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 of mean-field backward doubly stochastic systems driven by Itô-Lévy processes

ABSTRACT

In this paper, we introduce a new class of backward doubly stochastic differential equations (in short BDSDE) called mean-field backward doubly stochastic differential equations (in short MFBDSDE) driven by Itô-Lévy processes and study the partial information optimal control problems for backward doubly stochastic systems driven by Itô-Lévy processes of mean-field type, in which the coefficients depend on not only the solution processes but also their expected values. First, using the method of contraction mapping, we prove the existence and uniqueness of the solutions to this kind of MFBDSDE. Then, by the method of convex variation and duality technique, we establish a sufficient and necessary stochastic maximum principle for the stochastic system. Finally, we illustrate our theoretical results by an application to a stochastic linear quadratic optimal control problem of a mean-field backward doubly stochastic system driven by Itô-Lévy processes.     

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.     

Stochastic maximum principle for SPDEs with noise and control on the boundary

In this paper we prove necessary conditions for optimality of a stochastic control problem for a class of stochastic partial differential equations that is controlled through the boundary. This kind of problem can be interpreted as a stochastic control problem for an evolution system in a Hilbert space. The regularity of the solution of the adjoint equation, that is a backward stochastic equation in infinite dimension, plays a crucial role in the formulation of the maximum principle.     

Optimal control of stochastic FitzHugh–Nagumo equation

This paper is concerned with the existence and uniqueness of solution for the optimal control problem governed by the stochastic FitzHugh–Nagumo equation driven by a Gaussian noise. First-order conditions of optimality are also obtained.     

Verification Theorem Of Stochastic Optimal Control With Mixed Delay And Applications To Finance

This paper focuses on a general model of a controlled stochastic differential equation with mixed delay in the state variable. Based on the Itô formula, stochastic analysis, convex analysis, and inequality technique, we obtain a semi‐coupled forward‐backward stochastic differential equation with mixed delay and mixed initial‐terminal conditions and prove that such forward‐backward system admits a unique adapted solution. The verification theorem for an optimal control of a system with mixed delay is established. The obtained results generalize and improve some recent results, and they are more easily verified and applied in practice. As an application, we conclude with finding explicitly the optimal consumption rate from the wealth process of a person given by a stochastic differential equation with mixed delay which fit into our general model.     

Second-order Taylor expansion for backward doubly stochastic control system

We consider the second-order Taylor expansion for backward doubly stochastic control system. The results are obtained under no restriction on the convexity of control domain. Moreover, the control variable is allowed in the drift coefficient and the diffusion coefficient.     

带有随机跳跃干扰的线性二次随机最优控制问题

给出一类布朗运动和泊松过程混合驱动的正倒向随机微分方程解的存在唯一性结果,应用这一结果研究带有随机跳跃干扰的线性二次随机最优控制问题,并得到最优控制的显式形式,可以证明最优控制是唯一的.然后,引入和研究一类推广的黎卡提方程系统,讨论该方程系统的可解性并由该方程的解得到带有随机跳跃干扰的线性二次随机最优控制问题最优的线性反馈.     

The relaxed optimal control problem for Mean-Field SDEs systems and application

The present study deals with a new approach of optimal control problems where the state equation is a Mean-Field stochastic differential equation, and the set of strict (classical) controls need not be convex and the diffusion coefficient depends on the term control. Our consideration is based on only one adjoint process, and the necessary conditions as well as a sufficient condition for optimality in the form of a relaxed maximum principle are obtained, with application to Linear quadratic stochastic control problem with mean-field type.     

A stochastic maximum principle for systems with jumps, with applications to finance

We consider a stochastic control problem with linear dynamics with jumps, convex cost criterion, and convex state constraint, in which the control enters the drift, the diffusion, and the jump coefficients. We allow these coefficients to be random, and do not impose any L^p-bounds on the control.

We obtain a stochastic maximum principle for this model that provides both necessary and sufficient conditions of optimality. This is the first version of the stochastic maximum principle that covers the consumption–investment problem in which there are jumps in the price system.     

基于g-期望的部分可观测非零和随机微分博弈(英文)

本文研究了g-期望下的部分可观测非零和随机微分博弈系统,该系统的状态方程由It?-Lévy过程驱动,成本函数由g-期望描述.根据Girsanov定理和凸变分技巧,本文得到了最大值原理和验证定理.为对所获结果进行说明,本文讨论了关于资产负债管理的博弈问题.