Robust utility maximization for a diffusion market model with misspecified coefficients

The paper studies the robust maximization of utility from terminal wealth in a diffusion financial market model. The underlying model consists of a tradable risky asset whose price is described by a diffusion process with misspecified trend and volatility coefficients, and a non-tradable asset with a known parameter. The robust functional is defined in terms of a utility function. An explicit characterization of the solution is given via the solution of the Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation.


Introduction
The purpose of the present paper is to study the robust maximization of utility from terminal wealth in a diffusion financial market model where the trend and volatility of the asset price are uncertain.
The concept of robustness was introduced by P. Huber (see [19]) in the context of statistical estimation of an unknown distribution parameter. The essence of our approach is as follows. Suppose we need to estimate the mean of some symmetric distribution. If the estimation is based on "pure" observations, then the effective estimate is the sample mean. But if observations are contaminated by outliers, then the situation changes completely. Huber introduced the so-called gross error model (the contaminated neighborhood of a true distribution) and showed that an optimal estimate is a maximum likelihood estimate constructed for the so-called least favorable distribution. Analytically, this means that we need to solve a minimax problem analogous to the problem given by (2.1) below with the asymptotic mean square error as a risk function. In some limiting cases, an optimal estimate is a median, but not a sample mean. In mathematical finance, most approaches and settings implicitly suppose that the underlying asset model is fully specified: the parameters (trend and volatility) of the model are known. Actually, we have all the same to estimate these parameters and construct, say, confidence intervals for them. Hence we only know that a pair (μ, σ ) belongs with high probability to a rectangle [μ − , μ + ] × [σ − , σ + ]. In that case there arises a problem of construction of robust trading strategies, where an optimal strategy is the best strategy against the worst state of nature. If the risk function of the problem is the expected utility from terminal wealth, then our definition of the optimization problem (2.1) is an exact one.
In 2002, Chen and Epstein [7] introduced a continuous-time intertemporal version of a multiple-priors utility function for a Brownian filtration. In that case, beliefs are represented by a set P of probability measures and the utility is defined as the minimum, over the set P, of the expected utilities. Independently, Cvitanić and Karatzas [9] studied, for a given option, the hedging strategies which minimize the expected "shortfall", i.e., the difference between the payoff and the terminal wealth. They considered the problem of determining the "worst-case" modelQ, i.e., the model which maximizes a minimal shortfall risk over all possible priors Q ∈ P. It was shown that under certain assumptions their maximin problem could be written as a minimax problem. In 2004, Quenez [29] studied the problem of utility maximization in an incomplete multiple-priors model where asset prices are semimartingales. This problem corresponds to a maximin problem where the maximum is taken over the set of feasible wealths X (or portfolios) and the minimum over the set of priors P. The author showed that under suitable conditions, there exists a saddle point for this problem. Moreover, Quenez developed a dual approach which consists of solving a dual minimization problem over the set of priors and supermartingale measures, and showed how the solution of the dual problem leads to a solution of the primal problem.
The majority of the relevant published work is concerned with the case where one of the parameters is known exactly. For the case of an unknown drift coefficient, the existence of a saddle point of the corresponding minimax problem was established and the characterization of an optimal strategy obtained in [9,15,16]. For unknown volatility coefficients, the hedging strategy was constructed in [2-4, 6, 11, 24, 34].
The most difficult case is to characterize the optimal strategy of the maximin problem under uncertainty about both drift and volatility terms.
Talay and Zheng [32] applied a PDE-based approach to the minimax problem and characterized the value as a viscosity solution of the corresponding Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation. In general, such a problem does not contain a saddle point. Moreover, in robust maximization problems, the maximin should be taken instead of the minimax used by Talay and Zheng. Recently in the work of Denis and Kervarec [10], a general problem of utility maximization encompassing the case of uncertain volatility was studied, and a duality theory for robust utility maximization in this framework was established.
During the refereeing process, we have found the preprint of Matoussi et al. [27] which is also devoted to a robust utility maximization problem. To study the exponential, power and logarithmic utility maximization, the authors use 2BSDE theory (this theory was thoroughly developed by Cheridito, Soner, Touzi, Victoir and Zhang in [8,31]). They obtained explicit solutions in some particular cases, which is one of the tasks of our paper, too. Despite some advantages of their approach (non-Markovian models, the existence of a saddle point, a general contingent claim), we should say that that approach is not sufficiently general for our model. Namely, (a) only the volatility matrix is misspecified in their model. In our case both coefficients (drift and volatility) are misspecified, (b) the volatility matrix √ a t satisfies the condition a ≤ a t ≤ a, where a and a are given matrices, which does not cover our "partially misspecified volatility" case since in our paper the matrices a t = σ 2 t ρσ t ρσ t 1 , a = σ 2 − ρσ − ρσ − 1 and a = σ 2 + ρσ + ρσ + 1 are non-comparable to each other.
Moreover, in the non-Markovian case the BSDE corresponding to our problem will not be a 2BSDE (see Remark 3.5). And, besides, we cannot even get our BSDE as a particular case of the 2BSDE given in [27]. So we can conclude that [27] has little in common with our paper.
In this paper, we consider an incomplete diffusion financial market model which resembles the model considered by Schied [30], Hernández-Hernández and Schied [16,17]. We suppose that the market consists of a risk-free asset, a tradable risky asset with misspecified trend and volatility, and a non-tradable asset with known parameters. Differently from the approach of Quenez [29] and Schied [30], we solve the maximin problem using the HJBI equation which corresponds to the primal problem. When the trend and volatility coefficients are uncertain, such a maximin problem has no saddle point in general. We extend the set of model coefficients, i.e., carry out some "randomization", and obtain as a result a minimax problem with a saddle point. This makes it possible to replace the maximin problem by a minimax problem, which is easier to study using the HJBI equation properties. In particular, we have found a form of this equation that coincides with the equation derived by Hernández-Hernández and Schied [16] when the volatility is assumed to be known. We establish the solvability of the obtained equation in the classical sense and solve the HJBI equation explicitly for a specific drift coefficient. The saddle point (an optimal portfolio and optimal coefficients) of the considered maximin problem has been found as well. An explicit characterization of the optimal strategies of the maximin problem for the case of power and exponential utilities in terms of the solution of the HJBI equation is the main result of the paper.
To illustrate our approach, we present a simple quadratic hedging problem. Let T -measurable random variable. Denote by Π 2 the set of square-integrable predictable processes with respect to the filtration F . Let P([σ − , σ + ]) be the set of probability measures on [σ − , σ + ] and U ,Ũ denote the set of predictable processes with respect to the filtration F with values in [σ − , σ + ] and P, respectively. We use the notation f · ν for . The wealth process corresponding to a portfolio process π ∈ Π 2 and volatility σ ∈ U is defined as X t (π, σ ) = c + t 0 π s σ s dB s . (1.1) The problem is to find π * ∈ Π 2 minimizing the worst case mean-variance hedging error Such a π * is called a robust hedging strategy. Let us extend problem (1.2) as follows. For each ν ∈Ũ we define the processes where p, p 2 are the functions p(σ ) = σ , p 2 (σ ) = σ 2 , respectively. One can easily check that (W ν , W ν,⊥ ) is also a 2-dimensional Brownian motion and the equation It is clear that U ⊂Ũ and for ν ∈ U , W ν = B and (1.1) coincides with (1.4). Hence we can consider the minimax problem which is the extension of problem (1.2). For the sake of simplicity, it is assumed that c = EH and, using the stochastic integral representation We shall see below that this expression is strictly positive. Moreover, This means that a saddle point does not exist for the problem (1.2).
On the other hand, the function G defined on Π 2 ×Ũ by is convex in π and linear in ν. Then by the Neumann theorem (see Theorem 8 of [1], Chap. 6), there exists a saddle point (π * , σ * ) ∈ Π 2 ×Ũ . Therefore we have It is easy to see that the saddle point is 1 As we see, the extension of the problem allows us to find a robust strategy and the worst case mean-variance hedging error for the original problem (1.2). In Sect. 2, we shall obtain this result by means of the HJBI equation in the case of a terminal contingent claim H (B T ).
Notice that the problem (1.2) can also be solved directly, but in more general cases (e.g. for models with nonzero drift), such "explicit computations" are complicated and to our knowledge do not exist in the literature. The aim of this work is to show that the existence of a saddle point in the extended problem simplifies solving the original problem and enables us to find "explicit solutions".
The paper is organized as follows. In Sect. 2, we describe the model and consider the misspecified coefficients as generalized controls. Furthermore, we show the existence of a saddle point of the generalized maximin problem and derive the HJBI equation for the value function. Some examples are also discussed. In Sect. 3, we prove the solvability in the classical sense of the obtained PDE in the case of power and exponential utility and give an explicit PDE characterization of the robust maximization problem.

Generalized coefficients and the existence of a saddle point
Suppose that the financial market consists of a risk-free asset with r(y) ≥ 0 and a risky financial asset whose price is defined through the stochastic differential equation (SDE) Here W is a standard Brownian motion and Y denotes an economic factor process modeled by the SDE for some correlation factor ρ ∈ [0, 1] and a standard Brownian motion W ⊥ which is independent of W . Let (F t ) t∈[0,T ] denote the augmented filtration generated by W, W ⊥ . Denote b =b − r and assume that is the class of bounded continuous functions with bounded derivatives and C 0 (R) the class of continuous functions with compact support.
Introduce the set P(K) of probability distributions with support on the com- . Such a process is usually called a generalized control in control theory [35]. We identify the set of predictable K-valued processes U K with the subset ofŨ K assigning to each (μ t , σ t ) from U K the P(K)-valued process δ (μ t ,σ t ) .
By Π 2 we denote the set of predictable processes with finite L 2 ([0, T ] × Ω)norm. The objective of our economic agent is to find an optimal robust strategy for the problem where U (x, y) is a continuous function defined on R 2 and satisfying a quadratic growth condition.
with values in the set K defines an element ν ∈Ũ K by the formula P ((μ t , σ t ) ∈ B | F t ) = ν t (B). More precisely, denoting by p Y the predictable projection of a process Y (see [25]), we have the equalities p μ t = K μν t (dμ dσ ), p σ t = K σ ν t (dμ dσ ). Hence instead of (2.4) we can write where we write p σ 2 t for p (σ 2 t ).

Remark 2.2
The main theorems of the paper are valid if instead of Π 2 ×Ũ K we consider the set of Markovian strategies and coefficients, i.e., the set of Borel-measurable R × P(K)-valued functions (π(t, x, y), ν(t, x, y)) such that there exists a weak solution (X, Y ) of (2.5) satisfying the condition the generator of the process (X ν , Y ν ) can be given by H π,ν (x, y, p, q). It is obvious that

Proposition 2.3 For each fixed
Moreover, for each continuous function f on K, where (X t,x,y s (π, ν), Y t,x,y s (π, ν)), s ≥ t, denote the solution of (2.2) with initial condition (X t,x,y t , Y t,x,y t ) = (x, y). Notice that if ν t = δ (μ t ,σ t ) , then Y t,x,y s (π, ν) does not depend on x, π, ν and we can also use the notation Y t,y s . Since the Isaacs condition is satisfied (by virtue of Proposition 2.3), there exists, as we shall see below, a value of the differential game v ≡ v + = v − , which will be a solution of the HJBI equation The latter equation can be rewritten as x, y)π 2 + (p σ · ν)ρv xy (t, x, y)π where we suppose that q 11 < 0 and use the notation κ = ρq 12 For the sake of simplicity, we assume in addition that is a continuous, piecewise smooth function of z ∈ (−∞, ∞).
The proof is given in the Appendix.
Example 2.8 Let us consider the robust mean-variance hedging problem with zero drift and unknown volatility min Therefore we have U (x, y) = −(x − H (y)) 2 , (x, y) ∈ R 2 , μ − = μ + = 0, r (y) = 0. We assume that H is a continuous bounded function. By (2.12), we get and it follows from (2.10) that

This system admits the explicit solution
A(t, y) = e 2r(T −t) , The function B(t, y) = e 2r(T −t) EH (Y t,y T ) is a classical bounded solution of the corresponding linear parabolic equation with bounded continuous B y (t, y) and continuous B yy (t, y) (see [14], formulas (5.20)-(5.22) of Chap. VI). It is clear that In the case of an objective function U (x, y) defined on R + × R, it is convenient to determine the wealth process as a solution of the SDE dX t = r(Y t )X t dt + π t X t b(Y t ) + μ t dt + π t X t σ t dW t , X 0 = x, (2.17) The set of admissible strategies Π is now defined as the set of all predictable processes π such that π s dW s is a BMO-martingale (as regards BMO-martingales, see [20]). It is clear that for each (π, μ, σ ) ∈ Π × U K , π s σ s dW s is also a BMOmartingale, a solution of (2.17) is positive, and the maximin problem It is easy to see that the HJBI equation for the value v(t, x, y) of this problem is same as (2.10), but π * is now defined by (t, x, y))v x (t, x, y) +m(t, x, y)ρv xy (t, x, y) where the pair (¯ (t, x, y),m(t, x, y)) is defined by (2.12).

Power and exponential utility cases
Now let us consider the robust utility maximization problem with the power utility In this case, the HJBI equation (2.10), (2.11) becomes A solution of this equation is of the form v(t, x, y) = 1 q x q e u(t,y) , where u satisfies the equation The pair (p μ · ν * (t, x, y), p σ · ν * (t, x, y)) from Theorem 2.6 takes the form p μ · ν * (t, y), p σ · ν * (t, y) = ρu y (t, y) , m ρu y (t, y) , (3.5) where ( , m) is defined by (2.9).