Smooth Solutions to Optimal Investment Models with Stochastic Volatilities and Portfolio Constraints

Abstract. This paper deals with an extension of Merton's optimal investment problem to a multidimensional model with stochastic volatility and portfolio constraints. The classical dynamic programming approach leads to a characterization of the value function as a viscosity solution of the highly nonlinear associated Bellman equation. A logarithmic transformation expresses the value function in terms of the solution to a semilinear parabolic equation with quadratic growth on the derivative term. Using a stochastic control representation and some approximations, we prove the existence of a smooth solution to this semilinear equation. An optimal portfolio is shown to exist, and is expressed in terms of the classical solution to this semilinear equation. This reduction is useful for studying numerical schemes for both the value function and the optimal portfolio. We illustrate our results with several examples of stochastic volatility models popular in the financial literature.     

Mean–variance portfolio selection under a constant elasticity of variance model

This paper discusses a mean–variance portfolio selection problem under a constant elasticity of variance model. A backward stochastic Riccati equation is first considered. Then we relate the solution of the associated stochastic control problem to that of the backward stochastic Riccati equation. Finally, explicit expressions of the optimal portfolio strategy, the value function and the efficient frontier of the mean–variance problem are expressed in terms of the solution of the backward stochastic Riccati equation.     

Non-linear filtering and optimal investment under partial information for stochastic volatility models

This paper studies the question of filtering and maximizing terminal wealth from expected utility in partial information stochastic volatility models. The special feature is that the only information available to the investor is the one generated by the asset prices, and the unobservable processes will be modeled by stochastic differential equations. Using the change of measure techniques, the partial observation context can be transformed into a full information context such that coefficients depend only on past history of observed prices (filter processes). Adapting the stochastic non-linear filtering, we show that under some assumptions on the model coefficients, the estimation of the filters depend on a priori models for the trend and the stochastic volatility. Moreover, these filters satisfy a stochastic partial differential equations named “Kushner–Stratonovich equations”. Using the martingale duality approach in this partially observed incomplete model, we can characterize the value function and the optimal portfolio. The main result here is that, for power and logarithmic utility, the dual value function associated to the martingale approach can be expressed, via the dynamic programming approach, in terms of the solution to a semilinear partial differential equation which depends on the filters estimate and the volatility. We illustrate our results with some examples of stochastic volatility models popular in the financial literature.     

Portfolio Optimization with Stochastic Volatilities and Constraints: An Application in High Dimension

In this paper we are interested in an investment problem with stochastic volatilities and portfolio constraints on amounts. We model the risky assets by jump diffusion processes and we consider an exponential utility function. The objective is to maximize the expected utility from the investor terminal wealth. The value function is known to be a viscosity solution of an integro-differential Hamilton-Jacobi-Bellman (HJB in short) equation which could not be solved when the risky assets number exceeds three. Thanks to an exponential transformation, we reduce the nonlinearity of the HJB equation to a semilinear equation. We prove the existence of a smooth solution to the latter equation and we state a verification theorem which relates this solution to the value function. We present an example that shows the importance of this reduction for numerical study of the optimal portfolio. We then compute the optimal strategy of investment by solving the associated optimization problem.     

A PDE representation of the density of the minimal entropy martingale measure in stochastic volatility markets

Under general conditions stated in Rheinländer [An entropy approach to the stein/stein model with correlation. Preprint, 2003, ETH Zürich.], we prove that in a stochastic volatility market the Radon–Nikodym density of the minimal entropy martingale measure (MEMM) can be expressed in terms of the solution of a semilinear PDE. The semilinear PDE is suggested by the dynamic programming approach to the utility indifference pricing problem of contingent claims. One of our main results is the existence and uniqueness of a classical solution of the semilinear PDE in the case of a general stochastic volatility model with additive noise correlated with the asset price. Our results are applied to the Stein–Stein and Heston stochastic volatility models.     

Comparison Method of Stability Analysis of Semilinear Time-Delay Stochastic Evolution Equation

Abstract

In this paper, we set up the comparison theorem between the mild solution of semilinear time-delay stochastic evolution equation with general time-delay variable and the solution of a class (1-dimension) deterministic functional differential equation, by using the Razumikhin–Lyapunov type functional and the theory of functional differential inequalities. By applying this comparison theorem, we give various types of the stability comparison criteria for the semilinear time-delay stochastic evolution equations. With the aid of these comparison criteria, one can reduce the stability analysis of semilinear time-delay stochastic evolution equations in Hilbert space to that of a class (1-dimension) deterministic functional differential equations. Furthermore, these comparison criteria in special case have been applied to derive sufficient conditions for various stability of the mild solution of semilinear time-delay stochastic evolution equations. Finally, the theories are illustrated with some examples.     

On the multi-dimensional portfolio optimization with stochastic volatility

In a recent paper by Mnif [18], a solution to the portfolio optimization with stochastic volatility and constraints problem has been proposed, in which most of the model parameters are time-homogeneous. However, there are cases where time-dependent parameters are needed, such as in the calibration of financial models. Therefore, the purpose of this paper is to generalize the work of Mnif [18] to the time-inhomogeneous case. We consider a time-dependent exponential utility function of which the objective is to maximize the expected utility from the investor’s terminal wealth. The derived Hamilton-Jacobi-Bellman(HJB) equation, is highly nonlinear and is reduced to a semilinear partial differential equation (PDE) by a suitable transformation. The existence of a smooth solution is proved and a verification theorem presented. A multi-asset stochastic volatility model with jumps and endowed with time-dependent parameters is illustrated.     

Stochastic 2-D Navier—Stokes Equation

   Abstract. In this paper we prove the existence and uniqueness of strong solutions for the stochastic Navier—Stokes equation in bounded and unbounded domains. These solutions are stochastic analogs of the classical Lions—Prodi solutions to the deterministic Navier—Stokes equation. Local monotonicity of the nonlinearity is exploited to obtain the solutions in a given probability space and this significantly improves the earlier techniques for obtaining strong solutions, which depended on pathwise solutions to the Navier—Stokes martingale problem where the probability space is also obtained as a part of the solution.     

On Controlled Linear Diffusions with Delay in a Model of Optimal Advertising under Uncertainty with Memory Effects

We consider a class of dynamic advertising problems under uncertainty in the presence of carryover and distributed forgetting effects, generalizing the classical model of Nerlove and Arrow (Economica 29:129–142, 1962). In particular, we allow the dynamics of the product goodwill to depend on its past values, as well as previous advertising levels. Building on previous work (Gozzi and Marinelli in Lect. Notes Pure Appl. Math., vol. 245, pp. 133–148, 2006), the optimal advertising model is formulated as an infinite-dimensional stochastic control problem. We obtain (partial) regularity as well as approximation results for the corresponding value function. Under specific structural assumptions, we study the effects of delays on the value function and optimal strategy. In the absence of carryover effects, since the value function and the optimal advertising policy can be characterized in terms of the solution of the associated HJB equation, we obtain sharper characterizations of the optimal policy.     

最优投资组合模型研究

本文研究了在完备金融市场上 ,投资者最优投资组合的随机模型。在模型参数为常系数 ,效用函数为 (0 ,T],B[0 ,T])上的有界可测函数的情形下 ,得出其最大效用值函数是随机控制问题对应的 HJB方程的平滑解 ;最优策略被证明是存在的 ,并用反馈形式给出了最优投资组合策略。     

Maximum Principle for Risk-Sensitive Stochastic Optimal Control Problem and Applications to Finance

This article is concerned with a risk-sensitive stochastic optimal control problem motivated by a kind of optimal portfolio choice problem in the financial market. The maximum principle for this kind of problem is obtained, which is similar in form to its risk-neutral counterpart. But the adjoint equations and maximum condition heavily depend on the risk-sensitive parameter. This result is used to solve a kind of optimal portfolio choice problem and the optimal portfolio choice strategy is obtained. Computational results and figures explicitly illustrate the optimal solution and the sensitivity to the volatility rate parameter.     

Exponential utility maximization for an insurer with time-inconsistent preferences

This paper studies the optimal consumption–investment–reinsurance problem for an insurer with a general discount function and exponential utility function in a non-Markovian model. The appreciation rate and volatility of the stock, the premium rate and volatility of the risk process of the insurer are assumed to be adapted stochastic processes, while the interest rate is assumed to be deterministic. The object is to maximize the utility of intertemporal consumption and terminal wealth. By the method of multi-person differential game, we show that the time-consistent equilibrium strategy and the corresponding equilibrium value function can be characterized by the unique solutions of a BSDE and an integral equation. Under appropriate conditions, we show that this integral equation admits a unique solution. Furthermore, we compare the time-consistent equilibrium strategies with the optimal strategy for exponential discount function, and with the strategies for naive insurers in two special cases.     

The Mean-Variance Hedging of a Defaultable Option with Partial Information

Abstract

We consider the mean-variance hedging of a defaultable claim in a general stochastic volatility model. By introducing a new measure Q ⁰, we derive the martingale representation theorem with respect to the investors' filtration . We present an explicit form of the optimal-variance martingale measure by means of a stochastic Riccati equation (SRE). For a general contingent claim, we represent the optimal strategy and the optimal cost of the mean-variance hedging by means of another backward stochastic differential equation (BSDE). For the defaultable option, especially when there exists a random recovery rate we give an explicit form of the solution of the BSDE.     

Optimal Trade Execution Under Stochastic Volatility and Liquidity

Abstract

We study the problem of optimally liquidating a financial position in a discrete-time model with stochastic volatility and liquidity. We consider the three cases where the objective is to minimize the expectation, an expected exponential or a mean-variance criterion of the implementation cost. In the first case, the optimal solution can be fully characterized by a forward-backward system of stochastic equations depending on conditional expectations of future liquidity. In the other two cases, we derive Bellman equations from which the optimal solutions can be obtained numerically by discretizing the control space. In all three cases, we compute optimal strategies for different simulated realizations of prices, volatility and liquidity and compare the outcomes to the ones produced by the deterministic strategies of Bertsimas and Lo (1998; Optimal control of execution costs. Journal of Financial Markets, 1, 1–50) and Almgren and Chriss (2001; Optimal execution of portfolio transactions. Journal of Risk, 3, 5–33).     

Strong Solutions of Stochastic Generalized Porous Media Equations: Existence,Uniqueness, and Ergodicity

Explicit conditions are presented for the existence, uniqueness, and ergodicity of the strong solution to a class of generalized stochastic porous media equations. Our estimate of the convergence rate is sharp according to the known optimal decay for the solution of the classical (deterministic) porous medium equation.     

On a Class of Stochastic Semilinear PDEs

Abstract

We consider stochastic semilinear partial differential equations with Lipschitz nonlinear terms. We prove existence and uniqueness of an invariant measure and the existence of a solution for the corresponding Kolmogorov equation in the space L ²(H;ν), where ν is the invariant measure. We also prove the closability of the derivative operator and an integration by parts formula. Finally, under boundness conditions on the nonlinear term, we prove a Poincaré inequality, a logarithmic Sobolev inequality, and the ipercontractivity of the transition semigroup.     

Stochastic Differential Equations with Non-Lipschitz Coefficients in Hilbert Spaces

Abstract

In this article, we discuss the successive approximations problem for the solutions of the semilinear stochastic differential equations in Hilbert spaces with cylindrical Wiener processes under some conditions which are weaker than the Lipschitz one. We establish the existence and the uniqueness of the solution and additionally, in our framework we consider a limiting problem for the mild solution. It is shown that the mild solution tends to the solution of the stochastic differential equation of Itô type in finite dimensional space.     

特殊形式的倒向随机微分方程在金融市场中的应用

首先,针对一类线性倒向随机微分方程,给出了g-鞅同鞅之间相互联系所满足的充分条件.通过该条件得到了经典的Black-Scholes模型下未定权益的公平价格过程以及最优增长投资策略的价格过程.其次,引入了带惩罚的非线性倒向随机微分方程,并通过惩罚比率的不同取值来讨论相关的经济学意义.     

Variance-Optimal Martingale Measures for Diffusion Processes with Stochastic Coefficients

In this paper we present the solution of the optimal variance optimal martingale measure for stochastic volatility models, when the noises are correlated. It is proved that the value function of the dual problem is a classical solution of the corresponding Hamilton-Jacobi-Bellman equation. The method to develop our results is based on a Bernstein’s type of argument. The dual problem of the quadratic hedging problem is studied analyzing the expression obtained after a change of measure, which corresponds to some class of risk-sensitive control problems.     

Existence of Optimal Controls for Semilinear Parabolic Equations without Cesari-Type Conditions

   Abstract. Optimal control problems governed by semilinear parabolic partial differential equations are considered. No Cesari-type conditions are assumed. By proving the existence theorem and the Pontryagin maximum principle of optimal ``state-control" pairs for the corresponding relaxed problems, an existence theorem of optimal pairs for the original problem is established.