Necessary and sufficient conditions of risk‐sensitive optimal control and differential games for stochastic differential delayed equations

In this paper, we consider risk‐sensitive optimal control and differential games for stochastic differential delayed equations driven by Brownian motion. The problems are related to robust stochastic optimization with delay due to the inherent feature of the risk‐sensitive objective functional. For both problems, by using the logarithmic transformation of the associated risk‐neutral problem, the necessary and sufficient conditions for the risk‐sensitive maximum principle are obtained. We show that these conditions are characterized in terms of the variational inequality and the coupled anticipated backward stochastic differential equations (ABSDEs). The coupled ABSDEs consist of the first‐order adjoint equation and an additional scalar ABSDE, where the latter is induced due to the nonsmooth nonlinear transformation of the adjoint process of the associated risk‐neutral problem. For applications, we consider the risk‐sensitive linear‐quadratic control and game problems with delay, and the optimal consumption and production game, for which we obtain explicit optimal solutions.     

A class of non-zero-sum stochastic differential investment and reinsurance games

In this article, we provide a systematic study on the non-zero-sum stochastic differential investment and reinsurance game between two insurance companies. Each insurance company’s surplus process consists of a proportional reinsurance protection and an investment in risky and risk-free assets. Each insurance company is assumed to maximize his utility of the difference between his terminal surplus and that of his competitor. The surplus process of each insurance company is modeled by a mixed regime-switching Cramer–Lundberg diffusion approximation process, i.e. the coefficients of the diffusion risk processes are modulated by a continuous-time Markov chain and an independent market-index process. Correlation between the two surplus processes, independent of the risky asset process, is allowed. Despite the complex structure, we manage to solve the resulting non-zero sum game problem by applying the dynamic programming principle. The Nash equilibrium, the optimal reinsurance/investment, and the resulting value processes of the insurance companies are obtained in closed forms, together with sound economic interpretations, for the case of an exponential utility function.     

基于鲁棒控制的期权套期保值策略

在标的资产价格服从带有随机方差几何布朗运动的非完全市场假设条件下,应用随机微分对策方法,研究与标的资产有关的欧式期权的动态套期保值策略问题。建立了最优动态套期保值策略的随机微分对策数学模型,给出了基于鲁棒控制的均方复制误差最小的自融资动态套期保值策略。当方差为时间的确定性函数时,最优动态套期保值策略与用Black-Scholes套期比表示的delta套期保值策略是一致的。     

带Poisson跳的线性二次随机微分博弈及其在鲁棒控制中的应用

研究了一类带Poisson跳扩散过程的线性二次随机微分博弈,包括非零和博弈的Nash均衡策略与零和博弈的鞍点均衡策略问题．利用微分博弈的最大值原理,得到Nash均衡策略的存在条件等价于两个交叉耦合的矩阵Riccati方程存在解,鞍点均衡策略的存在条件等价于一个矩阵Riccati方程存在解的结论,并给出了均衡策略的显式表达及最优性能泛函值．最后,将所得结果应用于现代鲁棒控制中的随机H₂/H_∞控制与随机H_∞控制问题,得到了鲁棒控制策略的存在条件及显式表达,并验证所得结果在金融市场投资组合优化问题中的应用．     

An HMM approach for optimal investment of an insurer

In this paper, we introduce a Hidden Markov Model (HMM) for studying an optimal investment problem of an insurer when model uncertainty is present. More specifically, the financial price and insurance risk processes are modulated by a continuous‐time, finite‐state, hidden Markov chain. The states of the chain represent different modes of the model. The HMM approach is viewed as a ‘dynamic’ version of the Bayesian approach to model uncertainty. The optimal investment problem is formulated as a stochastic optimal control problem with partial observations. The innovations approach in the filtering theory is then used to transform the problem into one with complete observations. New robust filters of the chain and estimates of key parameters are derived. We discuss the optimal investment problem using the Hamilton–Jacobi–Bellman (HJB) dynamic programming approach and derive a closed‐form solution in the case of an exponential utility and zero interest rate. Copyright © 2011 John Wiley & Sons, Ltd.     

Stochastic differential games and inverse optimal control and stopper policies

In this paper, we consider a two-player stochastic differential game problem over an infinite time horizon where the players invoke controller and stopper strategies on a nonlinear stochastic differential game problem driven by Brownian motion. The optimal strategies for the two players are given explicitly by exploiting connections between stochastic Lyapunov stability theory and stochastic Hamilton–Jacobi–Isaacs theory. In particular, we show that asymptotic stability in probability of the differential game problem is guaranteed by means of a Lyapunov function which can clearly be seen to be the solution to the steady-state form of the stochastic Hamilton–Jacobi–Isaacs equation, and hence, guaranteeing both stochastic stability and optimality of the closed-loop control and stopper policies. In addition, we develop optimal feedback controller and stopper policies for affine nonlinear systems using an inverse optimality framework tailored to the stochastic differential game problem. These results are then used to provide extensions of the linear feedback controller and stopper policies obtained in the literature to nonlinear feedback controllers and stoppers that minimise and maximise general polynomial and multilinear performance criteria.     

基于策略迭代的连续时间系统的随机线性二次最优控制

针对模型参数部分未知的随机线性连续时间系统, 通过策略迭代算法求解无限时间随机线性二次(LQ) 最优控制问题. 求解随机LQ最优控制问题等价于求随机代数Riccati 方程(SARE) 的解. 首先利用伊藤公式将随机微分方程转化为确定性方程, 通过策略迭代算法给出SARE 的解序列; 然后证明SARE 的解序列收敛到SARE 的解, 而且在迭代过程中系统是均方可镇定的; 最后通过仿真例子表明策略迭代算法的可行性.

     

Non‐Zero Sum Differential Games of Backward Stochastic Differential Delay Equations Under Partial Information

In this paper, we study a new type of differential game problems of backward stochastic differential delay equations under partial information. A class of time‐advanced stochastic differential equations (ASDEs) is introduced as the adjoint process via duality relation. By means of ASDEs, we suggest the necessary and sufficient conditions called maximum principle for an equilibrium point of non‐zero sum games. As an application, an economic problem is putted into our framework to illustrate the theoretical results. In terms of the maximum principle and some auxiliary filtering results, an equilibrium point is obtained.     

具有异常波动市场的消费与投资策略

讨论了异常波动市场中容许借贷的消费与投资策略问题,阐述了随机最优控制理论应用于现代金融理论研究中的一种方法.首先给出了金融市场中不确定性的随机模型,利用It^o公式,得到了与消费及投资策略有关的财富过程的随机微分方程,并建立了最优消费与投资问题的随机控制模型.根据随机最优控制理论,导出了目标函数满足的Hamilton-Jacobi-Bellman(HJB)方程.通过对HJB方程的讨论,得到了最优消费与投资策略的分段表示函数,并就Hara效用函数进行讨论,得到了具体的消费与投资策略.     

Backward Stochastic H2/H∞ Control with Random Jumps

This paper is concerned with H₂/H_∞ control of a new class of stochastic systems. The most distinguishing feature, compared with the existing literature, is that the systems are described by backward stochastic differential equations (BSDEs) with Brownian motion and random jumps. It is shown that the backward stochastic H₂/H_∞ control under consideration is associated with the of the corresponding uncontrolled backward stochastic perturbed system. A necessary and sufficient condition for the existence of a unique solution to the control problem under consideration is derived. The resulting solution is characterized by the solution of an uncontrolled forward backward stochastic differential equation (FBSDE) with Brownian motion and random jumps. When the coefficients are all deterministic, the equivalent linear feedback solution involves a pair of Riccati‐type equations and an uncontrolled BSDE. In addition an uncontrolled forward stochastic differential equation (SDE) is given.     

不确定拟哈密顿系统的随机最优控制

本文提出了不确定拟哈密顿系统、基于随机平均法、随机极大值原理和随机微分对策理论的一种随机极大极小最优控制策略.首先,运用拟哈密顿系统的随机平均法,将系统状态从速度和位移的快变量形式转化为能量的慢变量形式,得到部分平均的It随机微分方程;其次,给定控制性能指标,对于不确定拟哈密顿系统的随机最优控制,根据随机微分对策理论,将其转化为一个极小极大控制问题;再根据随机极大值原理,建立关于系统与伴随过程的前向-后向随机微分方程,随机最优控制表达为哈密顿控制函数的极大极小条件,由此得到最坏情形下的扰动参数与极大极小最优控制;然后,将最坏扰动参数与最优控制代入部分平均的It随机微分方程并完成平均,求解与完全平均的It随机微分方程相应的Fokker-Planck-Kolmogorov(FPK)方程,可得受控系统的响应量并计算控制效果;最后,将上述不确定拟哈密顿系统的随机最优控制策略应用于一个两自由度非线性系统,通过数值结果说明该随机极大极小控制策略的控制效果.     

Maximum principle for the stochastic optimal control problem with delay and application

In this paper, we consider an optimal control problem for the stochastic system described by stochastic differential equations with delay. We obtain the maximum principle for the optimal control of this problem by virtue of the duality method and the anticipated backward stochastic differential equations. Our results can be applied to a production and consumption choice problem. The explicit optimal consumption rate is obtained.     

Optimal control problem for stochastic evolution equations in Hilbert spaces

In this article, we consider an optimal control problem in which the controlled state dynamics is governed by a stochastic evolution equation in Hilbert spaces and the cost functional has a quadratic growth. The existence and uniqueness of the optimal control are obtained by the means of an associated backward stochastic differential equations with a quadratic growth and an unbounded terminal value. As an application, an optimal control of stochastic partial differential equations with dynamical boundary conditions is also given to illustrate our results.     

A risk‐sensitive stochastic maximum principle for fully coupled forward‐backward stochastic differential equations with applications

In this paper, we are interested in the problem of optimal control where the system is given by a fully coupled forward‐backward stochastic differential equation with a risk‐sensitive performance functional. As a preliminary step, we use the risk neutral which is an extension of the initial control system where the admissible controls are convex, and an optimal solution exists.Then, we study the necessary as well as sufficient optimality conditions for risk sensitive performance. At the end of this work, we illustrate our main result by giving an example that deals with an optimal portfolio choice problem in financial market, specifically the model of control cash flow of a firm or project where, for instance, we can set the model of pricing and managing an insurance contract.     

Near-optimal control for stochastic recursive problems

It is well documented (e.g. Zhou (1998) [8]) that the near-optimal controls, as the alternative to the “exact” optimal controls, are of great importance for both the theoretical analysis and practical application purposes due to its nice structure and broad-range availability, feasibility as well as flexibility. However, the study of near-optimality on the stochastic recursive problems, to the best of our knowledge, is a totally unexplored area. Thus we aim to fill this gap in this paper. As the theoretical result, a necessary condition as well as a sufficient condition of near-optimality for stochastic recursive problems is derived by using Ekeland’s principle. Moreover, we work out an ε-optimal control example to shed light on the application of the theoretical result. Our work develops that of [8] but in a rather different backward stochastic differential equation (BSDE) context.     

Feasible solution to discrete-time linear quadratic stochastic Stackelberg difference game

This paper investigates the discrete-time linear quadratic (LQ) stochastic Stackelberg game, which has not been thoroughly addressed in previous literature. Firstly, we derive the maximum principle for the stochastic Stackelberg difference game using the variational method, and obtain the necessary and sufficient solvability conditions. However, due to the coupling between the two players and the presence of stochastic noise, obtaining explicit optimal leader and follower's strategies becomes challenging. Therefore, we present a feasible suboptimal control strategy instead. As a result, we derive a feasible suboptimal control strategy. To achieve this, we assume a linear homogeneous relationship to decouple the group of stochastic game forward-backward stochastic differential equations (SG-FBSDEs), which serves as a compromise for obtaining the optimal solution. With this approach, we derive a feasible solution to the stochastic Stackelberg difference game based on the solution to symmetric Riccati equations.     

Optimal control of a stochastic system with an exponential-of-integral performance criterion

An approach based on the theory of positive semigroups is introduced for the analysis of the infinite-horizon, exponential-of-integral optimal control problem for a stochastic system of the Ito form. Existence conditions for state-feedback admissible controllers are formulated and optimality conditions are derived. Connections between the exponential-of-integral optimal control problem and stochastic differential games are discussed.     

A Partially Observed Nonzero‐Sum Stochastic Differential Game with Delays and its Application to Finance

In this paper, we deal with a new kind of partially observed nonzero‐sum differential game governed by stochastic differential delay equations. One of the special features is that the controlled system and the utility functionals involve both delays in the state variable and the control variables under different observation equations for each player. We obtain a maximum principle and a verification theorem for the game problem by virtue of Girsanov's theorem and the convex variational method. In addition, based on the theoretical results and Malliavin derivative techniques, we solve a production and consumption choice game problem.     

Singular control of stochastic linear systems with recursive utility

We formulate a class of singular stochastic control problem with recursive utility where the cost function is determined by a backward stochastic differential equation. Some characteristics of the value function of the control problem are obtained by the method of approximation via penalization, and the optimal control process is constructed.     

Maximum principle for mean-field jump–diffusion stochastic delay differential equations and its application to finance

This paper investigates a stochastic optimal control problem with delay and of mean-field type, where the controlled state process is governed by a mean-field jump–diffusion stochastic delay differential equation. Two sufficient maximum principles and one necessary maximum principle are established for the underlying system. As an application, a bicriteria mean–variance portfolio selection problem with delay is studied to demonstrate the effectiveness and potential of the proposed techniques. Under certain conditions, explicit expressions are provided for the efficient portfolio and the efficient frontier, which are as elegant as those in the classical mean–variance problem without delays.