The maximum principles for partially observed risk-sensitive optimal controls of Markov regime-switching jump-diffusion system

ABSTRACT

This paper studies partially observed risk-sensitive optimal control problems with correlated noises between the system and the observation. It is assumed that the state process is governed by a continuous-time Markov regime-switching jump-diffusion process and the cost functional is of an exponential-of-integral type. By virtue of a classical spike variational approach, we obtain two general maximum principles for the aforementioned problems. Moreover, under certain convexity assumptions on both the control domain and the Hamiltonian, we give a sufficient condition for the optimality. For illustration, a linear-quadratic risk-sensitive control problem is proposed and solved using the main results. As a natural deduction, a fully observed risk-sensitive maximum principle is also obtained and applied to study a risk-sensitive portfolio optimization problem. Closed-form expressions for both the optimal portfolio and the corresponding optimal cost functional are obtained.     

General Maximum Principles for Partially Observed Risk-Sensitive Optimal Control Problems and Applications to Finance

This paper is concerned with partially observed risk-sensitive optimal control problems. Combining Girsanov’s theorem with a standard spike variational technique, we obtain some general maximum principles for the aforementioned problems. One of the distinctive differences between our results and the standard risk-neutral case is that the adjoint equations and variational inequalities strongly depend on a risk-sensitive parameter γ. Two examples are given to illustrate the applications of the theoretical results obtained in this paper. As a natural deduction, a general maximum principle is also obtained for a fully observed risk-sensitive case. At last, this result is applied to study a risk-sensitive optimal portfolio problem. An explicit optimal investment strategy and a cost functional are obtained. A numerical simulation result shows the influence of a risk-sensitive parameter on an optimal investment proportion; this coincides with its economic meaning and theoretical results. This work was partially supported by the National Natural Science Foundation (10671112), the National Basic Research Program of China (973 Program, No. 2007CB814904), the Natural Science Foundation of Shandong Province (Z2006A01) and the Doctoral Fund of the Education Ministry of China.     

Dynamic asset management with risk-sensitive criterion and non-negative factor constraints: a differential game approach

In this paper, we study continuous time portfolio optimization problem where individual securities are directly affected by economic factors. We consider the risk-sensitive criterion function as is familiar in the robust control literature. This is the natural setting for studying the infinite horizon case of the control problem arising in portfolio optimization. Our result extends earlier works by imposing explicitly the non-negativity constraint on the economic factors. This is achieved by using reflected diffusions. The risk-sensitive control problem with reflected diffusion is then converted into a stochastic differential game. The lower value of this game leads immediately to the desired optimal strategy. Also we prove the existence of unique strong solution to reflected diffusions with bounded measurable drift coefficient which is the first result of its kind for higher dimensional reflected diffusions.     

A RISK-SENSITIVE STOCHASTIC MAXIMUM PRINCIPLE FOR OPTIMAL CONTROL OF JUMP DIFFUSIONS AND ITS APPLICATIONS

A stochastic maximum principle for the risk-sensitive optimal control problem of jump diffusion processes with an exponential-of-integral cost functional is derived assuming that the value function is smooth, where the diffusion and jump term may both depend on the control. The form of the maximum principle is similar to its risk-neutral counterpart. But the adjoint equations and the maximum condition heavily depend on the risk-sensitive parameter. As applications, a linear-quadratic risk-sensitive control problem is solved by using the maximum principle derived and explicit optimal control is obtained.     

跳扩散最优控制的随机最大值原理及在金融中的应用

讨论了由金融市场中投资组合和消费选择问题引出的一类最优控制问题,投资者的期望效用是常数相对风险厌恶(CRRA)情形.在跳扩散框架下,利用古典变分法得到了一个局部随机最大值原理.结果应用到最优投资组合和消费选择策略问题,得到了状态反馈形式的显式最优解.     

Sufficient Stochastic Maximum Principle for the Optimal Control of Semi-Markov Modulated Jump-Diffusion with Application to Financial Optimization

The finite state semi-Markov process is a generalization over the Markov chain in which the sojourn time distribution is any general distribution. In this article, we provide a sufficient stochastic maximum principle for the optimal control of a semi-Markov modulated jump-diffusion process in which the drift, diffusion, and the jump kernel of the jump-diffusion process is modulated by a semi-Markov process. We also connect the sufficient stochastic maximum principle with the dynamic programming equation. We apply our results to finite horizon risk-sensitive control portfolio optimization problem and to a quadratic loss minimization problem.     

Maximum Principle for a Stochastic Optimal Control Problem and Application to Portfolio/Consumption Choice

We consider mainly an optimal control problem motivated by a portfolio and consumption choice problem in a financial market where the utility of the investor is assumed to have a given homogeneous form. A Pontryagin local maximum principle is obtained by using classical variational methods. We apply the result to make optimal portfolio and consumption decisions for the problem under consideration. The optimal selection coincides with the one obtained in Refs. 1 and 2, where the Bellman dynamic programming principle was used.     

Portfolio Optimization in a Semi-Markov Modulated Market

We address a portfolio optimization problem in a semi-Markov modulated market. We study both the terminal expected utility optimization on finite time horizon and the risk-sensitive portfolio optimization on finite and infinite time horizon. We obtain optimal portfolios in relevant cases. A numerical procedure is also developed to compute the optimal expected terminal utility for finite horizon problem. This work was supported in part by a DST project: SR/S4/MS: 379/06; also supported in part by a grant from UGC via DSA-SAP Phase IV, and in part by a CSIR Fellowship.     

一类证券市场中投资组合及消费选择的最优控制问题

研究一类证券市场中投资组合及消费选择的最优控制问题．在随机干扰源相互关联情形下,运用动态规划方法,对一类典型的效用函数CRRA(Constant Relative Risk Aversion,常数相对风险厌恶)情形,得到了最优投资组合及消费选择的显式解,并给出了最优解的经济解释和关于部分参数的灵敏度分析．     

Risk-Sensitive Dynamic Asset Management

This paper develops a continuous time portfolio optimization model where the mean returns of individual securities or asset categories are explicitly affected by underlying economic factors such as dividend yields, a firm's return on equity, interest rates, and unemployment rates. In particular, the factors are Gaussian processes, and the drift coefficients for the securities are affine functions of these factors. We employ methods of risk-sensitive control theory, thereby using an infinite horizon objective that is natural and features the long run expected growth rate, the asymptotic variance, and a single risk-aversion parameter. Even with constraints on the admissible trading strategies, it is shown that the optimal trading strategy has a simple characterization in terms of the factor levels. For particular factor levels, the optimal trading positions can be obtained as the solution of a quadratic program. The optimal objective value, as a function of the risk-aversion parameter, is shown to be the solution of a partial differential equation. A simple asset allocation example, featuring a Vasicek-type interest rate which affects a stock index and also serves as a second investment opportunity, provides some additional insight about the risk-sensitive criterion in the context of dynamic asset management. Accepted 10 December 1997     

随机波动率模型下的最优证券组合选择

在证券价格服从随机波动过程下 ,研究了自融资策略下的最优证券组合问题 ,得到了相应的最优投资组合及其效用的解析表达式 .     

Risk-Sensitive Portfolio Optimization Problems with Fixed Income Securities

We discuss a class of risk-sensitive portfolio optimization problems. We consider the portfolio optimization model investigated by Nagai (SIAM J. Control Optim. 41:1779–1800, 2003). The model by its nature can include fixed income securities as well in the portfolio. Under fairly general conditions, we prove the existence of an optimal portfolio in both finite-horizon and infinite-horizon problems.     

Remarks on Risk-Sensitive Control Problems

The main purpose of this paper is to investigate the asymptotic behavior of the discounted risk-sensitive control problem for periodic diffusion processes when the discount factor $\alpha$ goes to zero. If $u_\alpha(\theta,x)$ denotes the optimal cost function, $\theta$ being the risk factor, then it is shown that $\lim_{\alpha\to 0}\alpha u_\alpha(\theta,x)=\xi(\theta)$ where $\xi(\theta)$ is the average on $]0,\theta[$ of the optimal cost of the (usual) infinite horizon risk-sensitive control problem.     

Quantile Portfolio Optimization Under Risk Measure Constraints

This paper analyzes the problem of optimal portfolio choice with budget and risk constraints. The problem is formulated in terms of quantile functions and the risk is quantified through a large family of coherent risk measures. The solution is obtained analyzing the problem without constraints using Lagrange multipliers, getting a unique solution to the optimization problem.     

Composition of an efficient portfolio in the Bielecki and Pliska market model

We study the continuous time portfolio optimization model due to Bielecki and Pliska where the mean returns of individual securities or asset categories are explicitly affected by underlying economic factors. We introduce the functional Q _γ featuring the expected earnings yield of portfolio minus a penalty term proportional with a coefficient γ to the variance when we keep the value of the factor levels fixed. The coefficient γ plays the role of a risk-aversion parameter. We find the optimal trading positions that can be obtained as the solution to a maximization problem for Q _γ at any moment of time. The single-factor case is analyzed in more details. We present a simple asset allocation example featuring a Vasicek-type interest rate which affects a stock index and also serves as a second investment opportunity. Then we compare our results with the theory of Bielecki and Pliska where the authors employ the methods of the risk-sensitive control theory thereby using an infinite horizon objective featuring the long run expected growth rate, the asymptotic variance, and a risk-aversion parameter similar to γ.     

Estimation risk and stability of optimal portfolio composition

The mean-variance portfolio models indicate that for optimal investment decisions, the ‘true’ ex-ante values of the model parameters should be used. Instead, in practice, ex-post parameter estimates are used. If in the estimation process, the probability distribution of estimators is not known, there is a problem of estimation risk. This paper investigates the impact of estimation risk on the composition of optimal portfolios. As the multivariance distribution of the vector of optimal portfolio weights allocated to risky assets is analytically intractable, a use of the Monte Carlo simulation experimental is made. This study suggests that the composition of optimal portfolio is relatively more stable when the estimates of model parameters are obtained from longer series of historical observations or the expected portfolio return is low.     

Risk-sensitive and risk-neutral control for continuous-time hidden Markov models

In this paper the optimal control of a continuous-time hidden Markov model is discussed. The risk-sensitive problem involves a cost function which has an exponential form and a risk parameter, and is solved by defining an appropriate information state and dynamic programming. As the risk parameter tends to zero, the classical risk-neutral optimal control problem is recovered. The limits are proved using viscosity solution methods.The first author wishes to acknowledge the funding of the activities of the Cooperative Research Centre for Robust and Adaptive Systems by the Australian Commonwealth Government under the Cooperative Research Centers Program. The support of NSERC Grant A7964 is acknowledged by the second author, as is the hospitality of the Department of Systems Engineering and the Cooperative Research Centre for Robust and Adaptive Systems, Australian National University, in July 1993.     

状态和控制均含时滞的受外界扰动的线性系统的最优控制

主要研究一类受外界持续扰动且状态和控制含不同时滞的线性系统的最优控制,首先通过变量代换,将系统化为控制不含时滞的滞后型微分系统.接着使用最优控制的极大值原理的必要条件,得到含超前和滞后项的两点边值问题.为了得到最优控制律的解析解,引进一个灵敏参数ε,得到两点边值问题序列,通过迭代法,得到最优控制律的解析解.并对外界扰动状态构造降维观测器,来实现最优控制律的物理可实现性.最后实例验证了上述方法的有效性.     

Adaptive control of a continuous-time portfolio and consumption model

In this paper, an adaptive control problem is formulated and solved using Merton's stochastic differential equation for the wealth in a portfolio selection and consumption model. Since the asset prices are assumed to satisfy a log normal distribution, it suffices to consider two assets. It is assumed that the drift parameter for the price of the risky asset is unknown. A recursive family of estimators for this unknown parameter is defined and is shown to converge almost surely to the true value of the parameter. The controls in the equation for the wealth are obtained from the optimal controls where the estimates of the unknown parameter are substituted for the unknown parameter.This research was partially supported by NSF Grant No. ECS-84-03286-A01.The authors wish to thank P. Varaiya for some useful comments on this paper.     

Existence of Risk-Sensitive Optimal Stationary Policies for Controlled Markov Processes

In this paper we are concerned with the existence of optimal stationary policies for infinite-horizon risk-sensitive Markov control processes with denumerable state space, unbounded cost function, and long-run average cost. Introducing a discounted cost dynamic game, we prove that its value function satisfies an Isaacs equation, and its relationship with the risk-sensitive control problem is studied. Using the vanishing discount approach, we prove that the risk-sensitive dynamic programming inequality holds, and derive an optimal stationary policy. Accepted 1 October 1997