Robust utility maximization in non-dominated models with 2BSDEs

Robust Utility Maximization in Non-dominated Models with 2BSDEs∗

arXiv:1201.0769v4 [math.PR] 8 Feb 2012

Anis Matoussi†

Dylan Possamai‡

Chao Zhou§

January 26, 2012

Abstract In this article, we consider the problem of robust utility maximization in an incomplete market with volatility uncertainty. The set of all possible models (probability measures) considered here is non-dominated. We propose to study this problem in the framework of second order backward stochastic differential equations with quadratic growth generator. We solve the problem for exponential, power and logarithmic utility functions and prove existence of an optimal strategy. Finally we provide several examples which shed more light on the problem and its links with the classical utility maximization one.

Key words: Second order backward stochastic differential equation, quadratic growth, robust utility maximization, volatility uncertainty. AMS 2000 subject classifications: 60H10, 60H30

∗

Research supported by the Chair Financial Risks of the Risk Foundation sponsored by Soci´et´e G´en´erale, the Chair Derivatives of the Future sponsored by the F´ed´eration Bancaire Fran¸caise, and the Chair Finance and Sustainable Development sponsored by EDF and Calyon. † CMAP, Ecole Polytechnique, Paris, and Universit´e du Maine, Le Mans, [email protected]. ‡ CMAP, Ecole Polytechnique, Paris, [email protected]. § CMAP, Ecole Polytechnique, Paris, [email protected].

1

1

Introduction

Backward stochastic differential equations (BSDEs for short) were first introduced by Bismut [3] in the linear case, then generalized by Pardoux and Peng [33] to Lipschitz generator. Applications can be found in many domains such as finance, stochastic controls, stochastic diffirential games. On a filtered probability space (Ω, F, {Ft }0≤t≤T , P) generated by an Rd -valued Brownian motion B, a solution to a BSDE consists on finding a pair of progressively measurable processes (Y, Z) such that Z Z T

T

Zs dBs , t ∈ [0, T ], P − a.s.

fs (Ys , Zs )ds −

Yt = ξ +

t

t

where f (also called the driver) is a progressively measurable function and ξ is an FT -measurable random variable. Pardoux and Peng proved existence and uniqueness of the above BSDE provided that the function f is uniformly Lipschitz in y and z and that ξ and fs (0, 0) are square integrable. Since their pioneering work, many efforts have been made to relax the assumptions on the driver f ; for instance, Lepeltier and San Martin [30] have proved the existence of a solution when f is only continuous in (y, z) with linear growth, and Kobylanski [28] and Tevzadze [47] respectively obtained the existence and uniqueness of a solution when f is continuous and has quadratic growth in z and the terminal condition ξ is bounded. More recently, motivated by applications in financial mathematics and probabilistic numerical methods for PDEs (see [7], [21], [37] and [42]), Cheredito, Soner, Touzi and Victoir [8] introduced the notion of Second order BSDEs (2BSDEs), which are connected to the larger class of fully nonlinear PDEs. Then, Soner, Touzi and Zhang [43] provided a complete theory of existence and uniqueness for 2BSDEs under uniform Lipschitz conditions similar to those of Pardoux and Peng. Their key idea was to reinforce the condition that the 2BSDE must hold P − a.s. for every probability measure P in a non-dominated class of mutally singular measures (see Section (2) for precise definitions). The BSDE theory finds one of its application in the problem of utility maximization which can be formulated as follows V ξ (x) := sup inf EQ [U (XTπ − ξ)], π∈A˜ Q∈P

where A˜ is a given set of admissible trading strategies, P is the set of all considered possible probability measures, U is a utility function, XTπ is the liquidation value of a trading strategy π with positive initial capital X0π = x and ξ is a terminal liability, equal to 0 if U is only defined on R+ . In the standard problem of utility maximization, P contains only one probability measure P. This means that the investor knows the ”historical” probability P that describes the dynamics of the state process. In reality, the investor may have some uncertainty on this probability, which means that there can be several objective probability measures in P. In this case, we call the problem robust utility maximization. Many authors introduce a set of probability measures which is absolutely continuous with respect to a reference probability measure P. This is the case if we only take into account drift uncertainty. However, if we want to work in the framework of the uncertain volatility model introduced by Avellaneda, Levy and Paras. [2] and Lyons [31], the set of probability measures becomes non-dominated. 2

The usual approach for the standard utility maximization problem is due to Von Neumann and Morgenstern [48]. In the seminal paper [32], Merton was the first to study the problem of portfolio selection with utility maximization by stochastic optimal control techniques. Then in [29], Kramkov and Schachermayer studied the problem of utility maximization in a general semimartingale model by means of duality theory. Later, El Karoui and Rouge [16] considered the indifference pricing problem with exponential utility. They assumed that the admissible trading strategies set was closed and convex, and showed that the solution is related to a standard BSDE with quadratic growth. Following their ideas, Hu, Imkeller and M¨ uller, in [25], used a similar approach to extend their results to power and logarithmic utility functions. Moreover, they considered a set of admissible strategies which is only closed. In that case, the maximization problem was also found to be related to BSDEs with quadratic generator. In a more recent paper [27], Jeanblanc, Matoussi and Ngoupeyou studied the indifference price of an unbounded claim in an incomplete jump-diffiusion model by considering the risk aversion represented by an exponential utility function. Using the dynamic programming equation, they found that the price of an unbounded credit derivatives was solution of a quadratic BSDE with jumps. The problem of robust utility maximization with dominated models has been introduced by Gilboa and Schmeidler [22]. Anderson, Hansen and Sargent [1] and Hansen et al. [24] introduced and discussed the basic problem of robust utility maximization penalized by a relative entropy term of the model uncertainty Q ∈ P with respect to a given reference probability measure P0 . Inspired by these latter works, Bordigoni, Matoussi and Schweizer [6] solved the robust problem (the minimization part) in more general semimartingale framework by using stochastic control techniques and proved that the solution was related to a quadratic semimartingale BSDE. Then, more recently, Faidi, Matoussi and Mnif [19] proved a dynamic maximum principle for the maximization part of the same robust problem. See also Jeanblanc, Matoussi and Ngoupeyou [26] for a extension of these latter works in the case of jump models. Some results in the robust maximization problem have also been obtained in Gundel [23] , Quenez [38], Schied [39], Schied and Wu [40], Skiadas [41] in the case of continuous filtrations. The overall approach relies essentially on convex duality ideas. Robust utility maximization with non-dominated models, encompassing the case of the UVM model, has been studied for the first time by Denis and Kervarec [12] (see also [46] where this problem is analyzed under the framework of stochastic games and Hamilton-Jacobi-BellmanIsaacs equations). In this article, they first establish a duality theory for robust utility maximization and then show that there exists a least favorable probability. They also take into account uncertainty about the drift. The utility function U in their framework is supposed to be bounded and to satisfy Inada conditions. More recently, in [18], Epstein and Ji formulate a model of utility for a continuous-time framework that captures the decision-maker’s concern with ambiguity or model uncertainty, even though they do not study the maximization problem of robust utility per se. In the present framework, we study robust utility maximization with non-dominated models via 2BSDEs techniques. For this purpose, we recall the 2BSDEs framework in Section 2. Largely inspired by [16] and [25], in Sections 3, 4, 5 and 6 we solve the problem for robust exponential

3

utility, robust power utility and robust logarithmic utility, which, unlike in [12], are not bounded. In Section 7, we give some examples where we can explicitly solve the robust utility maximization problems by finding the solution of the associated 2BSDEs, and we provide some intuitions and comparisons with the classical dynamic programming approach adopted in the seminal work of Merton [32].

2

Preliminaries

We will start by recalling some notations and notions related to the theory of 2BSDEs, those objects being the main tool in our approach to the robust utility maximization problem.

2.1

Probability spaces

 Let Ω := ω ∈ C([0, 1])d : ω(0) = 0 be the canonical space, B the canonical process, and F the filtration generated by B. Let P0 be the Wiener measure. As recalled in [43], we can construct the quadratic variation of B and its density b a pathwise. Let P W denote the set of all local martingale measures P such that

hBit is absolutely continuous in t and b a takes values in S>0 d , P − a.s.

(2.1)

where S>0 d denotes the space of all d × d real valued positive definite matrices.

As in the paper [43], we concentrate on the subclass P S ⊂ P W consisting of all probability measures Pα := P0 ◦ (X α )−1 where Xtα :=

Z

0

t

α1/2 s dBs , t ∈ [0, 1], P0 − a.s.

for some F-progressively measurable process α taking values in S>0 d with

RT 0

(2.2)

|αt |dt < +∞, P0 −a.s.

Finally, we fix a ≤ a ∈ S>0 d and we define the class:  PH := P ∈ P S s.t. a ≤ b at ≤ a, dt × P − a.e. .

This reduces to a particular case of [43] where the two bounds on b a are independent of probability measures. Throughout this paper we assume that PH is not empty.

Definition 2.1. We say a property holds PH -quasi-surely (PH -q.s. for short) if it holds P-a.s. for all P ∈ PH .

2.2

Spaces and Norms

We now recall from Possama¨ı and Zhou [36] the spaces and norms which will be needed for the formulation of the quadratic second order BSDEs. L∞ H is the space of random variables which are bounded quasi-surely endowed with the norm kξkL∞ := sup kξkL∞ (P) . H

P∈PH

4

For p ≥ 1, HpH denotes the space of all F+ -progressively measurable Rd -valued processes Z with kZkpHp := sup EP H

P∈PH

" Z

T 0

1/2

|b at Zt |2 dt

 p2 #

< +∞.

BMO(PH ) denotes the space of all F+ -progressively measurable Rd -valued processes Z with

Z .



kZkBMO(PH ) := sup Zs dBs < +∞,

P∈PH

0

BMO(P)

where k·kBMO(P) is the usual BMO(P) norm under P.

+ D∞ H denotes the space of all F -progressively measurable R-valued processes Y with

PH − q.s. c` adl` ag paths, and kY kD∞ := sup kYt kL∞ < +∞. H

0≤t≤T

H

Finally, we denote by UCb (Ω) the collection of all bounded and uniformly continuous maps ξ : Ω → R with respect to the k·k∞ -norm, and we let L∞ H := the closure of UCb (Ω) under the norm k·kL∞ . H

2.3

The quadratic generator

We consider a map Ht (ω, z, γ) : [0, T ] × Ω × Rd × DH → R, where DH ⊂ Rd×d is a given subset containing 0. Define the corresponding conjugate of H w.r.t. γ by   1 Tr(aγ) − Ht (ω, z, γ) for a ∈ Sd>0 Ft (ω, z, a) := sup 2 γ∈DH Fbt (z) := Ft (z, b at ) and Fbt0 := Fbt (0).

We denote by DFt (z) the domain of F in a for a fixed (t, ω, z). As in [36], the generator F is supposed to verify either Assumption 2.1.

(i) The domain DFt (y,z) = DFt is independent of (ω, y, z).

(ii) For fixed (y, z, γ), F is F-progressively measurable. (iii) F is uniformly continuous in ω for the || · ||∞ norm. (iv) F is continuous in z and has the following growth property. There exists (α, β, γ) ∈ R+ × R+ × R∗+ such that γ 1/2 2 |Ft (ω, y, z, a)| ≤ α + β |y| + a z , for all (t, y, z, ω, a). 2 5

(v) F is C 1 in y and C 2 in z, and there are constants r and θ such that for all (t, ω, y, z, a), |Dy Ft (ω, y, z, a)| ≤ r, |Dz Ft (ω, y, z, a)| ≤ r + θ a1/2 z , 2 |Dzz Ft (ω, y, z, a)| ≤ θ.

or (i) The domain DFt (y,z) = DFt is independent of (ω, y, z).

Assumption 2.2.

(ii) For fixed (y, z, γ), F is F-progressively measurable. (iii) F is uniformly continuous in ω for the || · ||∞ norm. (iv) F is continuous in z and has the following growth property. There exists (α, β, γ) ∈ R+ × R+ × R∗+ such that 2 γ |Ft (ω, y, z, a)| ≤ α + β |y| + a1/2 z , for all (t, y, z, ω, a). 2 (v) We have the following ”local Lipschitz” assumption in z, ∃µ > 0 and a progressively mea′ surable process φ ∈ BMO(PH ) such that for all (t, y, z, z , ω, a),   ′ ′ ′ ′ Ft (ω, y, z, a) − Ft (ω, y, z , a) − φt .a1/2 (z − z ) ≤ µa1/2 z − z a1/2 z + a1/2 z .

(vi) We have the following uniform Lipschitz-type property in y ′ ′ ′ Ft (ω, y, z, a) − Ft (ω, y , z, a) ≤ C y − y , for all (y, y , z, t, ω, a).

We recall that Assumption 2.2 is weaker than Assumption 2.1, but is sufficient to have existence of the quadratic 2BSDE defined below only if the norm of the terminal condition ξ is small enough. Notice that this will always be the case with power and logarithmic utilities for which the terminal condition of the 2BSDE will be 0.

2.4

Quadratic 2BSDE

In the sequel we will have to deal with the following type of 2BSDEs Yt = ξ −

Z

T

t

Fb (Ys , Zs )ds −

Z

T

Zs dBs + KT − Kt , 0 ≤ t ≤ T, PH − q.s.

(2.3)

t

∞ 2 Definition 2.2. Given ξ ∈ L∞ H , we say (Y, Z) ∈ DH × HH is a solution to 2BSDE (2.3) if

• YT = ξ, PH − q.s. • For each P ∈ PH , the process K P defined below has nondecreasing paths, P-a.s.: KtP

:= Y0 − Yt +

Z

t 0

Fbs (Ys , Zs )ds + 6

Z

0

t

Zs dBs 0 ≤ t ≤ T, P − a.s.

(2.4)

• The familly of processes condition:



K P , P ∈ PH

′



defined in (2.4) satisfies the following minimum

′

KtP = ess inf P EtP [KTP ], P − a.s. for all P ∈ PH , t ∈ [0, T ]. P′ ∈PH (t+,P)

(2.5)

 Moreover, if the family K P , P ∈ PH can be aggregated into a universal process K, we call (Y, Z, K) a solution of 2BSDE (2.3). We recall here one of the results proved in [36] Theorem 2.1. Let ξ ∈ L∞ H . Under Assumption 2.1 or Assumption 2.2 with the addition that 2 the norm of ξ is small enough, there exists a unique solution (Y, Z) ∈ D∞ H × HH of the 2BSDE (2.3).

3

Robust utility maximization

We will now present the main problem of this paper and introduce a financial market with volatility uncertainty. The financial market consists of one bond with interest rate zero and d stocks. The price process is given by dSt = diag [St ] (bt dt + dBt ), PH − q.s. where b is an Rd -valued uniformly bounded stochastic process which is uniformly continuous in ω for the || · ||∞ norm. Remark 3.1. The volatility is implicitly embedded in the model. Indeed, under each P ∈ PH , 1/2 a1/2 plays the we have dBs ≡ b at dWtP where W P is a Brownian motion under P. Therefore, b role of volatility under each P and thus allows us to model the volatility uncertainty. We also note that we make the uniform continuity assumption for b to ensure that the 2BSDE obtained later satisfies Assumptions 2.1 or 2.2. We then denote π = (πt )0≤t≤T a trading strategy, which is a d-dimensional F-progressively ˜ In the following sections, measurable process, supposed to take its value in some closed set A. ˜ for each of the three utility we will define the set of admissible trading strategies, denoted by A, functions. The process πti describes the amount of money invested in stock i at time t, with 1 ≤ i ≤ d. πi The number of shares is Sti . So the liquidation value of a trading strategy π with positive initial t capital x is given by the following wealth process: Z t πs (dBs + bs ds), 0 ≤ t ≤ T, PH − q.s. Xtπ = x + 0

The problem of the investor in this financial market is to maximize his expected utility under model uncertainty from his total wealth XTπ − ξ where ξ is a liability at time T which is a 7

random variable assumed to be FT -measurable and in L∞ H . Then the value function V of the maximization problem can be written as V ξ (x) := sup inf EQ [U (XTπ − ξ)]. π∈A˜

Q∈PH

(3.1)

In the case where PH contains only one probability measure, the problem reduces to the classical utility maximization problem. Remark 3.2. Due to the construction of 2BSDE, we need the liability ξ to be in the class L∞ H . It is easy to see that ξ can be constant, deterministic or in the form of g(BT ) where g is a Lipschitz bounded function, such as a Put or a Call spread payoff function. However, we notice that vanilla options payoffs with underlying S may not be in L∞ H . Indeed, we have in the one-dimensioal framework Z T  1 bt dt − hBiT + BT , PH − q.s. ST = S0 exp 2 0

Since the quadratic variation of the canonical process can be written as follows 2 X  lim B i+1 (ω) − B , i (ω) n n n→+∞

i≤2n t

2

2

it is not too difficult to see that S can be approximated by a sequence of random variables in UCb (Ω). Besides, this sequence converges in L2H . However, we cannot be sure that it also converges in L∞ H , which is our space of interest here. Of course, in an uncertain volatility framework, this seems to be a major drawback. Nevertheless, to deal with these options, it suffices to redo the whole 2BSDE construction from scratch but taking exponential of the Brownian motion under the Wiener measure as the canonical process instead + of PH , of the Brownian motion itself. This would amount to restrict ourselves to the subset PH containing the local martingale measure which make the canonical process a positive continuous martingale. To find the value function V ξ and an optimal trading strategy π ∗ , we follow the ideas of the general martingale optimality principle approach as in [16] and [25], but adapt it here to a nonlinear framework. Let {Rπ } be a family of processes which satisfy the following properties Properties 3.1.

˜ (i) RTπ = U (XTπ − ξ) for all π ∈ A.

˜ (ii) R0π = R0 is constant for all π ∈ A. (iii) We have ′ ess inf P EPt [U (XTπ − ξ)] ≤ Rtπ , ∀π ∈ A˜

P′ ∈PH (t+ ,P)

∗ ′ ∗ ˜ P − a.s. for all P ∈ PH . Rtπ = ess inf P EPt [U (XTπ − ξ)] for some π ∗ ∈ A,

P′ ∈PH (t+ ,P)

8

Then it follows ∗

inf EP [U (XTπ − ξ)] ≤ R0 = inf EP [U (XTπ − ξ)] = V ξ (x).

P∈PH

P∈PH

(3.2)

In the following sections we will follow the ideas of Hu, Imkeller and M¨ uller [25] to construct such a family for our three utility functions.

4

Robust exponential utility

In this section, we will consider the exponential utility function which is defined as U (x) = −exp(−βx), x ∈ R for β > 0. In our context, the set of admissible trading strategies is defined as follows Definition 4.1. Let A˜ be a closed set in R1×d . The set of admissible trading strategies A˜ consists of all d-dimensional progressively measurable processes, π = (πt )0≤t≤T satisfying ˜ dt ⊗ PH − a.e. π ∈ BMO(PH ) and πt ∈ A, Remark 4.1. Many authors shed light on the natural link between BMO class, exponential uniformly integrable class and BSDEs with quadratic growth. See [5], [4] and [25] among others. In the standard utility maximization problem studied in [25], their trading strategies satisfy a uniform integrability assumption on the family (expXτπ )τ . Since the optimal strategy is a BMO martingale, it is easy to see that the utility maximization problem can also be solved if the uniform integrability assumption is replaced by a BMO assumption. However, at the end of the day, those two assumptions are deeply linked, as shown in the context of quadratic semimartingales in [4]. Nonetheless, in our framework, as explained below in Remark 4.3, we need to generalize the BMO martingale assumption instead of the uniform integrability assumption.

4.1

Characterization of the value function and existence of an optimal strategy

The investor wants to solve the maximization problem V ξ (x) := sup inf EQ [−exp (XTπ − ξ)] π∈A˜ Q∈PH

(4.1)

To construct Rπ , we set ˜ Rtπ = −exp(−β(Xtπ − Yt )), t ∈ [0, T ], π ∈ A. 2 where (Y, Z) ∈ D∞ H × HH the unique solution of the following 2BSDE with quadratic generator: Z T Z T Yt = ξ − Zs dBs − Fb(s, Zs )ds + KTP − KtP , P − a.s., ∀P ∈ PH . t

t

9

The generator Fb is chosen so that Rπ satisfies the Properties 3.1.

Remark 4.2. From Theorem 3.1 of [36], we have the following representation ′

Yt = ess supP ytP (T, ξ). P′ ∈PH (t+ ,P)

So that, in general, Y0 is only F0+ -measurable and therefore not a constant. But by Proposition 4.2 of [36], we know that the process Y is actually F-measurable (this is true when the teminal condition is in UCb (Ω) and by passing to the limit wheh the terminal condition is in L∞ H ). This and the above representation then imply easily that by the Blumenthal Zero-One law Y0 is a constant and ′ ′ Y0 = ess supP y0P (T, ξ) = sup y0P (T, ξ). P′ ∈PH

P′ ∈PH (0+ ,P)

Let us now define for all a ∈ S>0 d such that a ≤ a ≤ a the set Aa by n o e = a1/2 b : b ∈ A e . Aa := a1/2 A

e= The set Aa is still closed. Moreover, since A 6 ∅ and a ≤ a ≤ a, we have min {|r| , r ∈ Aa } ≤ k,

(4.2)

for some constant k independent of a. We can now state the main result of this section Theorem 4.1. Let Assumption 2.2, with the addition that the norm of ξ is small enough, or e is C 2 , hold. Then, the value function Assumption 2.1, with the addition that the closed domain A of the optimization problem (4.1) is given by V ξ (x) = −exp (−β (x − Y0 )) , 2 where Y0 is defined as the initial value of the unique solution (Y, Z) ∈ D∞ H × HH of the following 2BSDE Z T Z T (4.3) Zs dBs − Fbs (Zs )ds + KTP − KtP , P − a.s., ∀P ∈ PH . Yt = ξ − t

t

The generator is defined as follows

Fbt (ω, z, a) := Ft (ω, z, b at ),

where for all t ∈ [0, T ], z ∈ Rd and a ∈ S>0 d   β 1 ′ 1 2 1/2 Ft (ω, z, a) = − dist a z + θt (ω), Aa + z a1/2 θt (ω) + |θt (ω)|2 , for a ∈ Sd>0 , 2 β 2β were θt (ω) = a−1/2 bt (ω). 10

(4.4)

Moreover, there exists an optimal trading strategy π ∗ satisfying   1b 1/2 ∗ 1/2 b at πt ∈ ΠAabt b at Zt + θt , t ∈ [0, T ], PH − q.s. β

(4.5)

−1/2 bt . where θbt := b at

Proof. Step 1: We first show that the 2BSDE (4.3) has an unique solution. We need to verify that the generator Fb satisfies the conditions of Assumption 2.2 or 2.1. First of all, F defined above is a convex function of a, and thus F can be written as the Fenchel transform of a function   1 Tr(aγ) − Ft (ω, z, a) for γ ∈ Rd×d . Ht (ω, z, γ) := sup 2 a∈DF

That F satisfies the first two conditions of either Assumption 2.2 or 2.1 is obvious. For Assumptions 2.2(iii) and 2.1(iii), the assumption of boundedness and uniform continuity in ω on b implies that b2 is uniformly continuous in ω. Since b and b2 are the only non-deterministic terms in F , then F is also uniformly continuous in ω. Then, since we consider the distance function to a closed set, we know that it is attained for some element. From this, it is clear that the generator of this 2BSDE is purely quadratic. Besides, as recalled earlier in (4.2), there exists a constant k ≥ 0 such that min {|d| : d ∈ Abat } ≤ k

for dt ⊗ P − a.e., for all P ∈ PH .

Then we get, for all z ∈ Rd , t ∈ [0, T ], 2    1 b 1b 1/2 2 1/2 dist b at z + θt , Abat ≤ 2 b at z + 2 θt + k . β β 2

Thus, we get from the boundedness of θb 1/2 2 b at z , Ft (z) ≤ c0 + c1 b

that is to say that Assumptions 2.2(iv) and 2.1(iv) are satisfied. Finally, Assumption 2.2(v) is clear from the Lipschitz property of the distance function, and e in that case. Assumption 2.1(v) is also clear by our regularity assumption on A

b The terminal condition ξ is in L∞ H and we have proved that the generator F satisfies Assumption 2.2 or Assumption 2.1, therefore Theorem 2.1 states that the 2BSDE (4.3) has a unique solution 2 in D∞ H × HH .

Step 2: We first decompose Rπ as the product of a process M π and a non-decreasing process ˜ Aπ that is constant for some π ∗ ∈ A.

11

Define for all P ∈ PH Mtπ

−β(x−Y0 )

=e

We can then write

 Z t  Z 2 1 t 2 1/2 P β(πs − Zs )dBs − β b as (πs − Zs ) ds − βKt , P − a.s. exp − 2 0 0 Rπ = M π Aπ ,

with Aπt and

= −exp

Z



t

v(s, ps , Zs )ds , 0

2 1 1/2 v(t, π, z) : = −βπbt + β Fbt (z) + β 2 b at (π − z) . 2

Clearly, we may rewrite v(t, πt , Zt ) in the following form   ′ 1/2 β 1/2 2 1b β 1/2 2 b 1 1/2 v(t, πt , Zt ) = at at πt − βπtb a Zt + Ft (Zt ) b at Zt + θt + b b β 2 β 2 t =

β 2

  2 1/2 1 b 2 ′ 1/2 1/2 bt − 1 θbt + Fbt (Zt ). b Z + π − b a a θ − Z θ b a t t t t t t t β 2β

By a classical measurable selection theorem (see [10] (chapitre III) or [14] or Lemma 3.1 in [15]), we can define a progressively measurable process π ∗ satisfying (4.5). Then, it follows from the definition of Fb that PH − q.s. ˜ • v(t, πt , Zt ) ≥ 0 for all π ∈ A.

• v(t, πt∗ , Zt ) = 0, which implies that Aπ is always non-increasing for all π and is equal to −1 for π ∗ . Step 3: In this step, we show that the processes Z · Z · πs∗ dBs , Zs dBs , 0

0

are BMO(PH ) martingales. First of all, by Lemma 2.1 in [36], we know that

R· 0

Zs dBs is a BMO(PH ) martingale.

By the triangle inequality and the definition of π ∗ together with (4.2), we have   1/2 ∗ 1/2 1 1 1/2 ∗ 1/2 at Zt + θbt at π t − b at πt ≤ b at Zt + θbt + b b β β 2 1/2 1/2 ≤ 2 b at Zt + θbt + k ≤ 2 b at Zt + k1 , β b where k1 is a bound on θ.

12

Then, for every probability P ∈ PH and every stopping time τ ≤ T , Z T Z T   b ∗ 2 1/2 2 P P 2 Eτ 8 b at Zt dt + 2T k1 , θt πt dt ≤ Eτ τ

and therefore

τ

Z ·

πs∗ dBs

0

Z ·



≤ 8 Zs dBs

0

BMO(PH )

+ 2T k12 .

BMO(PH )

R·

This implies the BMO(PH ) martingale property of

0

∗

πs∗ dBs as desired. ∗

Step 4: We then prove that π ∗ ∈ A˜ and Rπ ≡ −M π satisfies Property (iii) of 3.1, that is to say i h ∗ ∗ ′ ess supP EPt MTπ = Mtπ P − a.s. ∀P ∈ PH . P′ ∈PH (t,P)

For a fixed P′ ∈ PH (t, P), we denote Z t Z 2 1 t 2 1/2 ∗ ′ ∗ β(πs − Zs )dBs + Lt := β b as (πs − Zs ) ds + βKtP , 2 0 0

then with Itˆo’s formula, we obtain, thanks to the BMO(PH ) property proved in Step 3

′ EPt

h

∗ MTπ

i

−

∗ Mtπ

=

′ −βEPt

′

Z

T

t



∗ ′ Msπ− dKsP

X

+ EPt 

t≤s≤T

First, we prove ess inf

P

P′ ∈PH (t,P)

′ EPt

Z

T t





e−Ls − e−Ls− + e−Ls− (Ls − Ls− ) .

∗ ′ Msπ− dKsP



= 0, P − a.s.

For every t and every P′ ∈ PH (t, P), we have P′

0 ≤ Et

Z

T

π∗ s−

M t

P′

dKs



P′

≤ Et

"

π∗

sup Ms

t≤s≤T

!

#  ′  ′ KTP − KtP .

′

Besides, since K P is non-decreasing, we obtain for all s ≥ t  Z s  ∗ (Zu − πu∗ ) dBu Msπ ≤ e−β(x−Y0 ) E β t

Then, again thanks to Step 3, we know that Z t (Zs − πs∗ ) dBs ∈ BMO(PH ), 0

13

(4.6)

and thus the exponential martingale above is a uniformly integrable martingale for all P and is in LrH for some r > 1 (see Lemma 2.2 in [36]). Thus, by H¨older inequality,

′ EPt

Z

T

t

∗ ′ Msπ− dKsP



′ eβ(Y0 −x) EPt

≤

"

 Z sup E β

s

r

t≤s≤T

(Zu − t

πu∗ ) dBu

With Doob’s maximal inequality, we have "  Z s #1/r   Z ′ r P ∗ P′ sup E β Et (Zu − πu ) dBu ≤ CEt E r β t≤s≤T

t

# 1r

T

(Zu − t

EPt

′

h

′

KTP − KtP

πu∗ ) dBu

1/r

′

q i 1 q

.

< +∞.

where C is an universal constant that can change value from line to line. Then by the Cauchy-Schwarz inequality, we get  h ′ q i1/q h ′ i ′  ′ 2q−1  2q1 P P P P′ P′ P′ P P′ P′ Et KT − Kt KT − Kt ≤ C Et KT − Kt Et

!1  i 1 2q−1  2q  ′ h ′ ′ ′ ′ 2q KTP − KtP . EPt KTP − KtP ess supP Et P′

≤C

P′ ∈PH (t,P)

Arguing as in the proof of Theorem 3.1 in [36] we know that !1  2q−1  2q ′ ′ ′ KTP − KtP ess supP EPt < +∞. P′ ∈PH (t,P)

Hence, we obtain 0 ≤ ess inf P EPt

′

P′ ∈PH (t,P)

Z

T t

∗

Msπ− dKsP

′



≤ C ess inf P

which means ess inf

P

P′ ∈PH (t,P)

Finally, we have  Z P′  P ess inf Et P′ ∈PH (t,P)

≤ ess inf P EPt P′ ∈PH (t,P)

′

T t

Z P′

t

− ess inf P Et  P′ ∈PH (t,P)

≤ 0,

∗

′

Msπ− dKsP −

X

t≤s≤T

T

∗

Msπ− dKsP

X

t≤s≤T

′



′ EPt

P′ ∈PH (t,P)

Z

T t

 ′ h ′ i 1 ′ 2q KTP − KtP EPt = 0,

∗ ′ Msπ− dKsP



= 0.



exp(−βLs ) − exp(−βLs− ) + βexp(−βLs− )(Ls − Ls− ) 

exp(−βLs ) − exp(−βLs− ) + βexp(−βLs− )(Ls − Ls− )

14

because the function x → exp(−x) is convex and the jumps of L are positive. Hence, using (4.6), we have

i h ∗ ∗ ′ ess supP EPt MTπ − Mtπ ≥ 0.

P′ ∈PH (t,P) ∗

But by definition M π is the product of a martingale and a positive decreasing process and is therefore a supermartingale. This implies that i h ∗ ∗ ′ ess supP EPt MTπ − Mtπ = 0. P′ ∈PH (t,P) ∗

Finally, π ∗ is an admissible strategy, Rπ satisfies Property 3.1(iii) and ∗ R0π

=

inf E

P

P∈PH







−exp −β x +

= −exp (−β (x − Y0 )) .

Z

0

T

πs∗ (dBs

+ θs ds) − ξ



˜ Rπ satisfies Property (iii) of 3.1, that is, Step 5: Next we will show that for all π ∈ A, ′

ess inf P EPt [−exp(−β(XTπ − ξ))] ≤ Rtπ , P − a.s.

P′ ∈PH (t+ ,P)

˜ the process Since π ∈ A,

is in BMO(PH ). Then the process

Z

t

(Zs − πs ) dBs , 0

  Z N = exp (−β(x − Y0 )) E −β (πs − Zs ) dBs , π

0

is a uniformly integrable martingale under each P ∈ PH .

As in the previous steps, we write Rπ as Rtπ = Mtπ Aπt , where Aπ is a negative non-increasing process. We then have ′

′

ess inf P EPs [Mtπ Aπt ] ≤ ess inf P EPs [Mtπ Aπs ] , P − a.s. P′ ∈PH (s,P)

P′ ∈PH (s,P)

′

= ess supP EPs [Mtπ ] Aπs , P − a.s. P′ ∈PH (s,P)

∗

because Aπ is negative. By the same arguments as in Step 3 for M π , we have ′

ess supP EPs [Mtπ ] = Msπ , P − a.s.

P′ ∈PH (s,P)

Therefore the following inequality holds ′

ess inf P EPs [Rtπ ] ≤ Rsπ , P − a.s.

P′ ∈PH (s,P)

which ends the proof.

⊔ ⊓

15

Remark 4.3. We see here why it is essential in our context to have strong integrability assump∗ tions on the trading strategies. Indeed, in the proof of the above property for M π , the fact that the stochastic integral Z ·

0

πs∗ dBs ,

is in BMO(PH ) allowed us to control the moments of its stochastic exponential, which in turn allowed us to deduce from the minimal property for K P a similar minimal property for Z · ∗ Msπ dKsP . 0

This term is new when compared with the context of [25]. To deal with it, we have to impose the BMO(PH ) property. Le us note however that since the optimal strategy already has that property, we do not lose a lot by restricting the strategies. Remark 4.4. We note that our approach still works when there are no constraints on trading strategies. In this case, the 2BSDE related to the maximization problem has an uniformly Lipschitz generator, and we are in the context of complete markets.

4.2

A min-max property

By comparing the value function of our robust utility maximization problem and the one presented in [25] for standard utility maximization problem, we are able to prove a min-max property similar to the one proved by Denis and Kervarec in [12]. We observe that we were only able to prove this property after having solved the initial problem, unlike in the approach of [12]. Theorem 4.2. Under the previous assumptions on the probability measures set PH and the ˜ the following min-max property holds. admissible strategies set A, sup inf EP [RTπ ] = inf sup EP [RTπ ] = inf sup EP [RTπ ] ,

P∈PH π∈A˜

P∈PH

π∈A˜

P∈PH

π P ∈A˜P

e and such that the process where A˜P is the set consisting of trading strategies which are in A R t is in BMO(P). 0 πs dBs 0≤t≤T

Proof. First note that we have D := sup inf EP [RTπ ] ≤ inf sup EP [RTπ ] ≤ inf sup EP [RTπ ] =: C. π∈A˜P∈PH

P∈PH π∈A˜

P∈PH π P ∈A˜P

Indeed, the first inequality is obvious and the second one follows from the fact that for all P, A˜ ⊂ A˜P . It remains to prove that C ≤ D. By the previous sections, we know that D = −exp (−β (x − Y0 )) . 16

Moreover, we know from [36] that we have a representation for Y0 , Y0 = sup y0P , P∈PH

where y0P is the solution of the standard BSDE with the same generator Fb. On the other hand, we observe from [25] that i h   , C = inf −exp −β x − y0P P∈PH

implying that C = D.

4.3

⊔ ⊓

Indifference pricing via robust utility maximization

It has been shown in [16] that in a market model with constraints on the portfolios, if we define the indifference price for a claim Φ as the smallest number p such that 

 sup E −exp −β X x+p,π − Φ ≥ sup E [−exp (−βX x,π )] , π

π

where X x,π is the wealth associated with the portfolio π and initial value x, then this problem turns into the resolution of a BSDE with quadratic growth generator. In our framework of uncertain volatility, the problem of indifference pricing of a contingent claim φ boils down to solve the following equation in p V 0 (x) = V Φ (x + p). Thanks to our results, we know that if ψ ∈ L∞ H then the two sides of the above equality can be calculated by solving 2BSDEs. The price p can therefore be calculated as soon as we are able to solve the 2BSDEs (explicitely or numerically). We provide two examples in Section 7.

5

Robust power utility

In this section, we will consider the power utility function 1 U (x) = − x−γ , x > 0 γ > 0. γ Here we shall use a different notion of trading strategy: ρ = (ρi )i=1,...,d denotes the proportion of wealth invested in stock i. The number of shares of stock i is given by

ρit Xt . Sti

Then the wealth process is defined as Xtρ = x +

Z tX d Xsρ ρi,s 0 i=1

Si,s

dSi,s = x +

Z

17

t 0

Xsρ ρs (dBs + bs ds) , PH − q.s.

(5.1)

and the initial capital x is positive. The wealth process X ρ can be written as  Z t ρ ρs (dBs + bs ds) , t ∈ [0, T ] , PH − q.s. Xt = xE 0

˜ the wealth process X ρ is a local P-martingale bounded from below, hence, Then for every ρ ∈ A, a P-supermartingale, for all P ∈ PH . In the present setting, the set of admissible strategies is defined as follows Definition 5.1. The set of admissible trading strategies A˜ consists of all Rd -valued progressively measurable processes ρ = (ρt )0≤t≤T satisfying ˜ dt ⊗ PH − a.e. ρ ∈ BMO(PH ) and ρ ∈ A, We suppose that there is no liability (ξ = 0). Then the investor faces the maximization problem   V (x) = sup inf EP U (XTρ ) . (5.2) ρ∈A˜ P∈PH

In order to find the value function and an optimal strategy, we apply the same method as in the exponential utility case. We therefore have to construct a stochastic process Rρ with terminal value   Z T ρ ρ dSs Xs ρs RT = U x + . Ss 0

satisfying Properties 3.1.

Then the value function will be given by V (x) = R0 . Applying the utility function to the wealth process yields   Z t Z t Z 1 t 1/2 2 1 1 −γ ρ −γ γρs bs ds + − (Xt ) = − x exp − γρs dBs − (5.3) as ρs ds . γ b γ γ 2 0 0 0

This equation suggests the following choice  Z t  Z t Z 1 −γ 1 t 1/2 2 ρ Rt = − x exp − γρs bs ds + γρs dBs − γ b as ρs ds + Yt , γ 2 0 0 0 ∞,κ where (Y, Z) ∈ DH × H2,κ H is the unique solution of the following 2BSDE

Yt = 0 −

Z

t

T

Zs dBs −

Z

T t

Fbs (Zs )ds + KT − Kt , t ∈ [0, T ] PH − q.s.

(5.4)

In order to get Property 3.1 (iii) for Rρ , we have to construct Fbt (z) such that, for t ∈ [0, T ] 2 1 1/2 1 1/2 2 b ˜ at ρt − Ft (Zt ) ≤ − b (5.5) at (γρt − Zt ) for all ρ ∈ A, γρt bt − γ b 2 2 18

˜ This is equivalent to with equality for some ρ∗ ∈ A. 2 1/2 2 b   Z + θ γ a −b t t 1/2 t 1 1 1 1/2 2 1 1/2 b b Ft (Zt ) ≥ − γ (1 + γ) b −b at Zt + θt − + b a Zt . at ρt − 2 1+γ 2 1+γ 2 t

Hence, the appropriate choice for Fb is γ(1 + γ) dist2 Fbt (z) = − 2

1/2 −b at z

+ θbt , Abat 1+γ

!

+

2 1/2 γ −b at z + θbt 2(1 + γ)

1 1/2 2 + b a z , 2 t

(5.6)

and a candidate for the optimal strategy must satisfy   1 1/2 ∗ 1/2 b b at ρt ∈ ΠAabt (−b at Zt + θt ) , t ∈ [0, T ]. 1+γ

We summarize the result above in the following Theorem.

Theorem 5.1. Let Assumption 2.2 or Assumption 2.1 with the addition that the closed domain e is C 2 hold. Then, the value function of the optimization problem (5.2) is given by A 1 V (x) = − x−γ exp(Y0 ) for x > 0 γ

2 where Y0 is defined as the unique solution (Y, Z) ∈ D∞ H × HH of the quadratic 2BSDE

Yt = 0 −

Z

T

Zs dBs −

t

with γ(1 + γ) Fbt (z) = − dist2 2



Z

T t

Fbs (Zs )ds + KT − Kt , t ∈ [0, T ] PH − q.s.

1 1/2 (−b at z + θbt ), Abat 1+γ



+

2 1/2 γ −b at z + θbt 2(1 + γ)

Moreover, there exists an optimal trading strategy ρ∗ ∈ A˜ with the property   1 1/2 ∗ 1/2 b b at ρt ∈ ΠAabt (−b at Zt + θt ) , t ∈ [0, T ]. 1+γ

+

(5.7)

1 1/2 2 a z . b 2 t (5.8)

Proof. The proof is very similar to the case of robust exponential utility. First we can show, with the same arguments, that the generator Fb satisfies the conditions of Assumption 2.1 or Assumption 2.2, hence there exists a unique solution to the 2BSDE (5.7).

Let then ρ∗ denote the progressively measurable process, constructed with a measurable selection theorem, which realizes the distance in the definition of Fb. The same arguments as in the case ˜ of robust exponential utility show that ρ∗ ∈ A. 19

Then with the choice we made for Fb, we have the following multiplicative decomposition  Z t   Z t  1 −γ ρ −γKtP Rt = − x E − (γρs − Zs ) dBs e exp − vs ds , γ 0 0 where

2 1 1/2 1 1/2 2 b vt = γρt bt − γ b at ρt − Ft (Zt ) + b at (γρt − Zt ) ≤ 0, dt ⊗ P−a.e. 2 2

Rt Then since the stochastic integral 0 (ρs − Zs )dBs is in BMO(PH ), the stochastic exponential above is a uniformly integrable martingale. By exactly the same arguments as before, we have ′

ess inf P EPs [Rtρ ] ≤ Rsρ , s ≤ t, P − a.s.

P′ ∈PH (s,P)

with equality for ρ∗ . Hence, the terminal value RTρ is the utility of the terminal wealth of the trading strategy ρ. Consequently, 
 1 ˜ ess inf P EP U XTρ ≤ R0 = − x−γ exp(Y0 ) for all ρ ∈ A. γ

P′ ∈PH (0,P)

⊔ ⊓ Remark 5.1. Of course, the min-max property of Theorem 4.2 still holds.

6

Robust logarithmic utility

In this section, we consider logarithmic utility function U (x) = log(x), x > 0. Here we use a the same notion of trading strategies as in the power utility case: ρ = (ρi )i=1,...,d denotes the part of the wealth invested in stock i. The number of shares of stock i is given by ρit Xt . Then the wealth process is defined as: Si t

Xtρ

Z t Z tX d Xsρ ρis i Xsρ ρs (dBs + bs ds) , PH − q.s. dSs = x + =x+ Ssi 0 0 i=1

and the initial capital x is positive. The wealth process X ρ can be written as  Z t ρ ρs (dBs + bs ds) , t ∈ [0, T ] , PH − q.s. Xt = xE 0

In this case, the set of admissible strategies is defined as follows 20

(6.1)

Definition 6.1. The set of admissible trading strategies A˜ consists of all Rd -valued progressively measurable processes ρ satisfying  Z T 1/2 P 2 sup E |b at ρt | dt < ∞, P∈PH

0

˜ dt ⊗ dP − a.s., ∀P ∈ PH . and ρ ∈ A, For the logarithmic utility, the agent has no liability at time T (ξ = 0). Then the optimization problem is given by V (x) = sup inf EP [log(XTρ )] P∈PH ρ∈A˜

= log(x) + sup inf E ρ∈A˜ P∈PH

P

Z

T

ρs dBs + 0

Z

T 0

 1 1/2 2 (ρs bs − |b a ρs | )ds . 2 s

(6.2)

We have the following theorem. Theorem 6.1. Let Assumption 2.2 or Assumption 2.1 with the addition that the closed domain e is C 2 hold. Then, the value function of the optimization problem (6.2) is given by A V (x) = log(x) − Y0 for x > 0,

2 where Y0 is defined as the unique solution (Y, Z) ∈ D∞ H × HH of the quadratic 2BSDE Z T Z T Fbs ds + KTP − KtP , t ∈ [0, T ], P − a.s., ∀P ∈ PH . Zs dBs − Yt = 0 −

(6.3)

t

t

The generator is defined by where

Fbs = Fs (b as ),

1 1 Fs (a) = − dist2 (θs , Aa ) + |θs |2 , for a ∈ Sd>0 . 2 2

Moreover, there exists an optimal trading strategy ρ∗ ∈ A˜ with the property   1/2 b at ρ∗t ∈ ΠAabt θbt .

(6.4)

Proof. The proof is very similar to the case of exponential and power utility. First we show that there exists an unique solution to the 2BSDE (6.3). We then write, for t ∈ [0, T ] Rtρ = Mtρ + Aρt , where Z t (ρs − Zs ) dBs + KtP , M ρ = log(x) − Y0 + 0  Z t 2 1 2 1 1/2 b ρ b b a ρs − θs + θs − Fs ds. A = − b 2 s 2 0 21

Then, we similarly prove that ρ∗ , which can be consructed by means of a classical measurable b b ˜ Note in particular that ρ∗ only depends on θ, selection argument, is in A. a1/2 and the closed set A˜ describing the constraints on the trading strategies.

Next, due to Definition (6.1), the stochastic integral in Rρ is a martingale under each P for all ˜ Moreover, Fb is chosen to make Aρ non-increasing for all ρ and a constant for ρ∗ . Thus, ρ ∈ A. the minimum condition of K P implies that Rρ satisfies the Property (iii) of 3.1. Furthermore, the initial value Y0 of the simple 2BSDE (6.3) satisfies Y0 = − sup E

P

P∈PH

0

Hence, V (x) =

∗ R0ρ (x)

Z

T

 b Fs ds .

= log(x) + sup E P∈PH

P

Z

0

T

 b Fs ds .

⊔ ⊓

Remark 6.1. Of course, the min-max property of Theorem 4.2 still holds.

7

Examples

In general, it is difficult to solve BSDEs and 2BSDEs explicitely. In this section, we will give some examples where we have an explicit solution. In particular, we show how the optimal probability measure is chosen. In all our examples, we will work in dimension one, d = 1. First, we deal with robust exponential utility. We consider the case where there are no constraints e = R. Then the associated 2BSDE has a generator which is linear on trading strategies, that is A in z. In the first example, we consider a deterministic terminal liability ξ and show that we can compare our result with the one obtained by solving the HJB equation in the standard Merton’s approach, working with the probability measure associated to the constant process a ¯. In the 2 second example, we show that with a random payoff ξ = −BT , where B is the canonical process, we end up with an optimal probability measure which is not of Bang-Bang type (Bang-Bang type means that, under this probability measure, the density of the quadratic variation b a takes only the two extreme values, a and a). We emphasize that this example does not have real financial significance, but shows nonetheless that one cannot expect the optimal probability measure to depend only on the two bounds for the volatility unlike with option pricing in the UVM model.

7.1

Example 1: Deterministic payoff

In this example, we suppose that b is a constant in R. From Theorem 4.1, we know that the value function of the robust maximization problem is given by V ξ (x) = −exp (−β (x − Y0 )) ,

22

where Y is the solution of a 2BSDE with quadratic generator. When there are no constraints, the 2BSDE can be written as follows Z T Z T Fbs (Zs )ds + KTP − KtP , P − a.s., ∀P ∈ PH . Zs dBs − Yt = ξ − t

t

and the generator is given by

Ft (ω, z, a) = bz +

b2 , for a ∈ Sd>0 . 2βa

Then we can solve explicitly the correpondent BSDEs with the same generator under each P. Let R t −1 R t 1 2 −1 Mt = e− 0 2 b bas ds− 0 bbas dBs . By applying Itˆo’s formula to ytP Mt , we have   Z b2 T −1 b as Ms ds . y0P = EP ξMT − 2β 0

Since a ≤ b a≤a ¯, we derive that

y0P ≤ ξ −

1 b2 T. 2β a ¯

Therefore, by the representation of Y , we have Y0 ≤ ξ −

1 b2 T. 2β a ¯

Moreover, under the specific probability measure Pa¯ ∈ PH , we have a ¯

y0P = ξ −

1 b2 T. 2β a ¯

a ¯

This implies that Y0 = y0P , which means that the robust utility maximization problem is degenerated and is equivalent to a standard utility maximization problem under the probability measure Pa . We give more details and intuitions about this result in Example 7.3 below.

7.2

Example 2 : Non-deterministic payoff

In this subsection, we consider a non-deterministic payoff ξ = −BT2 . As in the first example, there are no constraints on trading strategies. Then, the 2BSDE has a linear generator. We can verify that −BT2 can be written as the limit under the norm k·kL2 of a sequence which is H

in UCb (Ω), and thus is in L2,κ H , which is the terminal condition set for 2BSDE with Lipschitz generator. Here, we suppose that b is a deterministic continuous function of time t. By the same method as in the previous example, let Mt = e−

R 1 2 −1 a−1 as ds− 0t bs b s dBs 0 2 bs b

Rt

23

,

then we obtain y0P

=E

P



−MT BT2

−

Z

T

0

By applying Itˆo’s formula to Mt Bt , we have

 b2s −1 b a Ms ds . 2β s

dMt Bt = Mt dBt + Bt dMt − bt Mt dt. Since b is deterministic, by taking expectation under P and localizing if necessary, we obtain   Z T Z T P P E [MT BT ] = E − bt dt. bt Mt dt = − 0

0

Again, by applying Itˆo’s formula to −Mt Bt2 , we have −dMt Bt2 = −2Mt Bt dBt − Bt2 dMt − b at Mt dt + 2bt Mt Bt dt.

Therefore y0P can be rewritten as  Z t   Z T  Z T b2s P P bs ds dt. 2bt dt − at + y0 = E −Mt b 2βb at 0 0 0 2

2

bs bs , we know that g′ = 1− 2βx By analysing the map g : x ∈ R+ 7−→ x− 2βx 2 , then g is non-decreasing

when x2 ≥ such that

b2 2β .

Let t such that

Consider b a deterministic positive continuous non-decreasing function of time t b2 b20 ≤ a 2 ≤ a2 ≤ T . 2β 2β b2t 2β

= a and t such that

as in Example 7.1, we can show that Bang-Bang type.

7.3

b2t ∗ 2β = a, and a := ∗ Pa is an optimal

a10≤t≤t +

√bt 1 2β t≤t≤t

+ a1t≤t≤T , then

probability measure, which is not of

Example 3 : Merton’s approach for robust power utility

Here, we deal with robust power utility. As in Example 7.1, we suppose that b is a constant in R e = R. From Theorem 5.1, Fbt (z) can be rewritten and ξ = 0. First, we consider the case where A as 2 1/2 −1 at γ −b at z + bb 1 1/2 2 Fbt (z) = + b a z , 2(1 + γ) 2 t

which is quadratic and linear in z. According to BSDEs theory, we can solve explicitly the corresponding BSDEs with this generator under each probability measure P. We use an exponential transformation and let P P γ , y ′P := e−αyt , z ′P := e−αyt ztP . α := 1 + 1+γ By applying Itˆo’s formula, we know that (y ′P , z ′P ) is the solution of the following linear BSDE     γ ′P 2 −1 P ′P ′P b b a − 2bz dt + z dBt dyt = −αy 2(1 + γ) 24

with the terminal condition yT′P = 1. Let λt :=

2 Rt R t −1/2 ηs αγ γ b2 b a−1 , ηt := − 2bb a−1/2 , and Mt := e 0 λs − 2 ds+ 0 bas ηs dBs . 2(1 + γ) 2(1 + γ)

By applying Itˆo’s formula to yt′P Mt , we obtain

yt′P = EPt [MT /Mt ] , so y0P = − Since a ≤ b a≤a ¯, we derive that

y0P ≤ −

 1  P ln E [MT ] . α

b2 γ T. 2(1 + γ) a ¯

Thus by the representation of Y , we have Y0 ≤ −

γ b2 T. 2(1 + γ) a ¯

Moreover, under the specific probability measure Pa¯ ∈ PH , we have a ¯

y0P = −

γ b2 T. 2(1 + γ) a ¯

a ¯

This implies that Y0 = y0P . Finally, the value of the robust power utility maximization problem is 1 V (x) = − x−γ exp (Y0 ) . γ As in Example 7.1, the robust utility maximization problem is degenerate, and becomes a standard utility maximization problem under the probability measure Pa . In order to shed more light on this somehow surprising result, we first recall the HJB equation obtained by Merton [32] in the standard utility maximization problem −

h i ∂v − sup Lδ v(t, x) = 0, ∂t δ∈A˜

together with the terminal condition

v(T, x) = U (x) := −

x−γ , x ∈ R+ , γ > 0, γ

2

∂v ∂ v where Lδ v(t, x) = xδb ∂x + 21 x2 δ2 σ 2 ∂x 2 , with a constant volatility σ.

It turns out that, when A˜ = R, the value function is given by   2 b −γ (T − t) U (x), (t, x) ∈ [0, T ] × R+ . v(t, x) = exp 2σ 2 (1 + γ) Let σ 2 = a, we have v(0, x) = V (x), the result given by our 2BSDE method. Intuitively and formally speaking (in the case of controls taking values in compact sets, it has actually been 25

proved under other technical conditions in [46] that the solution to the stochastic game we consider is indeed a viscosity solution of the equation below), the HJB equation for the robust maximization problem should then be −

h i ∂v − sup inf Lδ,α v(t, x) = 0 ∂t δ∈A˜ α∈[a,¯a]

together with the terminal condition v(T, x) = U (x), x ∈ R+ , where Lδ,αv(t, x) = xδb

∂v 1 ∂2v + x2 δ 2 α 2 . ∂x 2 ∂x

Note that the value function we obtained from our 2BSDE approach solves the above PDE. Now consider the case A˜ = R, if the second derivative of v is positive, then the term h i inf Lδ,a,¯a v(t, x) ˜ δ∈A

becomes infinite, so the above PDE has no meaning. This implies that v should be concave. Then a is the minimizer. This explains why the robust utility maximization problem degenerates in the case A˜ = R. However, it is clear that when, for instance, we impose no short-sale and no large sales conditions (that is to say A˜ is a segment), then the problem should not degenerate and the optimal probability measure switches between the two bounds a and a. Finally, notice that using the language of G-expectation introduced by Peng in [34], if we let G(Γ) =


1 1 sup αΓ = a (Γ)+ − a (Γ)− , 2 a≤α≤a 2

then the above PDE can be rewritten as follows h i ∂v + inf Lδ,a,¯a v(t, x) = 0, − ∂t δ∈A˜ where L

a δ,a,¯



∂2v v(t, x) = x δ G − 2 ∂x 2 2



.

Then, our PDE plays the same role for Merton’s PDE as the Black-Scholes-Barenblatt PDE plays for the usual Black-Scholes PDE, by replacing the second derivative terms by their non-linear versions.

References [1] Anderson E., Hansen L. P. , and Sargent T. (2003): A quartet of semigroups for model specification, robustness, prices of risk, and model detection. Journal of the European Economic Association, 1, 68–123 . [2] Avellaneda, M., Levy, A. and Paras, A. (1995). Pricing and hedging derivative securities in markets with uncertain volatilities. Applied Mathematical Finance, 2:73–88. 26

[3] Bismuth, J.M. (1973). Conjugate convex functions in optimal stochastic control, J. Math. Anal. Appl., 44:384–404. [4] Barrieu, P., El Karoui, N. (2011). Monotone stability of quadratic semimartingales with applications to general quadratic BSDEs and unbounded existence result, preprint. [5] Briand, Ph., Hu, Y. (2006). BSDE with quadratic growth and unbounded terminal value, Probab. Theory Relat. Fields, 136:604–618. [6] Bordigoni, G., Matoussi, A., and Schweizer, M. (2007). A stochastic control approach to a robust utility maximization problem, ”Stochastic analysis and applications”, Abel Symp, 2:125–151. Springer, Berlin. [7] C ¸ etin, U., Soner, H.M., and Touzi, N. (2007). Option hedging under liquidity costs, Finance and Stochastics, 14:317–341. [8] Cheridito, P., Soner, H.M., Touzi, N., and Victoir, N. (2007). Second Order Backward Stochastic Differential Equations and Fully Non-Linear Parabolic PDEs, Communications on Pure and Applied , 60(7): 1081–1110. [9] Dellacherie, C. (1972). ”Capacit´es et Processus Stochastiques”. Springer Verlag. [10] Dellacherie, C., and Meyer, P.-A. (1975) ” Probabilit´es et potentiel. Chapitre I ` a IV”, Hermann, Paris. [11] Denis, L., Martini, C. (2006). A theoretical framework for the pricing of contingent claims in the presence of model uncertainty, Annals of Applied Probability, 16(2): 827–852. [12] Denis, L., and Kervarec, M. (2007). Utility functions and optimal investment in nondominated models, preprint. [13] Denis, L., Hu, M., Peng, S. (2011). Function spaces and capacity related to a Sublinear Expectation: application to G-Brownian Motion Paths, Potential Analysis, 34: 139–161. [14] El Karoui, N. (1981). Les aspects probalilistes du contrˆ ole stochastique, Ecole d’Et´e de Probabilit´es de Saint-Flour IX-1979, Lecture Notes in Mathematics, 876:73-238. Springer, Berlin. [15] El Karoui, N., Peng, S. and Quenez, M.C. (1994). Backward stochastic differential equations in finance, Mathematical Finance, 7 (1): 1–71. [16] El Karoui, N., and Rouge, R. (2000). Princing via utility maximization and entropy, Mathematical Finance, 10:259-276. [17] El Karoui, N., Matoussi, A. and Ngoupeyou, A. (2011). Quadratic Backward Stochastic Differential Equations In jump Diffusion Model. preprint. [18] Epstein, L.G. and Ji, S. (2011). Ambiguous volatility, possibility and utility in continuous time, preprint.

27

[19] Faidi, W., Matoussi, A., and Mnif, M. (2011). Maximization of Recursive Utilities: A Dynamic Maximum Principle Approach, SIAM J. Financial Math., Vol. 2, pp. 1014-1041. [20] F¨ollmer, H. (1981). Calcul d’Itˆo sans probabilit´es, Seminar on Probability XV, Lecture Notes in Math., 850:143–150. Springer, Berlin. [21] Fahim, A., Touzi, N., and Warin, X. (2008). A probabilistic numerical scheme for fully nonlinear PDEs, Ann. Proba., to appear. [22] Gilboa, I. and Schmeidler, D. (1989). Maximin expected utility with a non-unique prior. Journal of Mathematical Economics, 18: 141–153. [23] Gundel, A. (2005). Robust utility maximization for complete and incomplete market models, Finance and Stochastics, 9: 151–176. [24] Hansen, L.P., Sargent,T.J., Turmuhambetova, G.A., Williams,N. : Robust control and model misspecification, Journal of Economic Theory, 128, 45-90 (2006). [25] Hu, Y., Imkeller, P., and M¨ uller, M. (2005). Utility maximization in incomplete markets, Ann. Appl. Proba., 15(3):1691–1712. [26] Jeanblanc, M., Matoussi, A. and Ngoupeyou, A. (2010). Robust utility maximization in a discontinuous filtration, preprint. [27] Jeanblanc, M., Matoussi, A. and Ngoupeyou, A. (2011). Indifference pricing of unbounded credit derivatives, preprint. [28] Kobylanski, M. (2000). Backward stochastic differential equations and partial differential equations with quadratic growth, Ann. Prob. 28:259–276. [29] Kramkov, D. and Schachermayer, W. (1999). The asymptotic elasticity of utility functions and optimal investement in incomplete markets. The Annals of Applied Probability, 9(3): 904–950. [30] Lepeltier, J. P. and San Martin, J. (1997). Backward stochastic differential equations with continuous coefficient, Statistics & Probability Letters, 32 (5): 425–430. [31] Lyons, F. (1995). Uncertain volatility and the risk-free synthesis of derivatives.Journal of Applied Finance, 2: 117–133. [32] Merton, R. (1969). Lifetime portfolio selection under uncertainty: the continuous time case, Rev. Econ. Stat., 51: 239–265. [33] Pardoux, E. and Peng, S. (1990). Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14:55–61. [34] Peng, S. (2010). Nonlinear expectations and stochastic calculus under uncertainty, preprint.

28

[35] Possama¨ı, D. (2010). Second order backward stochastic differential equations with continuous coefficient, preprint. [36] Possama¨ı, D., Zhou, C. (2012). Second order backward stochastic differential equations with quadratic growth , arXiv:1201.1050. [37] Possama¨ı, D., Soner, H.M., and Touzi, N. (2011). Large liquidity expansion of the superhedging costs, Asymptotic Analysis: Theory, Methods and Appl., to appear. [38] Quenez, M. (2004). Optimal portfolio in a multiple-priors model.Progress in Probability, 58: 291–321. [39] Schied, A. (2005). Optimal investments for risk- and ambiguity-averse preferences: a duality approach, preprint, TU Berlin, September 2005, [40] Schied, A., and Wu, C-T. (2005). Duality theory for optimal investments under model uncertainty, preprint. [41] Skiadas, C. (2003). Robust control and recursive utility, Finance and Stochastics, 7: 475– 489. [42] Soner, H.M., and Touzi, N. (2007). The dynamic programming equation for second order stochastic target problems, SIAM J. on Control and Optim., 48(4):2344-2365. [43] Soner, H.M., Touzi, N., Zhang J. (2010). Wellposedness of second order BSDE’s, Probability Theory and Related Fields, to appear. [44] Soner, H.M., Touzi, N., Zhang J. (2010). Dual formulation of second order target problems, preprint. [45] Soner, M., Touz, N. and Zhang, J. (2011). Martingale Representation Theorem for the G-Expectation. Stochastic Processes and their Applications 121, 265-287. [46] Talay, D. and Zheng, Z. (2002). Worst case model risk management.Finance and Stochastics, 6: 517–537. [47] Tevzadze, R. (2008). Solvability of backward stochastic differential equations with quadratic growth, Stoch. Proc. and their App., 118:503–515. [48] Von Neumann, J. and Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton University Press.

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