , satisfies X t ≥ , Y t ≥ a.s. Given
U ∈ A wi ( x, y ), define J ( U ) = E (cid:2)R ∞ e − ρs U ( c s ) ds (cid:3) , and the value function v i ( x, y ) = sup U∈A wi ( x,y ) J ( U ) . Proposition A.1
For every finite stopping time τ and initial conditions i, x, y , v i ( x, y ) = sup ( ζ,c ) ∈A wi ( x,y ) E (cid:20)Z τ e − ρt U ( c t ) dt + e − ρτ v I τ ( X τ , Y τ ) (cid:21) . (A.1)Before proving this proposition we state some technical lemmas. Lemma A.1
Given (Ω , F , P , F = ( F t ) , W, I, N ) satisfying the conditions of Definition A.1,define F = ( F t ) t ≥ , where F t = σ ( W s , I s , N s ; s ≤ t ) . Then if ( c t ) is F -progressively mea-surable (resp. predictable), there exists c F -progressively measurable (resp. predictable)such that c = c d P ⊗ dt a.e.. Proof.
We only give a sketch as the arguments is standard. We first use Lemma 3.2.4 page133 in [11] to find, for each n ∈ N , an approximating F t -simple process c n converging to c in the L ( dt ⊗ d P ) norm. Then, using Lemma 1.25 page 13 in [10], we can change every c n on a null-set and find a sequence of F t, s -simple process c n ( t ) that again converges to c in the L ( dt ⊗ d P ) norm. We now extract a subsequence (denoted again by c n ) such that c n → c a.e. and we define c := lim inf n → + ∞ c n . This is F t, s -progressively measurable and c = c , dt ⊗ d P a.e. on [0 , + ∞ ) × Ω. This concludes the proof. ✷ Remark A.1
With the notations of the previous lemma, it is easy to check that ( X c ′ ,ζ ′ , Y c ′ ,ζ ′ ) ∼ ( X c,ζ , Y c,ζ ) in law. Hence without loss of generality we can assume that c is F -progressively measurable and ζ is F -predictable.27efine W as the space of continuous functions on R + , I the space of cadlag I d -valuedfunctions, N the space of nondecreasing cadlag N -valued functions. On W × I × N , definethe filtration ( B t ) t ≥ , where B t is the smallest σ -algebra making the coordinate mappingsfor s ≤ t measurable, and define B t + = T s>t B s . Lemma A.2 If c is F -progressively measurable (resp. F -predictable), there exists a B t + -progressively measurable (resp. B t -predictable) process f c : R + × W × I × N → R , suchthat c t = f c ( t, W . ∧ t , I . ∧ t , N . ∧ t ) , for P − a.e ω, for all t ∈ R + Proof.
For the progressively measurable part one can see e.g. Theorem 2.10 in [22]. For c predictable, notice that this is true if c = X ( t,s ] , where X is F t -measurable, and concludewith a monotone class argument. ✷ Proof of Proposition A.1.
Let V i ( x, y ) be the right hand side of (A.1). Step 1. v i ( x, y ) ≤ V i ( x, y ): Take U ∈ A wi ( x, y ). Then E (cid:20)Z ∞ e − ρt U ( c t ) dt |F τ (cid:21) = Z τ e − ρt U ( c t ) dt + e − ρτ E (cid:20)Z ∞ e − ρs U ( c τ + s ) ds |F τ (cid:21) . (A.2)By Remark A.1, w.l.o.g. we can assume that c is F -progressively measurable (resp. ζ F -predictable). For ω ∈ Ω, define the shifted control ˜ U ω = (Ω , ˜ F τ , P ω , ˜ F τt , ˜ W , ˜ I, ˜ N , ˜ c, ˜ ζ ),where : • P ω = P ( . |F τ )( ω ) • ˜ W t = W τ + t − W τ • ˜ I t = I τ + t • N ′ t = N τ + t − N τ • ˜ F τ is the augmentation of F by the P ω -null sets, and ˜ F τt is the augmented filtrationgenerated by ( ˜ W , ˜ I, ˜ N ). • ˜ c t = c t + τ , ˜ ζ t = ζ t + τ Then we can check that for almost all ω , ˜ U ω satisfies the conditions of Definition A.1(with initial conditions ( I τ ( ω ) , X τ ( ω ) , Y τ ( ω ))) : 2. comes from the independence of W and ( I, N ) and the strong Markov property, and 4. is verified because for almost all ω F t + τ ⊂ ˜ F τt .Moreover, there is a modification ( X ′ , Y ′ ) of ( X, Y ) s.t. ( X ′ τ + t , Y ′ τ + t ) is ˜ F τ -adapted,and a solution of (2.3)-(2.2) for ( ˜ W , ˜ I, ˜ N ). Hence ˜ U ω ∈ A wI τ ( ω ) ( X τ ( ω ) , Y τ ( ω )), and E (cid:20)Z ∞ e − ρs U ( c τ + s ) ds |F τ (cid:21) ( ω ) = J ( ˜ U ω ) ≤ v I τ ( X τ , Y τ )( ω ) . ω in (A.2), E (cid:20)Z ∞ e − ρt U ( c t ) dt (cid:21) ≤ E (cid:20)Z τ e − ρt U ( c t ) dt + e − ρτ v I τ ( X τ , Y τ ) (cid:21) , and taking the supremum over U , we obtain v i ( x, y ) ≤ V i ( x, y ). Step 2. v i ( x, y ) ≥ V i ( x, y ): Recall that in the proof of Proposition 3.2 we only needed theDPP to prove the continuity of v i up to the boundary. Hence we know a priori that v i iscontinuous on Int ( R ), and that the restriction of v i to the boundary is continuous. Onecan then find a countable sequence ( U k ) k ≥ s.t.(i) ( U k ) k is a partition of R ,(ii) ∀ ( x, y ) , ( x ′ , y ′ ) ∈ U k , ∀ i, | v i ( x, y ) − v i ( x ′ , y ′ ) | ≤ ε ,(iii) U k contains its bottom-left corner ( x k , y k ) = (cid:0) min ( x,y ) ∈ U k x, min ( x,y ) ∈ U k y (cid:1) .Indeed, we can construct such a partition in the following way: v i is continuous on theboundary so we can partition each of the boundary lines into a countable number of seg-ments verifying (ii) and (iii). Then in the interior we have first a partition in “squaredrings” : Int ( R ) = ∪ n ≥ K n , where K n = [1 / ( n + 1) , n + 1] \ [1 /n, n ] . Since v i is continu-ous on the interior, we can partition each K n into a finite number of squares verifying (ii)and (iii). By taking the union of the line segments and the squares for each K n , we obtaina sequence ( U k ) satisfying (i)-(iii).Notice that (iii) implies the inclusion A i ( x k , y k ) ⊂ A i ( x, y ), for all ( x, y ) ∈ U k . Foreach k , take U i,k = (Ω i,k , F i,k , P i,k , F i,k , W i,k , I i,k , N i,k , c i,k , ζ i,k ) ε -optimal for ( i, x k , y k ),and f i,kc , f i,kζ associated to ( c i,k , ζ i,k ) by Lemma A.2. Then for each ( c, ζ ) ∈ A i ( x, y ), let usdefine ˜ c, ˜ ζ by :˜ c t = ( c t when t < τf i,kc ( t − τ, ˜ W ( . ∧ ( t − τ )) , ˜ I ( . ∧ ( t − τ )) , ˜ N ( . ∧ ( t − τ ))) when t ≥ τ, I τ = i, ( X τ , Y τ ) ∈ U k . Then ˜ c (resp. ˜ ζ ) is F - progressively measurable (resp. predictable). Furthermore, for almostall ω , with i = I τ ( ω ) and ( X τ , Y τ )( ω ) ∈ U k , L P ω ( ˜ W , ˜ I, ˜ N , (˜ c t + τ ) , (˜ ζ t + τ )) = L P i,k ( W i,k , I i,k , N i,k , c i,k , ζ i,k ) , and since A i ( x k , y k ) ⊂ A I τ ( ω ) ( X τ ( ω ) , Y τ ( ω )), this implies X ˜ c, ˜ ζt , Y ˜ c, ˜ ζt ≥ c, ˜ ζ ) ∈A i ( x, y ). We also have E (cid:20)Z ∞ e − ρs U (˜ c τ + s ) ds |F τ (cid:21) ( ω ) = E i,k (cid:20)Z ∞ e − ρs U ( c i,ks ) ds (cid:21) ≥ v i ( x k , y k ) − ε ≥ v I τ ( X τ , Y τ )( ω ) − ε. By taking expectation in (A.2), we have E (cid:20)Z ∞ e − ρt U (˜ c t ) dt (cid:21) ≥ E (cid:20)Z τ e − ρt U ( c t ) dt + e − ρτ v I τ ( X τ , Y τ ) (cid:21) − ε. U , and letting ε go to 0, we obtain v i ( x, y ) ≥ V i ( x, y ). ✷ Remark A.2
Actually the weak value function is equal to the value function defined in(2.14) for any (Ω , F , P , F , W, I, N ) satisfying (1)-(3) in Definition A.1. Indeed, given any U ′ = (Ω ′ , F ′ , P ′ , F ′ , W ′ , I ′ , N ′ ) ∈ A wi ( x, y ), letting f c ′ and f ζ ′ being associated to c ′ and ζ ′ byLemmas A.1 and A.2, and defining (almost surely) c t = f c ′ ( t, W, I, N ), ζ t = f ζ ′ ( t, W, I, N ),by the same arguments as in the Proof of Proposition A.1, U := (Ω , F , P , F , W, I, N, c, ζ ) ∈A wi ( x, y ), and J ( U ) = J ( U ′ ). Hencesup U ′ ∈A wi ( x,y ) J ( U ′ ) = sup ( c,ζ ) ∈A i ( x,y ) E (cid:20)Z ∞ e − ρs U ( c s ) ds (cid:21) . Appendix B: Viscosity characterization
We first prove the viscosity property of the value function to its dynamic programmingsystem (4.2), written as: F i ( x, y, v i ( x, y ) , Dv i ( x, y ) , D v i ( x, y )) + G i ( x, y, v ) = 0 , ( x, y ) ∈ (0 , ∞ ) × R + , for any i ∈ I d , where F i is the local operator defined by: F i ( x, y, u, p, A ) = ρu − b i yp − σ i y a − ˜ U ( p )for ( x, y ) ∈ (0 , ∞ ) × R + , u ∈ R , p = ( p p ) ∈ R , A = a a a a ! ∈ S (the set ofsymmetric 2 × G i is the nonlocal operator defined by: G i ( x, y, w ) = − X j = i q ij (cid:2) w j ( x, y (1 − γ ij )) − w i ( x, y ) (cid:3) − λ i (cid:2) ˆ w i ( x + y ) − w i ( x, y ) (cid:3) for w = ( w i ) i ∈ I d d -tuple of continuous functions on R . Proposition B.1
The value function v = ( v i ) i ∈ I d is a viscosity solution of (E). Proof.
Viscosity supersolution : Let ( i, ¯ x, ¯ y ) ∈ I d × (0 , ∞ ) × R + , ϕ = ( ϕ i ) i ∈ I d , C testfunctions s.t. v i (¯ x, ¯ y ) = ϕ i (¯ x, ¯ y ), and v ≥ ϕ . Take some arbitrary e ∈ ( − ¯ y, ¯ x ), and c ∈ R + .Since ¯ x >
0, there exists a strictly positive stopping time τ > ζ, ¯ c ) defined by: ¯ ζ t = e t ≤ τ , ¯ c t = c t ≤ τ , t ≥ , (B.1)with associated state process ( ¯ X, ¯ Y , I ) starting from ( x, y, i ) at time 0, satisfies ¯ X t ≥ Y t ≥
0, for all t . Thus, (¯ ζ, ¯ c ) ∈ A i ( x, y ). Let V be a compact neighbourhood of ( x, y, i )30n (0 , ∞ ) × R + × I d , and consider the sequence of stopping time: θ n = θ ∧ h n , where θ = inf (cid:8) t ≥ X t , ¯ Y t , I t ) / ∈ V (cid:9) , and ( h n ) is a strictly positive sequence converging tozero. From the dynamic programming principle (4.1), and by applying Itˆo’s formula to e − ρt ϕ ( ¯ X t , ¯ Y t , I t ) between 0 and θ n , we get: ϕ (¯ x, ¯ y, i ) = v ( x, y, i ) ≥ E (cid:20)Z θ n e − ρt U (¯ c t ) dt + e − ρθ n v ( ¯ X θ n , ¯ Y θ n , I θ n ) (cid:21) ≥ E h Z θ n e − ρt U (¯ c t ) dt + e − ρθ n ϕ ( ¯ X θ n , ¯ Y θ n , I θ n ) i = ϕ (¯ x, ¯ y, i ) + E h Z θ n e − ρt (cid:16) U (¯ c t ) − ρϕ − ¯ c t ∂ϕ∂x + b I t − ¯ Y t − ∂ϕ∂y + 12 σ I t − ¯ Y t − ∂ ϕ∂y + X j = I t − q It − j [ ϕ ( ¯ X t − , ¯ Y t − (1 − γ It − j ) , j ) − ϕ ( ¯ X t − , ¯ Y t − , I t − )]+ λ It − (cid:2) ϕ ( ¯ X t − − ¯ ζ t , ¯ Y t − + ¯ ζ t , I t − ) − ϕ ( ¯ X t − , ¯ Y t − , I t − ) (cid:3)(cid:17) dt i , and so E h h n Z θ n e − ρt (cid:16) ρϕ − U (¯ c t ) + ¯ c t ∂ϕ∂x − b I t − ¯ Y t − ∂ϕ∂y − σ I t − ¯ Y t − ∂ ϕ∂y − X j = I t − q It − j [ ϕ ( ¯ X t − , ¯ Y t − (1 − γ It − j ) , j ) − ϕ ( ¯ X t − , ¯ Y t − , I t − )] − λ It − (cid:2) ϕ ( ¯ X t − − ¯ ζ t , ¯ Y t − + ¯ ζ t , I t − ) − ϕ ( ¯ X t − , ¯ Y t − , I t − ) (cid:3)(cid:17) dt i ≥ n large enough, θ ≥ h n , i.e. θ n = h n , so that by using also(B.1) 1 h n Z θ n e − ρt (cid:16) ρϕ − U (¯ c t ) + ¯ c t ∂ϕ∂x − b I t − ¯ Y t − ∂ϕ∂y − σ I t − ¯ Y t − ∂ ϕ∂y − X j = I t − q It − j [ ϕ ( ¯ X t − , ¯ Y t − (1 − γ It − j ) , j ) − ϕ ( ¯ X t − , ¯ Y t − , I t − )] − λ It − (cid:2) ϕ ( ¯ X t − − ¯ ζ t , ¯ Y t − + ¯ ζ t , I t − ) − ϕ ( ¯ X t − , ¯ Y t − , I t − ) (cid:3)(cid:17) dt i −→ ρϕ i (¯ x, ¯ y ) − U ( c ) + c ∂ϕ i ∂x (¯ x, ¯ y ) − b i ¯ y ∂ϕ i ∂y (¯ x, ¯ y ) − σ i ¯ y ∂ ϕ i ∂y (¯ x, ¯ y ) − X j = i q ij [ ϕ j (¯ x, ¯ y (1 − γ ij )) − ϕ i (¯ x, ¯ y )] − λ i [ ϕ i (¯ x − e, ¯ y + e ) − ϕ i (¯ x, ¯ y )] , a.s. when n goes to infinity. Moreover, since the integrand of the Lebesgue integral term in(B.2) is bounded for t ≤ θ , one can apply the dominated convergence theorem in (B.2),which gives: ρϕ i (¯ x, ¯ y ) − U ( c ) + c ∂ϕ i ∂x (¯ x, ¯ y ) − b i ¯ y ∂ϕ i ∂y (¯ x, ¯ y ) − σ i ¯ y ∂ ϕ i ∂y (¯ x, ¯ y ) − X j = i q ij [ ϕ j (¯ x, ¯ y (1 − γ ij )) − ϕ i (¯ x, ¯ y )] − λ i [ ϕ i (¯ x − e, ¯ y + e ) − ϕ i (¯ x, ¯ y )] ≥ . c and e are arbitrary, we obtain the required viscosity supersolution inequality bytaking the supremum over c ∈ R + and e ∈ ( − ¯ y, ¯ x ). Viscosity subsolution : Let (¯ i, ¯ x, ¯ y ) ∈ I d × (0 , ∞ ) × R + , ϕ = ( ϕ i ) i ∈ I d , C test functions s.t. v (¯ x, ¯ y, ¯ i ) = ϕ (¯ x, ¯ y, ¯ i ), and v ≤ ϕ . We can also assume w.l.o.g. that v < ϕ outside (¯ x, ¯ y, ¯ i ).We argue by contradiction by assuming that ρϕ ¯ i (¯ x, ¯ y ) − b ¯ i ¯ y ∂ϕ ¯ i ∂y (¯ x, ¯ y ) − σ i ¯ y ∂ ϕ ¯ i ∂y (¯ x, ¯ y ) − ˜ U (cid:16) ∂ϕ ¯ i ∂x (¯ x, ¯ y ) (cid:17) − X j =¯ i q ¯ ij [ ϕ j (¯ x, ¯ y (1 − γ ¯ ij )) − ϕ ¯ i (¯ x, ¯ y )] − λ ¯ i [ ˆ ϕ ¯ i (¯ x + ¯ y ) − ϕ ¯ i (¯ x, ¯ y )] > . By continuity of ϕ , and of its derivatives, there exist some compact neighbourhood ¯ V of(¯ x, ¯ y, ¯ i ) in (0 , ∞ ) × R + × I d , and ε >
0, such that ρϕ i ( x, y ) − b i y ∂ϕ i ∂y ( x, y ) − σ i y ∂ ϕ i ∂y ( x, y ) − ˜ U (cid:16) ∂ϕ i ∂x ( x, y ) (cid:17) (B.3) − X j = i q ij [ ϕ j ( x, y (1 − γ ij )) − ϕ i ( x, y )] − λ i [ ˆ ϕ i ( x + y ) − ϕ i ( x, y )] ≥ ε, ∀ ( x, y, i ) ∈ ¯ V . Since v < ϕ outside (¯ x, ¯ y, ¯ i ), there exists some δ > v < ϕ − δ outside of ¯ V . We canalso assume that ε ≤ δρ . By the DPP (4.1), there exists ( ζ, c ) ∈ A ¯ i (¯ x, ¯ y ) s.t. v (¯ x, ¯ y, ¯ i ) − ε − e − ρ ρ ≤ E (cid:20)Z θ ∧ e − ρt U ( c t ) dt + e − ρ ( θ ∧ v ( X θ ∧ , Y θ ∧ , I θ ∧ ) (cid:21) , where ( X, Y, I ) is controlled by ( ζ, c ), and we take θ = inf (cid:8) t ≥ X t , Y t , I t ) / ∈ ¯ V (cid:9) . Wethen get: ϕ (¯ x, ¯ y, ¯ i ) − ε − e − ρ ρ = v (¯ x, ¯ y, ¯ i ) − ε − e − ρ ρ ≤ E (cid:20)Z θ ∧ e − ρt U ( c t ) dt + e − ρ ( θ ∧ ϕ ( X θ ∧ , Y θ ∧ , I θ ∧ ) − e − ρθ δ { θ< } (cid:21) = ϕ (¯ x, ¯ y, ¯ i ) + E h Z θ ∧ e − ρt (cid:16) U ( c t ) − ρϕ − c t ∂ϕ∂x + b I t − Y t − ∂ϕ∂y + 12 σ I t − Y t − ∂ ϕ∂y + X j = I t − q It − j [ ϕ ( X t − , Y t − (1 − γ It − j ) , j ) − ϕ ( X t − , Y t − , I t − )]+ λ It − (cid:2) ϕ ( X t − − ζ t , Y t − + ζ t , I t − ) − ϕ ( X t − , Y t − , I t − ) (cid:3)(cid:17) dt − e − ρθ δ { θ< } i ≤ ϕ (¯ x, ¯ y, ¯ i ) + E (cid:20)Z θ ∧ − εe − ρt dt − e − ρθ δ { θ< } (cid:21) − ε − e − ρ ρ ≤ E (cid:20)Z θ ∧ − εe − ρt dt − e − ρθ δ { θ< } (cid:21) = E (cid:20) − ερ + ερ e − ρ ( θ ∧ − e − ρθ δ { θ< } (cid:21) ≤ − ερ (1 − e − ρ ) , since ε/ρ ≤ δ , and we get the required contradiction. ✷ Let us now prove comparison principle for our dynamic programming system. As usual,it is convenient to formulate an equivalent definition for viscosity solutions to (4.2) in termsof semi-jets. We shall use the notation X = ( x, y ) for R + × R + -valued vectors. Given w = ( w i ) i ∈ I d a d-tuple of continuous functions on R , the second-order superjet of w i at X ∈ R is defined by: P , + w i ( X ) = n ( p, A ) ∈ R × S s.t. w i ( X ′ ) ≤ w i ( X ) + (cid:10) p, X ′ − X (cid:11) + 12 (cid:10) A ( X ′ − X ) , X ′ − X (cid:11) + o (cid:16)(cid:12)(cid:12) X ′ − X (cid:12)(cid:12) (cid:17) as X ′ → X o , and its closure P , + w i ( X ) as the set of elements ( p, A ) ∈ R × S for which there ex-ists a sequence ( X m , p m , A m ) m of R × P , + w i ( X m ) satisfying ( X m , p m , A m ) → ( X, p, A ).We also define the second-order subjet P , − w i ( X ) = −P , + ( − w i )( X ), and P , − w i ( X ) = −P , + ( − w i )( X ). By standard arguments (see e.g. [2] for equations with nonlocal terms),one has an equivalent definition of viscosity solutions in terms of semijets:A d -tuple w = ( w i ) i ∈ I d of continuous functions on R is a viscosity supersolution (resp.subsolution) of (4.2) if and only if for all ( i, x, y ) ∈ I d × (0 , ∞ ) × R + , and all ( p, A ) ∈P , − w i ( x, y ) (resp. P , + w i ( x, y )): F i ( x, y, w i ( x, y ) , p, A ) + G i ( x, y, w ) ≥ , ( resp. ≤ . We then prove the following comparison theorem.
Theorem B.1
Let V = ( V i ) i ∈ I d (resp. W = ( W i ) i ∈ I d ) be a viscosity subsolution (resp.supersolution) of (4.2) , satisfying the growth condition (3.4) , and the boundary conditions V i (0 , ≤ V i (0 , y ) ≤ E i (cid:20) ˆ V Iiτ (cid:0) y S τ S (cid:1)(cid:21) , ∀ y > , (B.5) (resp. ≥ for W ). Then V ≤ W . Proof.
Step 1 : Take p ′ > p such that k ( p ′ ) < ρ , and define ψ i ( x, y ) = ( x + y ) p ′ , i ∈ I d .Let us check that W n = W + n ψ is still a supersolution of (E). Notice that P , − W ni =33 , − W i + n ( Dψ i , D ψ i ), and we have for all ( p, A ) ∈ P , − W i ( x, y ): F i (cid:0) x, y, W ni ( x, y ) , p + 1 n Dψ i , A + 1 n D ψ i (cid:1) + G i ( x, y, W n )= F i (cid:0) x, y, W i ( x, y ) , p, A ) + G i ( x, y, W )+ 1 n ( x + y ) p ′ (cid:16) ρ − p ′ b i yx + y + p ′ (1 − p ′ ) σ i (cid:18) yx + y (cid:19) − X j = i q ij ((1 − yx + y γ ij ) p ′ − (cid:17) + ˜ U ( p ) − ˜ U (cid:0) p + 1 n p ′ x p ′ − (cid:1) (B.6) ≥ . Indeed, the three lines in the r.h.s. of (B.6) are nonnegative: the first one since W is asupersolution, the second one by k ( p ′ ) < ρ , and the last one since ˜ U is nonincreasing.Moreover, by the growth condition (3.4) on V and W , we have:lim r →∞ max i ∈ I d ( ˆ V i − ˆ W ni )( r ) = −∞ . (B.7)In the next step, our aim is to show that for all n ≥ V ≤ W n , which would imply that V ≤ W . We shall argue by contradiction. Step 2 : Assume that there exists some n ≥ M := sup i ∈ I d , ( x,y ) ∈ R ( V i − W ni )( x, y ) > . By (B.7), there exists i ∈ I d , some compact subset C of R , and X = ( x, y ) ∈ C such that M = max C ( V i − W ni ) = ( V i − W ni )( x, y ) . (B.8)Note that by (B.4), ( x, y ) = (0 , • Case 1 : x = 0. Notice that the boundary condition (B.5) implies the viscosity subsolutionproperty for V i also at ¯ X = (0 , ¯ y ): F i ( ¯ X, V i ( ¯ X ) , p, A ) + G i ( ¯ X, V ) ≤ , ∀ ( p, A ) ∈ P , + V i ( ¯ X )However the viscosity supersolution property fot W n does not hold at (0 , ¯ y ). Let ( X k ) k =( x k , y k ) k be a sequence converging to X , with x k >
0, and ε k := (cid:12)(cid:12) X k − X (cid:12)(cid:12) . We thenconsider the functionΦ k ( X, X ′ ) = V i ( X ) − W ni ( X ′ ) − ψ k ( X, X ′ ) ,ψ k ( x, y, x ′ , y ′ ) = x + ( y − y ) + | X − X ′ | ε k + (cid:18) x ′ x k − (cid:19) − Since Φ k is continuous, there exists ( b X k , b X ′ k ) ∈ C s.t. M k := sup C Φ k = Φ k ( b X k , b X ′ k ) , b X k , b X ′ k ), converging to some ( b X, b X ′ ) as k goes to ∞ . Bywriting that Φ k ( X, X k ) ≤ Φ k ( b X k , b X ′ k ), we have : V i ( X ) − W ni ( X k ) − (cid:12)(cid:12) X − X k (cid:12)(cid:12) ≤ V i ( b X k ) − W ni ( b X ′ k ) − (ˆ x k + (ˆ y k − y ) ) − R k (B.10) ≤ V i ( b X k ) − W ni ( b X ′ k ) − (ˆ x k + (ˆ y k − y ) ) , (B.11)where we set R k = (cid:12)(cid:12)(cid:12) b X k − b X ′ k (cid:12)(cid:12)(cid:12) ε k + (cid:18) ˆ x ′ k x k − (cid:19) − Since V i and W ni are bounded on C , we deduce by inequality (B.10) the boundedness ofthe sequence ( R k ) k ≥ , which implies b X = c X ′ . Then by sending k to infinity in (B.9) and(B.11), with the continuity of V i and W ni , we obtain M = V i ( X ) − W ni ( X ) ≤ V i ( b X ) − W ni ( b X ) − (ˆ x k + (ˆ y k − y ) ), and by definition of M this shows b X = b X ′ = X (B.12)Sending again k to infinity in (B.9)-(B.10)-(B.11), we obtain M ≤ M − lim sup k R k ≤ M ,and so (cid:12)(cid:12)(cid:12) b X k − b X ′ k (cid:12)(cid:12)(cid:12) ε k + (cid:18) b x ′ k x k − (cid:19) − → , (B.13)as k goes to infinity. In particular for k large enough ˆ x ′ k ≥ x k >
0. We can then applyIshii’s lemma (see Theorem 3.2 in [3]) to obtain
A, A ′ ∈ S s.t.( p, A ) ∈ P , + V i ( b X k ) , (cid:0) p ′ , A ′ (cid:1) ∈ P , − W ni ( b X ′ k ) (B.14) A − A ′ ! ≤ D + ε k D , (B.15)where p = D X ψ k ( b X k , c X ′ k ) , p ′ = D X ′ ψ k ( b X k , b X ′ k ) , D = D X,X ′ ψ k ( b X k , c X ′ k ) . Now, we write ρM ≤ ρM k ≤ ρ ( V i ( ˆ X k ) − W ni ( c X ′ k ))= F i (cid:0) b X k , V i ( b X k ) , p, A (cid:1) − F i (cid:0) b X k , W ni ( c X ′ k ) , p, A (cid:1) = F i (cid:0) b X k , V i ( b X k ) , p, A (cid:1) + G i ( b X k , V ) (B.16) − F i (cid:0) b X ′ k , W ni ( b X ′ k ) , p ′ , A ′ (cid:1) − G i ( b X ′ k , W n )+ G i ( b X ′ k , W n ) − G i ( b X k , V )+ F i (cid:0) b X ′ k , W ni ( b X ′ k ) , p ′ , A ′ (cid:1) − F i (cid:0) b X k , W ni ( b X ′ k ) , p, A (cid:1) V at ˆ X k , and the viscosity supersolution prop-erty for W n at ˆ X ′ k , the first two lines in the r.h.s. of (B.16) are nonpositive. For the thirdline, by sending k to infinity, we have: G i ( b X ′ k , W n ) − G i ( b X k , V ) → G i ( X, W n ) − G i ( X, V )= X j = i q ij h ( V j − W nj ) (cid:16) x, y (1 − γ ij ) (cid:17) − ( V i − W ni )( x, y ) i + λ i h(cid:0) ˆ V i − ˆ W ni (cid:1) ( x + y ) − ( V i − W ni )( x, y ) i ≤ F i (cid:0) b X ′ k , W ni ( b X ′ k ) , p ′ , A ′ (cid:1) − F i (cid:0) b X k , W ni ( b X ′ k ) , p, A (cid:1) = b i (ˆ y k p − ˆ y ′ k p ′ ) + ˜ U ( p ) − ˜ U ( p ′ ) + σ i (cid:0) ˆ y k a − ( b y ′ k ) a ′ (cid:1) Now ˆ y k p − ˆ y ′ k p ′ = b y k (cid:18) y k − y ) + ˆ y k − ˆ y ′ k ε k (cid:19) − ˆ y ′ k (cid:18) ˆ y k − ˆ y ′ k ε k (cid:19) ≤ y k (ˆ y k − y ) + (cid:12)(cid:12)(cid:12) b X k − ˆ x ′ k (cid:12)(cid:12)(cid:12) ε k → , as k → ∞ , by (B.12) and (B.13). Moreover,˜ U ( p ) − ˜ U ( p ′ ) = ˜ U (cid:18) ˆ x k − ˆ x ′ k ε k + 4ˆ x k (cid:19) − ˜ U ˆ x k − ˆ x ′ k ε k − x k (cid:18) ˆ x ′ k x k − (cid:19) − ! ≤ , since ˜ U is nonincreasing. Finally,ˆ y k a − (ˆ y ′ k ) a ′ = (cid:16) y k y ′ k (cid:17) A − A ′ ! y k y ′ k ≤ (cid:16) y k y ′ k (cid:17) (cid:0) D + ε k D (cid:1) y k y ′ k by (B.15). Since D ψ k ( x, y, x ′ , y ′ ) = x − ε k
00 12( y − y ) + ε k − ε k − ε k ε k + x k (cid:16) x ′ x k − (cid:17) − − ε k − ε k ,
36 direct calculation gives (cid:16) y k y ′ k (cid:17) (cid:0) D + ε k D (cid:1) y k y ′ k = 3 ε k (ˆ y k − ˆ y ′ k ) − y k − y ) ˆ y k ˆ y ′ k + (cid:0) y k − y ) + ε k (cid:0) y k − y ) (cid:1)(cid:1) ˆ y k → , as k → ∞ , where we used again (B.12) and (B.13), and the boundedness of ( b y k , b y ′ k ).Finally by letting k go to infinity in (B.16) we obtain ρM ≤
0, which is the requiredcontradiction. • Case 2 : x >
0. This is the easier case, and we can obtain a contradiction similarly as inthe first case, by considering for instance the functionΦ k ( X, X ′ ) = V i ( X ) − W ni ( X ′ ) − ( x − x ) − ( y − y ) − k | X − X ′ | . ✷ References [1] Ang A., and G. Bekaert (2002) : “International Asset Allocation with Regime Shifts”,
TheReview of Financial Studies , , 4, 1137-1187.[2] Arisawa M. (2008): “A remark on the definitions of viscosity solutions for the integro-differentialequations with L´evy operators”, Journal de Math´ematiques Pures et Appliqu´ees , , 6, 567-574.[3] Crandall M., Ishii H. and P.L. Lions (1992) : “User’s Guide to Viscosity Solutions of SecondOrder Partial Differential Equations”, Bull. Amer. Math. Soc. , , 1-67.[4] Cretarola A., Gozzi F., Pham H. and P. Tankov (2011): “Optimal consumption policies inilliquid markets”, Finance and Stochastics , , 85-115.[5] Diesinger P., Kraft H. and F. Seifried (2009): “Asset allocation and liquidity breakdowns: whatif your broker does not answer the phone?”, to appear in Finance and Stochastics .[6] Di Giacinto M., Federico S. and Gozzi F. (2011): “Pension funds with minimum guarantee: astochastic control approach”.
Finance and Stochastics . , 2, 297-342.[7] Gassiat P. (2011): PhD thesis of University Paris Diderot.[8] Gassiat P., Pham H. and M. Sirbu (2010): Optimal investment on finite horizon with randomdiscrete order flow in illiquid markets, to appear in International Journal of Theoretical andApplied Finance .[9] Hamilton, J. (1989): A New Approach to the Economic Analysis of Nonstationary Time Seriesand the Business Cycle,
Econometrica , , 357384.[10] Kallenberg O. (2002): Foundations of modern probability , second ed., Probability and its Ap-plications, Springer-Verlag, New York, 2002.
11] Karatzas I. and S. Shreve (1988): Brownian motion and stochastic calculus, Springer Verlag;New York.[12] Keller H.(1992) :
Numerical methods for two-point boundary-value problems , Dover Publica-tions Inc., New York.[13] Ladyzhenskaya O., and N. Uralseva (1968): Linear and quasilinear elliptic equations, Academicpress, New York.[14] Ludkovski M. and H. Min (2010): “Illiquidity effects in optimal consumption-investment prob-lems”, Preprint available on arXiv: 1004.1489[15] Matsumoto K. (2006): “Optimal portfolio of low liquid assets with a log-utility function”,
Finance and Stochastics , , 121-145.[16] Merton R. (1971): “Optimum consumption and portfolio rules in a continuous-time model”, Journal of Economic Theory , , 373-413.[17] Pham H. and P. Tankov (2008): “A model of optimal consumption under liquidity risk withrandom trading times”, Mathematical Finance , , 613-627.[18] Protter, P. (2004): “Stochastic Integration and Differential Equations”, Springer-Verlag.[19] Rogers C. and O. Zane (2002) : “A simple model of liquidity effects”, in Advances in Financeand Stochastics: Essays in Honour of Dieter Sondermann , eds. K. Sandmann and P. Schoen-bucher, pp 161–176.[20] Sotomayor L.R. and A. Cadenillas (2009) : “Explicit solutions of consumption-investmentproblems in financial markets with regime switching”,
Mathematical Finance , , 251-279.[21] Pirvu T. and H. Zhang (2011): “On investment-consumption with regime switching”, Preprintavailable on arXiv: 1107.1895[22] Yong J. and X.Y. Zhou (1999): Stochastic controls, Hamiltonian systems and HJB equations,Springer Verlag.[23] Zariphopoulou T. (1992): “Investment-consumption models with transaction fees and Markov-chain parameters”, SIAM J. Control and Optimization , , 613-636., 613-636.