A game theoretical approach to homothetic robust forward investment performance processes in stochastic factor models
aa r X i v : . [ q -f i n . P M ] M a y A game theoretical approach to homotheticrobust forward investment performance processesin stochastic factor models
Juan Li ∗ Wenqiang Li † Gechun Liang ‡ May 22, 2020
Abstract
This paper studies an optimal forward investment problem in an in-complete market with model uncertainty, in which the dynamics of theunderlying stocks depends on the correlated stochastic factors. The un-certainty stems from the probability measure chosen by an investor toevaluate the performance. We obtain directly the representation of thepower robust forward performance process in factor-form by combiningthe zero-sum stochastic differential game and ergodic BSDE approach.We also establish the connections with the risk-sensitive zero-sum stochas-tic differential games over an infinite horizon with ergodic payoff criteria,as well as with the classical power robust expected utility for long timehorizons.
Keywords : forward performance process; model uncertainty; self-generatingstochastic differential game; ergodic BSDE; ergodic risk-sensitive stochas-tic differential game.
The aim of this paper is to study optimal investment evaluated by a forwardperformance criterion in a stochastic factor market model, in which the prob-ability measure that models future stock price evolutions is ambiguous. The ∗ School of Mathematics and Statistics, Shandong University, Weihai, China.( [email protected] ). † School of Mathematics and Information Sciences, Yantai University, Yantai,China.( [email protected] ). Corresponding author. ‡ Department of Statistics, The University of Warwick, Coventry, U.K.( [email protected] ). π E P [ U ( X πT )] , where π is the portfolio choice, P is a probability measure that is used to measurethe evolutions of stock prices, T is the terminal horizon, and U is the relatedutility function at time T . However, the paradigm of expected utility clearlyhas some deficiencies: it is not satisfactory in dealing with model uncertainty(also called Knightian uncertainty) as predicted by the famous Ellsberg paradox.Based on this fact, the robust utility was introduced to account for uncertaintyaversion and it can be numerically represented by the following form X → inf P ∈P E P [ U ( X )] , where P is a family of probability measures which describe all the possibleprobabilities of future scenarios and the infimum means the worst-case scenariois implemented. For the worst-case scenario approach in the optimal investmentproblems, we refer to [6, 14, 15, 16, 21, 22, 39, 43] and the references therein.We consider the ambiguity of the probability measure under the framework offorward performance processes in (possibly) incomplete markets. We propose aframework that solves directly the above problem in a unified manner, combiningthe zero-sum stochastic differential game and ergodic backward stochastic dif-ferential equation (BSDE) theory. The concept of robust forward performanceprocesses was recently introduced in [26], by using a penalty function to weightrelatively the stochastic models such that they are more in line with the actualmarket. They obtained the characterization of the robust forward criteria viathe duality approach. See also [8] for the extension to the case with uncertainparameters. However, both only consider robust forward performance processeswith zero volatility, in particular, the Markovian case for the stochastic factormodel is not covered. In this paper, we consider the
Markovian robust forwardperformance process in stochastic factor models. The approach is different fromthe duality approach used in [26] and the saddle point method used in [8]. Next,we briefly introduce our framework and the main contributions.We construct the robust forward performance process via a zero-sum stochas-tic differential game. In our model, the ambiguity of the probability measure2s described via a family of equivalent probability measures parameterised by adensity process u in a compact and convex set (see (6)). To robustify the optimalinvestment, assuming the existence of the mother nature who acts maliciously tominimize the expected forward preference by choosing the worst-case scenario,whereas the investor aims to select the best investment portfolio that is least af-fected by the mother nature’s choice. This leads to a stochastic differential gamebetween the investor and the mother nature. It is well-known that the conceptof “strategy” corresponding to the “control” plays a key role in order to ensurethe existence of the game value (see [7, 13]). Utilizing the idea of “strategy”,we give a new characterization of the robust forward performance process (see(11)-(14)). Specifically, both the optimal investment “strategy” correspondingto each scenario and the worst-case scenario “strategy” corresponding to eachportfolio selection are given in our characterization. Compared to the saddle-point argument used in [21, 22, 43] in the classical framework and [8] in theforward performance process framework, our characterization relies on the in-vestor’s reflections to each scenario and portfolio choice, and moreover, it isoften relatively easy to compute the optimal strategies, as they only involvemaximization/minimization problems rather than maxmin/minmax problems.The second component to construct the robust forward performance process infactor form is an ergodic BSDE. The stochastic PDE (SPDE) approach, intro-duced in [35] to characterize the forward performance processes (without modelambiguity), may not be applied directly to our model. The form of the relatedSPDE is not easy to derive due to the introduction of model uncertainty. More-over, it is difficult to obtain the existence of the solution of the SPDE in thegeneral case even we know its form. In order to get the representation of therobust forward performance process in stochastic factor form, we apply directlythe ergodic BSDE approach, which was first proposed in [17] to study ergodiccontrol problems. The ergodic BSDE approach was first exploited in [30] tostudy the representation of the homothetic forward performance process in theabsence of model uncertainty. We first characterize the power robust forwardperformance process in terms of the solution of some Isaacs type equation. Al-though the solution of this Isaacs equation can not be directly obtained, it offers(1) the construction of the optimal portfolio “strategy”, the worst-case scenario“strategy”, and the related optimal portfolio choice and the worst-case scenario;(2) the hint of the driver form of the corresponding ergodic BSDE (32). Then,we obtain the representation of the robust power forward performance processby using the Markovian solution of this ergodic BSDE. The associated optimalportfolio and worst-case scenario “strategy” and “control” are also obtained infeedback form of the stochastic factor.Another contribution of this paper is establishing a connection between theconstant λ appearing in the solution of the ergodic BSDE (32) and a class ofzero-sum risk-sensitive stochastic differential game over an infinite horizon withergodic payoff criteria. Risk-sensitive optimal control has been widely appliedto optimal investment problems (see, [4, 11, 12, 20] and references therein).The corresponding risk-sensitive stochastic differential games are studied in [2,3, 5, 27] via PDE approach and in [9] via BSDE approach. In this paper, weapply directly the ergodic BSDE approach to address the zero-sum risk-sensitivestochastic differential game with ergodic payoff criteria over an infinite horizon.Thus, we provide a new method to obtain the value of the risk-sensitive gameproblem and give the robust optimal investment policy which generalises theresults in [11, 12] to stochastic factor model with uncertainty. To obtain thisconnection, we prove a comparison result for a class of ergodic BSDE whosedrivers are only local Lipschitz continuous.In addition, we also develop a connection between the robust forward perfor-mance processes and classical robust expected preferences. Optimal investmentproblems for classical robust expected utilities have been studied via differentmethods, among others, by the duality approach [14, 39], the stochastic con-trol approach based on BSDE [6, 21] and stochastic differential game approachbased on PDE [42]. With the help of the relation established in [25] on thesolution of a finite horizon BSDE and the solution of associated ergodic BSDE,we prove that an appropriately discounted lower value function associated withthe classical power robust expected utility will converge to the power robustperformance process as the trading horizon tends to infinity.This paper is organized as follows. In section 2, we introduce the market modelwith uncertainty and the notion of robust forward performance processes. Insections 3, we focus on the power utility case and construct the robust forwardperformance process in factor-form. Two examples are given in Section 4 toillustrate the applications in complete market and incomplete market. Then, wepresent the connection with risk-sensitive game problem and classical expectedutility in Sections 5 and 6, respectively. Let (Ω , F , F = {F t } t ≥ , P ) be a filtered probability space satisfying the usualcondition, on which the process W = ( W , · · · , W d ) T is a standard d -dimensionalBrownian motion. Here, the superscript T denotes the matrix transpose. Sup-pose the market consists of a risk-free bond and n risky stocks. The bondis assumed to be zero interest rate and the discounted (by the bond) individ-ual stock price S it , t ≥ , affected by the stochastic factor process V , has thefollowing form, for i = 1 , ..., n,dS it S it = b i ( V t ) dt + d X j =1 σ ij ( V t ) dW jt , (1)4ith S i > , where the factor process V = ( V , · · · , V d ) T satisfies, for i =1 , ..., d, dV it = η i ( V t ) dt + d X j =1 κ ij dW jt , (2)with V i ∈ R . The market price of risk vector θ ( v ) , v ∈ R d , is defined as θ ( v ) = σ ( v ) T [ σ ( v ) σ ( v ) T ] − b ( v ) , (3)so it solves the market price of risk equation σ ( v ) θ ( v ) = b ( v ). We introduce thebasic assumptions on the above model. Assumption 1 (H1) The coefficients b : R d → R n , σ : R d → R n × d , are uniformly bounded and the volatility matrix σ ( v ) has full row rank n .(H2) The drift coefficient η satisfies the following dissipative condition ( η ( v ) − η (¯ v )) T ( v − ¯ v ) ≤ − C η | v − ¯ v | , (4) for any v, ¯ v ∈ R d and a constant C η large enough. The volatility matrix κ = ( κ ij ) , ≤ i, j ≤ d , is a constant matrix with κκ T positive definite andnormalized to | κ | = 1 .(H3) The price of risk vector θ ( v ) , v ∈ R d , is uniformly bounded and Lipschitzcontinuous. The “large enough” requirement of the constant C η in (4) will be refined intothe assumption C η > C >
0, where C is the constant derived from the conditionthat the driver of upcoming ergodic BSDE (32) satisfies, i.e. the constant in thefirst inequality of (34). The dissipative condition (4) is introduced to ensure theexistence and the uniqueness of the invariant measure of the stochastic factorprocess V . One possible extension to a more general case for the stochasticfactor is letting κ = κ ( v ) under the following dissipative condition2( η ( v ) − η (¯ v )) T ( v − ¯ v ) + | κ ( v ) − κ (¯ v ) | ≤ − C | v − ¯ v | , which also implies the ergodicity property of the factor process (see Theorem6.3.2 in [38] or more recently [24]).We consider an investor starting at time t = 0 with initial wealth level x > π = (˜ π , · · · , ˜ π n ) T be theproportions of her total (discounted by the bond) wealth in the individual stockaccounts. Then, due to the self-financing policy, the cumulative wealth process X π satisfies dX πt = n X i =1 ˜ π it X πt S it dS it = X πt ˜ π Tt ( b ( V t ) dt + σ ( V t ) dW t ) .
5s in [30], using the investment proportions rescaled by the volatility of stockprices, namely, π Tt = ˜ π Tt σ ( V t ) , we get dX πt = X πt π Tt ( θ ( V t ) dt + dW t ) , (5)with X π = x ∈ R + .Next, we consider the model uncertainty, i.e., the ambiguity of the proba-bility measure which models the investor’s expectation. We denote by ˜ u =(˜ u , · · · , ˜ u d ) T the parameters reflecting the possible future scenarios. For con-venience, we will work throughout with the scenario parameters rescaled by thevolatility of the stochastic factors, i.e., u = κ T ( κκ T ) − ˜ u. As a result, the investor will apply P u to measure her preference instead ofprobability measure P , where the probability measure P u is an equivalent prob-ability measure with respect to P and introduced by the the following measuretransformation d P u d P (cid:12)(cid:12)(cid:12) F t = E (cid:18)Z t u Ts dW s (cid:19) := exp { Z t u Ts dW s − Z t | u s | ds } . (6)In fact, this characterization of model uncertainty, admitting an entire class { P u | u ∈ U} of possible prior models, is a common approach applied in theclassical robust expected utility, see [15, 21].We introduce admissible spaces ˜Π and U for the rescaled investment proportions π and scenario parameters u , respectively. Definition 1
Let Π ⊂ R d be convex and closed. For any t ≥ , a process π : Ω × [0 , t ] → Π is an admissible investment proportion for an investor in thetrading interval [0 , t ] , if π ∈ L BMO [0 , t ] , where L BMO [0 , t ] = n ( π s ) s ∈ [0 ,t ] : π is F -progressively measurable, E P ( Z tτ | π s | ds |F τ ) ≤ C, a.s., for some constant C and all F -stopping times τ ≤ t o . The set of all admissible investment proportions in the trading interval [0 , t ] isdenoted by Π [0 ,t ] . Moreover, we define the set of admissible proportions for alltime horizons as ˜Π := ∪ t ≥ Π [0 ,t ] . Definition 2
The admissible space of the scenario parameters is defined as U = { ( u t ) t ≥ : U -valued , F -progressively measurable, essentially bounded process } , where the set U ⊆ R d is convex and compact. u ∈ U , the process W u defined as dW ut = − u t dt + dW t , (7)is a Brownian motion under the probability measure P u . Moreover, if π ∈ Π [0 ,t ] ,then under P u , we also have ess sup τ E P u (cid:18) Z tτ | π s | ds (cid:12)(cid:12)(cid:12)(cid:12) F τ (cid:19) < ∞ . The investor will evaluate her investment via forward performance processes,the concept of which was first introduced and developed in [31]-[35]. Since theinvestor is uncertain about the probability measure she uses, she will seek for anoptimal investment proportion that is least affected by model uncertainty. Thisleads to the so called robust forward performance processes as first introducedin [26] and later extended to the case with uncertain parameters in [8].
Definition 3
A process U ( x, t ) , ( x, t ) ∈ R + × [0 , ∞ ) , is a robust forward per-formance process if i) for each x ∈ R + , U ( x, t ) is F -progressively measurable; ii) for each t ≥ , the mapping x U ( x, t ) is strictly increasing and strictlyconcave; iii) the process U ( x, t ) satisfies the self-generating property (dynamic program-ming principle), i.e. for all s ≥ t ≥ , ess sup π ∈ ˜Π ess inf u ∈U E P u [ U ( X πs , s ) |F t , X πt = x ] = U ( x, t ) , a.s. (8)In [26], a duality method is developed to construct U ( x, t ) and its associatedoptimal investment proportion π ∗ , while in [8], a saddle point method is em-ployed to further find the worst case scenario u ∗ . However, both only considerforward performance processes with zero volatility, in particular, the Markovian case for the stochastic factor model is not covered. Herein, we aim to constructa class of
Markovian robust forward performance processes with explicit depen-dency on the stochastic factor process V . Our approach is based on stochasticdifferential games . To robustify the optimal investment, the inner part of theabove optimization problem (8) is played by a so called mother nature who actsmaliciously to minimize the expected forward preference by choosing the worst-case scenario, whereas the investor aims to select the best investment proportionthat is least affected by the mother nature’s choice. This leads to a stochasticdifferential game between the investor and the mother nature. In addition to the representation of the robust forward performance process, wealso aim to provide both the optimal investment proportion for each scenario and7he worst-case scenario for each investment proportion. The investment propor-tion (resp. worst-case scenario) responding to each scenario (resp. investmentproportion) can be exactly expressed as the “strategy to control” in the setupof stochastic differential games (see [13, 7]). Thus, we next give the definitionsof two admissible “strategies” associated with their respective “controls”.
Definition 4
An admissible investment strategy responding to each scenarioparameter for an investor is a mapping α : [0 , ∞ ) × Ω × U → ˜Π satisfying thefollowing two properties: (i) For each u ∈ U , α is F -progressively measurable; (ii) Non-anticipative property, that is, for all t > and all u , u ∈ U , with u = u , dsd P -a.e., on [0 , t ] , it holds that α ( · , u ) = α ( · , u ) , dsd P -a.e., on [0 , t ] .An admissible scenario parameter strategy responding to each investment pro-portion for an investor, β : [0 , ∞ ) × Ω × ˜Π → U , is defined similarly. The set ofall admissible investment strategies for the investor is denoted by A , while theset of all admissible scenario parameter strategies is denoted by B . Herein, the non-anticipative property is natural as explained in differentialgames, implying that a rational investor will take the same investment action ifthe future scenario does not change.We consider a zero-sum stochastic differential game, where the state dynamicis given by the wealth equation (5). Furthermore, let U ( x, t ) be a stochasticprocess satisfying i) and ii) in Definition 3. For any fixed s >
0, the objectivefunctional is given by J ( t, x, π, u ) = E P u [ U ( X πs , s ) |F t , X πt = x ] . The lower and upper value of the game are then defined as U ( x, t ) = ess inf β ∈B ess sup π ∈ ˜Π J ( t, x, π, β ( · , π )) , a.s., (9)and ¯ U ( x, t ) = ess sup α ∈A ess inf u ∈U J ( t, x, α ( · , u ) , u ) , a.s., (10)respectively. Note that if U ( x, t ) = U ( x, t ), i.e. the objective functional of thestochastic differential game “self generates” the lower value of the game, thenit is clear that U ( x, t ) becomes a robust forward performance process satisfyingi)-iii) in Definition 3.Thus, we say the game is self-generating and its value exists if U ( x, t ) = ¯ U ( x, t ) = U ( x, t ) , which will in turn provide a robust forward performance process.To solve the above stochastic differential game, and in turn to construct the as-sociated forward performance process, we will construct a control pair ( π ∗ , u ∗ ) ∈ × U , a strategy pair ( α ∗ , β ∗ ) ∈ A × B , and a process U ( x, t ) satisfying themartingale properties: For any ( π, u ) ∈ ˜Π × U , a.s.,ess inf β ∈B J ( t, x, π, β ( · , π )) = J ( t, x, π, β ∗ ( · , π )) ≤ U ( x, t ) ; (11) J ( t, x, π ∗ , β ∗ ( · , π ∗ )) = U ( x, t ) ; (12)ess sup α ∈A J ( t, x, α ( · , u ) , u ) = J ( t, x, α ∗ ( · , u ) , u ) ≥ U ( x, t ); (13) J ( t, x, α ∗ ( · , u ∗ ) , u ∗ ) = U ( x, t ) . (14)Note that (11) and (12) are the martingale characterization of the lower valueof the game in (9), whereas (13) and (14) characterize the upper value of thegame in (10). Remark 5
When there is no model uncertainty (i.e., u ≡ , for all u ∈ U ), thenonly the conditions (13) and (14) are relevant, which is precisely the definitionof forward performance processes introduced in [31]-[35].On the other hand, if π ∗ = α ∗ ( · , u ∗ ) and u ∗ = β ∗ ( · , π ∗ ) , then the martingaleconditions (11)-(14) imply that J ( t, x, π ∗ , u ) ≥ J ( t, x, π ∗ , β ∗ ( · , π ∗ ))= J ( t, x, α ∗ ( · , u ∗ ) , u ∗ ) ≥ J ( t, x, π, u ∗ ) , so the control pair ( π ∗ , u ∗ ) is a saddle point for the stochastic differential gamewith the value J ( t, x, π ∗ , u ∗ ) = U ( x, t ) . As opposed to the saddle point method,the advantage of the stochastic differential game approach is to provide, in ex-plicit form, the optimal investment choice for the investor not only under theworst-case scenario but also for each scenario, as well as the worst case scenariofor each investment choice not only the optimal one. Moreover, it is often rel-atively easy to compute the optimal strategy pair ( α ∗ , β ∗ ) , as they only involvemaximization/minimization problems rather than maxmin/minmax problems. In this section, we focus on a class of homothetic robust forward performanceprocesses that are homogenous in the degree of δ ∈ (0 , U ( x, t ) = x δ δ e f ( V t ,t ) , (15)where f : R d × [0 , ∞ ) → R is a deterministic function to be specified. We callsuch a robust forward performance process a power robust forward performanceprocess . 9 roposition 6 Assume that f ( v, t ) , ( v, t ) ∈ R d × [0 , ∞ ) , is a classical solution(with enough regularity) of the semilinear PDE f t + 12 T race (cid:0) κκ T ∇ f (cid:1) + η ( v ) T ∇ f + G ( v, κ T ∇ f ) = 0 , (16) where G ( v, z ) = inf u ∈ U sup π ∈ Π F ( v, z, π, u ) , (17) with F ( v, z, π, u ) = − δ (1 − δ ) | π | + δπ T ( θ ( v ) + z + u ) + z T u + 12 | z | . (18) Then, U ( x, t ) = x δ δ e f ( V t ,t ) is a power robust forward performance process. Proof.
Since U ( x, t ) obviously satisfies i) and ii) in Definition 3, it is sufficientto examine iii) in Definition 3. Step 1.
From (17) and (18), we have G ( v, z ) = inf u ∈ U sup π ∈ Π F ( v, z, π, u ) = inf u ∈ U F ( v, z, α ∗ ( v, z, u ) , u ) , (19)with α ∗ ( v, z, u ) = argmax π ∈ Π F ( v, z, π, u ) = P roj Π ( θ ( v ) + z + u − δ ) . (20)Using the Lipschitz continuity of the projection operator on the convex set Π,there exists a Borel measurable mapping u ∗ : R d × R d → U such that u ∗ ( v, z ) = argmin u ∈ U F ( v, z, α ∗ ( v, z, u ) , u ) . (21)Then, from (19) and (21), we have G ( v, z ) = F ( v, z, π ∗ ( v, z ) , u ∗ ( v, z )) , (22)with π ∗ ( v, z ) := α ∗ ( v, z, u ∗ ( v, z )) . (23)We claim that, for any u ∈ U , F ( v, z, π ∗ ( v, z ) , u ) ≥ F ( v, z, π ∗ ( v, z ) , u ∗ ( v, z )) . (24)If (24) holds, thensup π ∈ Π inf u ∈ U F ( v, z, π, u ) ≥ inf u ∈ U F ( v, z, π ∗ ( v, z ) , u ) (25) ≥ F ( v, z, π ∗ ( v, z ) , u ∗ ( v, z )) (26)= G ( v, z ) = inf u ∈ U sup π ∈ Π F ( v, z, π, u ) ,
10o both (25) and (26) become equalities. In turn, π ∗ ( v, z ) in (23) and u ∗ ( v, z )in (21) satisfy, respectively, π ∗ ( v, z ) = argmax π ∈ Π inf u ∈ U F ( v, z, π, u ) , and u ∗ ( v, z ) = argmin u ∈ U F ( v, z, π ∗ ( v, z ) , u ) . (27)On the other hand, there exists a U -valued Borel measurable mapping ¯ β ∗ ( v, z, π )such that F ( v, z, π, u ) attains the minimum, i.e.inf u ∈ U F ( v, z, π, u ) = F ( v, z, π, ¯ β ∗ ( v, z, π )) . Then, from (27), the mapping β ∗ ( v, z, π ) defines as β ∗ ( v, z, π ) = (cid:26) u ∗ ( v, z ) , if π = π ∗ ( v, z );¯ β ∗ ( v, z, π ) , otherwise, (28)also minimizes F ( v, z, π, u ) over u ∈ U , and moreover, π ∗ ( v, z ) = argmax π ∈ Π F ( v, z, π, β ∗ ( v, z, π )) . (29) Step 2.
We are left to prove the inequality (24). We omit the variables ( v, z ) in π ∗ ( v, z ) and u ∗ ( v, z ), and write them as π ∗ and u ∗ in this step. For any u ∈ U and λ ∈ (0 ,
1) let u := λu + (1 − λ ) u ∗ . Set π := α ∗ ( v, z, u ) and recall from (23) that π ∗ = α ∗ ( v, z, u ∗ ). Then, itfollows from (21) that F ( v, z, π ∗ , u ∗ ) ≤ F ( v, z, π , u )= λF ( v, z, π , u ) + (1 − λ ) F ( v, z, π , u ∗ ) ≤ λF ( v, z, π , u ) + (1 − λ ) F ( v, z, π ∗ , u ∗ ) . where we used F ( v, z, π, u ) ≤ F ( v, z, α ∗ ( v, z, u ) , u ) in the last inequality. Thus, F ( v, z, π ∗ , u ∗ ) ≤ F ( v, z, π , u ) = F ( v, z, α ∗ ( v, z, u ) , u )for any u ∈ U . Sending λ → α ∗ ( v, z, u ) in u , wehave α ∗ ( v, z, u ) → α ∗ ( v, z, u ∗ ) = π ∗ . Then, the inequality (24) follows by thecontinuity of F ( v, z, π, u ) in π . Step 3.
Using the homothetic form (15) and applying Itˆo’s formula to U ( X πs , s ),we get dU ( X πs , s )= U ( X πs , s ) (cid:2) f s + 12 T race (cid:0) κκ T ∇ f (cid:1) + η ( V s ) T ∇ f + F ( V s , κ T ∇ f, π s , u s ) (cid:3) ds + U ( X πs , s )( δπ Ts + ∇ f T κ ) dW us . s ≥ t ≥
0, from (16), we further get E P u [ U ( X πs , s ) |F t , X πt = x ] − U ( x, t )= J ( t, x, π, u ) − U ( x, t )= E P u (cid:2) Z st U ( X πr , r ) (cid:0) F ( V r , κ T ∇ f, π r , u r ) − G ( V r , κ T ∇ f ) (cid:1) dr |F t , X πt = x (cid:3) . (30)We set π ∗ t = π ∗ ( V t , κ ∇ f ( V t , t )) , u ∗ t = u ∗ ( V t , κ ∇ f ( V t , t )) ,α ∗ ( t, u t ) = α ∗ ( V t , κ ∇ f ( V t , t ) , u t ) , β ∗ ( t, π t ) = β ∗ ( V t , κ ∇ f ( V t , t ) , π t ) , (31)with the mappings ( π ∗ , u ∗ , α ∗ , β ∗ ) given in (23), (21), (20) and (28), respec-tively. Then, it is easy to check that U ( x, t ) satisfies the martingale conditions(11)-(14), which implies that U ( x, t ) = x δ δ e f ( V t ,t ) is a power robust forwardperformance process, with the optimal control pair ( π ∗ , u ∗ ) and the optimalstrategy pair ( α ∗ , β ∗ ). Remark 7
It is worth to point out that the strategies α ∗ and β ∗ we constructedin the above proof are also called “counterstrategies”; the reader can refer toChapter 10, Section 1 in [28] for more details.Since, by our construction, π ∗ t = α ∗ ( t, u ∗ t ) and u ∗ t = β ∗ ( t, π ∗ t ) , it follows fromRemark 5 that ( π ∗ , u ∗ ) is actually a saddle point for the associated game. How-ever, compared to the classical saddle point argument such as Sion’s MinimaxTheorem (see, for example, [8, 43]), our formulae are more explicit and is con-structed via their corresponding counterstrategies. Note that the semi-linear PDE (16) is ill-posed with no known solutions todate. A similar difficulty also appears in [36], [37] and [41] for the constructionof forward processes without model ambiguity, where the Widder’s theorem isemployed. Nevertheless, the form of PDE (16) motivates us how to construct theoptimal investment proportion, worst-case scenario parameter and the relatedoptimal strategies for different situations, which will be used in the followingTheorem 9. In order to give the specific form of the process f ( V t , t ), we bypassPDE (16) by directly using the Markovian solution of an ergodic BSDE whosedriver has the form (17). This approach was first introduced in [30] to studythe forward performance process in the absence of model uncertainty. We firstgive the existence and uniqueness of the Markovian solution of the associatedergodic BSDE. Lemma 8
Assume the function G has the form (17) . Then, the ergodic BSDE dY t = ( − G ( V t , Z t ) + λ ) dt + Z Tt dW t , (32) admits a unique Markovian solution ( Y t , Z t , λ ) , t ≥ , i.e., there exist a uniqueconstant λ and functions y : R d → R , z : R d → R d such that Y t = y ( V t ) , Z t =12 ( V t ) . Here, the function y ( · ) is unique up to a constant and has at most lineargrowth, and z ( · ) is bounded. Proof.
Using the Lipschitz continuity of the projection operator, it follows from(18) and (20) that | F ( v, z, α ∗ ( v, z, u ) , u ) − F (¯ v, z, α ∗ (¯ v, z, u ) , u ) | ≤ C (1 + | z | ) · | v − ¯ v | , | F ( v, z, α ∗ ( v, z, u ) , u ) − F ( v, ¯ z, α ∗ ( v, ¯ z, u ) , u ) | ≤ C (1 + | z | + | ¯ z | ) · | z − ¯ z | , | F ( v, , α ∗ ( v, , u ) , u ) | ≤ C. (33)Here the constant C > u . Then, from(19) we obtain | G ( v, z ) − G (¯ v, z ) | ≤ C (1 + | z | ) · | v − ¯ v | , | G ( v, z ) − G ( v, ¯ z ) | ≤ C (1 + | z | + | ¯ z | ) · | z − ¯ v | , | G ( v, | ≤ C. (34)Therefore, from Proposition 3.1 and Appendix A in [30] we obtain the desiredresult.We next present the specific form of the process f ( V t , t ) by using the solutionof the ergodic BSDE (32). Theorem 9
Let ( Y t , Z t , λ ) = ( y ( V t ) , z ( V t ) , λ ) , t ≥ , be the unique Markoviansolution of (32). Then, the process U ( x, t ) , ( x, t ) ∈ R + × [0 , ∞ ) , given by U ( x, t ) = x δ δ e y ( V t ) − λt , (35) is a power robust forward performance process. Moreover, the optimal portfolioweight π ∗ , the worst-case scenario parameter u ∗ and the optimal strategies α ∗ , β ∗ responding to each scenario parameter u and portfolio weight π are given asfollows π ∗ t = π ∗ ( V t , z ( V t )) , u ∗ t = u ∗ ( V t , z ( V t )) ,α ∗ ( t, u t ) = α ∗ ( V t , z ( V t ) , u t ) , β ∗ ( t, π t ) = β ∗ ( V t , z ( V t ) , π t ) , (36) where the mappings ( π ∗ , u ∗ , α ∗ , β ∗ ) are given in (23) , (21) , (20) and (28) , re-spectively.In addition, the associated wealth process X ∗ under the worst-case scenario isgiven by X ∗ t = X E (cid:16) Z t ( π ∗ s ) T · [( θ ( V s ) + u ∗ s ) ds + dW u ∗ s ] (cid:17) . Proof.
It is easy to check that the process given by (35) is F -progressivelymeasurable, strictly increasing and strictly concave in x . We only need to show13hat the martingale conditions (11)-(14) hold. For this, from (5), (7) and (32)we get, for all s ≥ t ≥
0, ( π, u ) ∈ ˜Π × U , X πs = X πt · exp n Z st π Tr ( θ ( V r ) + u r ) − | π r | dr + Z st π Tr dW ur o , ( Y s − λs ) = ( Y t − λt ) − Z st G ( V r , Z r ) − Z Tr u r dr + Z st Z Tr dW ur . Thus, we have U ( X πs , s ) = ( X πs ) δ δ e Y s − λs = U ( X πt , t ) · E (cid:16) Z st ( δπ Tr + Z Tr ) dW ur (cid:17) · exp n Z st F ( V r , Z r , π r , u r ) − G ( V r , Z r ) dr o . Therefore, E P u [ U ( X πs , s ) |F t , X t = x ] − U ( x, t )= J ( t, x, π, u ) − U ( x, t )= U ( x, t ) · E P u (cid:18) M s M t · exp n Z st F ( V r , Z r , π r , u r ) − G ( V r , Z r ) dr o(cid:12)(cid:12)(cid:12) F t (cid:19) − U ( x, t ) , where, for t ∈ [0 , s ], M t := E (cid:16) R t ( δπ Tr + Z Tr ) dW ur (cid:17) , is a uniformly integrable ex-ponential martingale (since π satisfies the BMO-condition and z ( · ) is bounded).Similar to the argument in the proof of Lemma 6, we get the the martingaleconditions (11)-(14) from the above equality. Remark 10
The probability measure P u ∗ associated with u ∗ given in Theorem9 has the following form d P u ∗ d P (cid:12)(cid:12)(cid:12) F T = E Z T ( u ∗ s ) T dW s ! . Thus, as a byproduct, we obtain a specific formula for the least favorable mar-tingale measure as considered in [14].
Remark 11
Similar to Proposition 3.4 in [30], it is easy to check that f ( v, t ) = y ( v ) − λt is a classical solution of the semilinear PDE (16) with the initial condition f ( v,
0) = y ( v ) , where ( y ( V t ) , z ( V t ) , λ ) is the solution of ergodic BSDE (32) . For ρ >
0, we consider the following infinite horizon BSDE dY ρt = ( − G ( V t , Z ρt ) + ρY ρt ) dt + ( Z ρt ) T dW t , (37)14here the driver G ( · , · ) is given in (32). Then, this BSDE admits a uniqueMarkovian solution ( Y ρt , Z ρt ) = ( y ρ ( V t ) , z ρ ( V t )). Moreover, there exists a subse-quence, denoted by ρ n , such that y ( v ) = lim ρ n ↓ y ρ n ( v ) , z ( v ) = lim ρ n ↓ z ρ n ( v ) , λ = lim ρ n ↓ ρ n y ρ n ( v ) , where ( y ( V t ) , z ( V t ) , λ ) is the solution of ergodic BSDE (32) and v ∈ R d is anarbitrary given reference point. These results were first obtained in [17] withLipschitz driver and then extended to the quadratic driver in [30].Similar to the proof of Theorem 9, we can examine that the process U ρ ( x, t )given by (38) is still a power robust forward performance process and it convergesin an appropriate discounted manner to the process U ( x, t ) as ρ tends to 0. Corollary 12
The process U ρ ( x, t ) , ( x, t ) ∈ R + × [0 , ∞ ) , given by U ρ ( x, t ) = x δ δ e y ρ ( V t ) − R t ρy ρ ( V s ) ds (38) is a power robust forward performance process and the optimal portfolio strategy α ∗ ,ρt for each scenario parameter u is given by α ∗ ,ρt ( u ) = P roj Π (cid:18) θ ( V t ) + z ρ ( V t ) + u t − δ (cid:19) . Furthermore, there exists a subsequence ρ n ↓ such that, for ( x, t ) ∈ R + × [0 , ∞ ) , lim ρ n ↓ U ρ n ( x, t ) e − y ρn ( v ) U ( x, t ) = 1 . (39) and the associated optimal portfolio strategies α ∗ ,ρ n and α ∗ satisfy lim ρ n ↓ E P Z t | α ∗ ,ρ n ( s, u s ) − α ∗ ( s, u s ) | ds = 0 , for t ≥ , u ∈ U . (40) We apply Theorem 9 to analyse two specific examples. The first example iswhen the value spaces of the investment proportion and the scenario parameterare large enough, namely, in a complete market with uncertainty. Then, weconsider a single stock and single stochastic factor case in an incomplete marketwith uncertain model as the second example. In both two examples, optimalinvestment policies and the worst-case scenario parameters for the power ro-bust forward performance processes are given in the feedback form of stochasticfactors. 15 .1 Complete market model
We consider the case that two spaces Π and U are large enough in the sensethat the mappings α ∗ in (20) and u ∗ in (21) have the following form α ∗ ( v, z, u ) = P roj Π ( θ ( v ) + z + u − δ ) = θ ( v ) + z + u − δ ,u ∗ ( v, z ) = argmin u ∈ U F ( v, z, α ∗ ( v, z, u ) , u ) = − θ ( v ) − δ z. Then, the mappings π ∗ in (23) and β ∗ in (28) take the form π ∗ ( v, z ) = α ∗ ( v, z, u ∗ ( v, z )) = − δ z,β ∗ ( v, z, π ) = (cid:26) − θ ( v ) − δ z, if π = π ∗ ( v, z ); argmin u ∈ U ( δπ + z ) T u, otherwise.In this case, the ergodic BSDE (32) becomes dY t = ( 12 δ | Z t | + Z Tt θ ( V t ) + λ ) dt + Z Tt dW t . (41)In turn, from Theorem 9, we obtain the following result. Proposition 13
Denote by ( y ( V t ) , z ( V t ) , λ ) the Markovian solution of (41) .Then, the process U ( x, t ) given by U ( x, t ) = x δ δ e y ( V t ) − λt ,is a power robust forward performance process. Moreover, the optimal controlpair ( π ∗ , u ∗ ) ∈ ˜Π × U and optimal strategy pair ( α ∗ , β ∗ ) ∈ A × B , have thefollowing feedback form π ∗ t = − δ z ( V t ) , u ∗ t = − θ ( V t ) − δ z ( V t ) ,α ∗ ( t, u t ) = θ ( V t ) + z ( V t ) + u t − δ ,β ∗ ( t, π t ) = (cid:26) − θ ( V t ) − δ z ( V t ) , if π = π ∗ ( v, z ); argmin u t ∈ U ( δπ t + z ( V t )) T u t , otherwise. One may consider a special case of the above model in the following sense:(i) The market has only single risky asset and single stochastic factor (i.e., n = d = 1);(ii) the coefficients of the stock price have the form b ( v ) = a + 12 σ − b · v, σ ( v ) ≡ σ, a > b > σ > V t is viewed as the logarithm of the tradable stock price S t , then SDEs (1)-(2)become dS t = ( a + 12 σ − b · ln S t ) S t dt + σS t dW t ,dV t = ( a − b · V t ) dt + σdW t , (42)with η ( v ) = a − b · v, κ = σ . In fact, (42) has been studied in [40] as Model1 , which models the spot price of commodity. In this situation, we can obtainthe robust forward performance process and robust investment choice for anambiguity-aversion investor from Proposition 13, which extends the results ofSection 4.1 in [36] to the market with model uncertainty.
We consider a single stock and single stochastic factor model. In this situation,we suppose n = 1 and d = 2 in the state equations (1) and (2), i.e., dS t = b ( V t ) S t dt + σ ( V t ) S t dW t , dV t = η ( V t ) dt + ρdW t + p − ρ dW t and dV t = 0 , (43)with constant ρ ∈ (0 ,
1) and σ ( · ) bounded by a positive constant. Supposethat the stochastic factor cannot be traded directly so that the market modelis typically incomplete.Let Π = R × { } (which means π t ≡ U = { ( u , u ) : − R ≤ u ≤ u ≤ R } (a rectangle domain in R ) with some given constant R >
0. For simplicity ofcomputation, we here assume
R > M >
0, where M is refined later. Then,the wealth equation (5) reduces to dX πt = X πt π t (cid:0) θ ( V t ) dt + dW t (cid:1) with θ ( V t ) = b ( V t ) /σ ( V t ) , and the driver of (32) takes the form G ( v, z , z ) = − δ | z | − θ ( v ) z + (cid:16) δ − δ z − δ z − θ ( v ) (cid:17) z I { z ≥ } + ( 12 z + R ) z I { z < } . (44)Then, from Theorem 9, we have the following result. Proposition 14
Suppose that ( Y ( t ) , Z ( t ) , Z ( t ) , λ ) = ( y ( V t ) , z ( V t ) , z ( V t ) , λ ) is the Markovian solution of ergodic BSDE (32) with the driver (44) . Then, theprocess U ( x, t ) given by U ( x, t ) = x δ δ e y ( V t ) − λt , s a power robust forward performance process. Moreover, the optimal portfolioweights and worst-case scenario parameters are given by π ∗ ( t ) = − δ { Z ( t ) + Z ( t ) · I { Z ( t ) ≥ } } , π ∗ ( t ) = 0 ,u ∗ ( t ) = − n θ ( V t ) + 1 δ Z ( t ) o − − δδ Z ( t ) · I { Z ( t ) ≥ } ,u ∗ ( t ) = − n θ ( V t ) + 1 δ Z ( t ) + 1 − δδ Z ( t ) o · I { Z ( t ) ≥ } + R · I { Z ( t ) < } . The optimal portfolio weight strategies for each scenario u ∈ U and the worstcase scenario strategies for each investment weight π ∈ ˜Π are given as follows α ∗ ( t, u ( t )) = 11 − δ [ θ ( V t ) + Z ( t ) + u ( t )] , α ∗ ( t, u ( t )) = 0 ,β ∗ ( t, π ( t )) = (cid:26) u ∗ ( t ) , if π ( t ) = π ∗ ( t ) , − R · sgn ( a ( t )) , otherwise, β ∗ ( t, π ( t )) = (cid:26) u ∗ ( t ) , if π ( t ) = π ∗ ( t ) , − R · sgn ( a ( t )) · I { Z ( t ) ≥ } + R · I { Z ( t ) < } , otherwise,where a ( t ) := δπ ( t ) + Z ( t ) + Z ( t ) · I { Z ( t ) ≥ } . From the boundedness of the functions z ( · ) and z ( · ), we know the scenarioparameter u ∗ ( t ) shown in the above Proposition is bounded and we denote by M its bound.When the stock price is not affected by the stochastic factor, i.e., the coefficients b and σ in (43) are constants, the processes Z and Z in the solution of theergodic BSDE (32) will equal to 0. Then, from Proposition 14, it is easy tocheck that the worst-case scenario parameters u ∗ and u ∗ will choose the valuesclosest to − θ (= − bσ ) for any given R > π ∗ shows that there will be no investment action into the stock. However, once thestochastic factor influences the price of stock, there will be a nontrivial robustinvestment opportunity π ∗ as shown in our result. In addition, we observethat the sign of z ( V t )(= Z ( t )) has an important impact on the the worst-casescenario, albeit not shown explicitly in the form of the power robust forwardperformance process U ( x, t ). It seems interesting to observe that the sign of z ( V t )(= Z ( t )) only affects the worst-case scenario strategies β ∗ and β ∗ , not tothe optimal investment policy strategies α ∗ and α ∗ responding to each scenario.A similar situation occurs if one consider a general compact and convex subset U ⊂ R (e.g. U = { ( u , u ) : − R ≤ u i ≤ R, i = 1 , } ); the only differenceis that for this general case the form of worst-case scenario parameters dependalso on the sign of some process involving in z ( V t ). Therefore, one may deducethat the Z ’s part of the solution of ergodic BSDE (32) carries on the importantinformation on the worst-case scenario.We remark that the above incomplete market model with uncertainty has alsobeen studied in [21] in the framework of classical robust expected utility. They18ive an explicit PDE characterization for the lower value function of a robustutility maximization problem combining the duality approach and the stochasticcontrol approach. On the other hand, when we do not consider the modeluncertainty, the above model will reduce to the case that has been studied in[30] (Section 3.1.3 therein). We establish the connection between the constant λ appearing in the solutionof ergodic BSDE (32) and a zero-sum risk-sensitive stochastic differential gameover the infinite horizon with ergodic payoff criteria. It turns out the constant λ is the value of the zero-sum risk-sensitive game problem. Thus, we providea new interpretation for the value of the zero-sum risk-sensitive game problemsassociated with the forward processes. We first give the comparison theoremfor ergodic BSDE (32). Lemma 15
Suppose that G i , i = 1 , , satisfy the following conditions | G i ( v, z ) − G i (¯ v, z ) | ≤ C (1 + | z | ) · | v − ¯ v | , | G i ( v, z ) − G i ( v, ¯ z ) | ≤ C (1 + | z | + | ¯ z | ) · | z − ¯ z | , | G i ( v, | ≤ C. (45) For i = 1 , , let ( Y i , Z i , λ i ) be the unique Markovian solution of the ergodicBSDE (32) with driver G i ( v, z ) . If G ( v, z ) ≥ G ( v, z ) , then we have λ ≥ λ . Proof.
Denote γ t = ( G ( V t ,Z t ) − G ( V t ,Z t ) | Z t − Z t | ( Z t − Z t ) , if Z t = Z t , , otherwise.Then, from the boundedness of Z and Z , we know γ is a bounded process.We define the probability measure Q as follows dQdP (cid:12)(cid:12)(cid:12) F T = E ( Z T γ t dW t ) . Using the notations ˆ Y = Y − Y , ˆ Z = Z − Z , ˆ λ = λ − λ , we getˆ Y − ˆ Y T = Z T G ( V t , Z t ) − G ( V t , Z t ) + γ Tt ˆ Z t dt − ˆ λT − Z T ˆ Z Tt dW t = Z T G ( V t , Z t ) − G ( V t , Z t ) dt − ˆ λT − Z T ˆ Z Tt dW Qt , W Q defined via dW Qt = − γ t dt + dW t is a Brownian motion under theprobability measure Q . Therefore, we get1 T E Q [ ˆ Y − ˆ Y T ] + ˆ λ = 1 T E Q [ Z T G ( V t , Z t ) − G ( V t , Z t ) dt ] . (46)Notice that there exist mappings y i , i = 1 , , such that Y it = y i ( V t ) , i = 1 , y i , i = 1 , , are of linear growth, there exists a constant C independent of T such that E Q | ˆ Y T | ≤ C (1 + E Q | V T | ) ≤ C, (47)where the last inequality is derived from the dissipative condition (4). It followsfrom (46) and G ( v, z ) ≥ G ( v, z ) thatˆ λ = lim sup T →∞ T E Q [ Z T G ( V t , Z t ) − G ( V t , Z t ) dt ] ≥ , which completes the proof. Theorem 16
For any ( π, u ) ∈ ˜Π × U with feedback forms, i.e. ( π s , u s ) =( π ( V s ) , u ( V s )) for some Borel measurable mappings ( π ( · ) , u ( · )) , we define thefunctional L ( v, π s , u s ) := − δ (1 − δ ) | π s | + δπ Ts [ θ ( V s ) + u s ] , s ≥ , and the probability measure P π,u as follows d P π,u d P (cid:12)(cid:12)(cid:12)(cid:12) F t = E (cid:18)Z t ( δπ Tr + u Tr ) dW r (cid:19) . (48) Let ( y ( V t ) , z ( V t ) , λ ) , t ≥ , be the unique Markovian solution of the ergodicBSDE (32), and X π solve the wealth equation (5). Furthermore, if the set Π isalso assumed to be bounded, then λ is the value of the associated risk-sensitivegame problem, namely, λ = inf u ∈U sup π ∈ ˜Π lim sup T ↑∞ T ln E P π,u (cid:16) e R T L ( V s ,π s ,u s ) ds (cid:17) = sup π ∈ ˜Π inf u ∈U lim sup T ↑∞ T ln E P π,u (cid:16) e R T L ( V s ,π s ,u s ) ds (cid:17) . (49) Moreover, the supremum and infimum in (49) can be attainable by choosing π ∗ and u ∗ as in (36) . Proof.
From (18), we have | F ( v, z, π, u ) − F (¯ v, z, π, u ) | ≤ C | π | · | v − ¯ v | , | F ( v, z, π, u ) − F ( v, ¯ z, π, u ) | ≤ C (1 + | π | + | z | + | ¯ z | ) · | z − ¯ v | , | F ( v, , π, u ) | ≤ C | π | + C | π | . (50)20hen, similar to the proof of Lemma 8, from (33) and (50), the following twoergodic BSDEs dY ut = ( − sup π t ∈ Π F ( V t , Z ut , π t , u t ) + λ u ) dt + ( Z ut ) T dW t ,dY πt = ( − inf u t ∈ U F ( V t , Z πt , π t , u t ) + λ π ) dt + ( Z πt ) T dW t , (51)have unique Markovian solutions ( Y u , Z u , λ u ) and ( Y π , Z π , λ π ), respectively,for each ( u, π ) ∈ U × ˜Π with feedback forms. Step 1.
We first show that λ = inf u ∈U λ u = sup π ∈ ˜Π λ π . (52)Since inf u t ∈ U F ( V t , Z t , π t , u t ) ≤ G ( V t , Z t ) ≤ sup π t ∈ Π F ( V t , Z t , π t , u t ) , Lemma 15 then implies that λ π ≤ λ ≤ λ u , for all ( π, u ) ∈ ˜Π × U with feedback forms . (53)On the other hand, from the uniqueness of the solution of ergodic BSDE (32),we know λ = λ u ∗ = λ π ∗ with u ∗ and π ∗ given in (36). Thus, we have established(52). Step 2.
We show that, for each ( u, π ) ∈ U × ˜Π with feedback forms, λ u = sup π ∈ ˜Π lim sup T ↑∞ T ln E P π,u (cid:16) e R T L ( V s ,π s ,u s ) ds (cid:17) , (54) λ π = inf u ∈U lim sup T ↑∞ T ln E P π,u (cid:16) e R T L ( V s ,π s ,u s ) ds (cid:17) . (55)We only prove the relation (54), and the proof of (55) is analogous.For arbitrary but fixed u ∈ U , from (51) we get, for every ˜ π ∈ ˜Π, dY ut = ( − sup π t ∈ Π F ( V t , Z ut , π t , u t ) + λ u ) dt + ( Z ut ) T dW t = (cid:16) − sup π t ∈ Π F ( V t , Z ut , π t , u t ) + λ u + ( Z ut ) T ( δ ˜ π t + u t ) (cid:17) dt + ( Z ut ) T dW ˜ π,ut , (56)where W ˜ π,u defined via dW ˜ π,u = − ( δ ˜ π t + u t ) dt + dW t is a Brownian motionunder probability measure P ˜ π,u (see (48)). We observe that the function F (see(18)) in (56) can be written as F ( V t , Z ut , π t , u t ) = L ( V t , π t , u t ) + ( Z ut ) T ( δπ t + u t ) + 12 | Z ut | . Y u − Y uT + λ u T = Z T sup π t ∈ ˜Π (cid:0) L ( V t , π t , u t ) + ( Z ut ) T δπ t (cid:1) − ( Z ut ) T δ ˜ π t + 12 | Z ut | dt − Z T ( Z ut ) T dW ˜ π,ut , which follows that, for arbitrary ˜ π ∈ ˜Π, e λ u T + Y u e − Y uT E (cid:16) Z T ( Z ut ) T dW ˜ π,ut (cid:17) = exp (cid:16) Z T sup π t ∈ ˜Π (cid:0) L ( V t , π t , u t ) + ( Z ut ) T δπ t (cid:1) − L ( V t , ˜ π t , u t ) − ( Z ut ) T δ ˜ π t dt (cid:17) · e R T L ( V t , ˜ π t ,u t ) dt ≥ e R T L ( V t , ˜ π t ,u t ) dt . Then, we obtain e λ u T + Y u E P ˜ π,u h e − Y uT E (cid:16) Z T ( Z ut ) T dW ˜ π,ut (cid:17)i ≥ E P ˜ π,u h e R T L ( V t , ˜ π t ,u t ) dt i . (57)We define the probability measure Q ˜ π,u as follows dQ ˜ π,u d P (cid:12)(cid:12)(cid:12)(cid:12) F t = E (cid:18)Z t ( δ ˜ π r + u r + Z ur ) T dW r (cid:19) . Using the measure Q ˜ π,u , from (57) we get e λ u T + Y u E Q ˜ π,u h e − Y uT i ≥ E P ˜ π,u h e R T L ( V t , ˜ π t ,u t ) dt i . Thus, it holds λ u + Y u T + 1 T ln E Q ˜ π,u h e − Y uT i ≥ T ln E P ˜ π,u h e R T L ( V t , ˜ π t ,u t ) dt i . (58)Similar to the proof of estimate (47), from the boundedness of Π and Jensen’sinequality, there exists a constant C independent of T such that1 C ≤ e − E Q ˜ π,u Y uT ≤ E Q ˜ π,u (cid:16) e − Y uT (cid:17) ≤ C, (59)where the last inequality is obtained using Lemma 3.1 in [10]. It follows from(58) and (59) that, for any ˜ π ∈ ˜Π, λ u ≥ lim sup T ↑∞ T ln E P ˜ π,u h e R T L ( V t , ˜ π t ,u t ) dt i . with equality choosing ˜ π t = π ∗ t , where π ∗ t is given in (36). Step 3.
Finally, we readily obtain (49) from (52) in Step 1 and (54) and (55) inStep 2. 22 emark 17
Notice that E P π,u (cid:16) e R T L ( V s ,π s ,u s ) ds (cid:17) = E P u (cid:16) e R T − δ | π s | + δπ Ts θ ( V s ) ds + R T δπ Ts dW s (cid:17) = E P u [ ( X πT ) δ δ ] · δx δ , then, from (49) , it is easy to check that λ is also the value for the followinggame problem λ = inf u ∈U sup π ∈ ˜Π lim sup T ↑∞ T ln E P u (cid:20) ( X πT ) δ δ (cid:21) = sup π ∈ ˜Π inf u ∈U lim sup T ↑∞ T ln E P u (cid:20) ( X πT ) δ δ (cid:21) . Therefore, Theorem 16 can be viewed as an optimal investment model in whichthe goal is to maximize the long-term growth rate of expected utility of wealthwith model uncertainty. A similar problem has been treated in [27] using theduality method.
We establish the link between the power robust forward process U ( x, t ) and thelong-time behaviour of the lower value function of the classical power robustexpected utility. For the latter, let [0 , T ] be an arbitrary trading horizon andwe introduce the lower value function as follows w T ( x, v ) = sup π ∈ Π [0 ,T ] inf u ∈U [0 ,T ] E P u (cid:20) ( X πT ) δ δ | X π = x, V = v (cid:21) , ( x, v ) ∈ R + × R d , (60)where the wealth process X πs , s ∈ [0 , T ] , solving (5) with X π = x , the stochasticfactor process V s , s ∈ [0 , T ], solving (2) with V = v , and u ∈ U [0 ,T ] implies that u belongs to U and is restricted to the time horizon [0 , T ].We recall that the optimal investment problem for the classical robust expectedutility has been considered in [6] via the stochastic control approach basedon BSDE, in [39] via the duality approach, and in [21] combining these twomethods. Proposition 18
Let U ( x, t ) = x δ δ e y ( V t ) − λt be the power robust forward perfor-mance process as in (35). Then, there exists a constant L ∈ R , independent ofthe initial states X π = x and V = v , such that, for ( x, v ) ∈ R + × R d , lim T ↑∞ w T ( x, v ) e − λT − L U ( x,
0) = 1 . roof. Since the maxmin problem (60) is standard in the literature (see, forexample, [43]), we only demonstrate its main steps briefly. To this end, for each π ∈ Π [0 ,T ] and u ∈ U [0 ,T ] , we introduce the objective functional w T ( x, v, π, u ) = E P u (cid:20) ( X πT ) δ δ | X π = x, V = v (cid:21) . We aim to find a saddle point ( π ∗ , u ∗ ) ∈ Π [0 ,T ] × U [0 ,T ] such that w T ( x, v, π, u ∗ ) ≤ w T ( x, v, π ∗ , u ∗ ) ≤ w T ( x, v, π ∗ , u ) . Then, it is clear that w T ( x, v ) = w T ( x, v, π ∗ , u ∗ ). We claim that w T ( x, v ) = x δ δ e ¯ Y , (61) π ∗ t = π ∗ ( V t , ¯ Z t ) , u ∗ t = u ∗ ( V t , ¯ Z t ) , t ∈ [0 , T ] , (62)with the mappings ( π ∗ , u ∗ ) given in (23) and (21), and ( ¯ Y , ¯ Z ) being the uniquesolution of the following BSDE¯ Y t = Z Tt G ( V r , ¯ Z r ) dr − Z Tt (cid:0) ¯ Z r (cid:1) T dW r , (63)where the driver G is defined in (17). The proof follows along similar argumentsas in Proposition 6 and Theorem 9, and thus omitted.From Theorem 4.4 in [25], there exists a constant L ∈ R such thatlim T ↑∞ ( ¯ Y − λT − Y ) = L, (64)where ( Y, Z, λ ) is the solution of ergodic BSDE (32). Finally, from (35), (61)and (64) we have lim T ↑∞ w T ( x, v ) e − λT − L U ( x,
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