Backward Stochastic Riccati Equations and Infinite Horizon L-Q Optimal Control with Infinite Dimensional State Space and Random Coefficients

We study the Riccati equation arising in a class of quadratic optimal control problems with infinite dimensional stochastic differential state equation and infinite horizon cost functional. We allow the coefficients, both in the state equation and in the cost, to be random. In such a context backward stochastic Riccati equations are backward stochastic differential equations in the whole positive real axis that involve quadratic non-linearities and take values in a non-Hilbertian space. We prove existence of a minimal non-negative solution and, under additional assumptions, its uniqueness. We show that such a solution allows to perform the synthesis of the optimal control and investigate its attractivity properties. Finally the case where the coefficients are stationary is addressed and an example concerning a controlled wave equation in random media is proposed.     

Coupled nonlinear stochastic differential equations

Systems of n coupled linear or nonlinear differential equations which may be deterministic or stochastic are solved by methods of the first author and his co-workers. Examples include two coupled Riccati equations, coupled linear equations, stochastic coupled equations with product terms, and n coupled stochastic differential equations.     

Numerical solution of the finite horizon stochastic linear quadratic control problem

The treatment of the stochastic linear quadratic optimal control problem with finite time horizon requires the solution of stochastic differential Riccati equations. We propose efficient numerical methods, which exploit the particular structure and can be applied for large‐scale systems. They are based on numerical methods for ordinary differential equations such as Rosenbrock methods, backward differentiation formulas, and splitting methods. The performance of our approach is tested in numerical experiments.     

Explicit Solutions of Quadratic FBSDEs Arising From Quadratic Term Structure Models

We provide explicit solutions of certain forward-backward stochastic differential equations (FBSDEs) with quadratic growth. These particular FBSDEs are associated with quadratic term structure models of interest rates and characterize the zero-coupon bond price. The results of this paper are naturally related to similar results on affine term structure models of Hyndman (Math. Financ. Econ. 2(2):107–128, 2009) due to the relationship between quadratic functionals of Gaussian processes and linear functionals of affine processes. Similar to the affine case a sufficient condition for the explicit solutions to hold is the solvability in a fixed interval of Riccati-type ordinary differential equations. However, in contrast to the affine case, these Riccati equations are easily associated with those occurring in linear-quadratic control problems. We also consider quadratic models for a risky asset price and characterize the futures price and forward price of the asset in terms of similar FBSDEs. An example is considered, using an approach based on stochastic flows that is related to the FBSDE approach, to further emphasize the parallels between the affine and quadratic models. An appendix discusses solvability and explicit solutions of the Riccati equations.     

Almost-periodic solutions for Riccati equations of stochastic control

We consider an average quadratic cost criteria for affine stochastic differential equations with almost-periodic coefficients. Under stabilizability and detectability conditions we show that the Riccati equation associated with the quadratic control problem has a unique almost-periodic solution. In the periodic case the corresponding result is proved in [4].     

Linear–quadratic stochastic two-person nonzero-sum differential games: Open-loop and closed-loop Nash equilibria

In this paper, we consider a linear–quadratic stochastic two-person nonzero-sum differential game. Open-loop and closed-loop Nash equilibria are introduced. The existence of the former is characterized by the solvability of a system of forward–backward stochastic differential equations, and that of the latter is characterized by the solvability of a system of coupled symmetric Riccati differential equations. Sometimes, open-loop Nash equilibria admit a closed-loop representation, via the solution to a system of non-symmetric Riccati equations, which could be different from the outcome of the closed-loop Nash equilibria in general. However, it is found that for the case of zero-sum differential games, the Riccati equation system for the closed-loop representation of an open-loop saddle point coincides with that for the closed-loop saddle point, which leads to the conclusion that the closed-loop representation of an open-loop saddle point is the outcome of the corresponding closed-loop saddle point as long as both exist. In particular, for linear–quadratic optimal control problem, the closed-loop representation of an open-loop optimal control coincides with the outcome of the corresponding closed-loop optimal strategy, provided both exist.     

Application of rational expansion method for stochastic differential equations

By means of the Hermite transformation, a new general ansätz and the symbolic computation system Maple, we apply the Riccati equation rational expansion method [24] to uniformly construct a series of stochastic non-traveling wave solutions for stochastic differential equations. To illustrate the effectiveness of our method, we take the stochastic mKdV equation as an example, and successfully construct some new and more general solutions. The method can also be applied to solve other nonlinear stochastic differential equation or equations.     

Riccati equation arising in the boundary control of stochastic hyperbolic systems

A Riccati equation of stochastic control theory is studied directly. The equation arises in the synthesis of a linear quadratic regulator problem for systems governed by stochastic partial differential equations of hyperbolic type, with contorl acting on the boundary through Dirichlet of Neumann conditions     

A random differential transform method: Theory and applications

In this article, a Differential Transform Method (DTM) based on the mean fourth calculus is developed to solve random differential equations. An analytical mean fourth convergent series solution is found for a nonlinear random Riccati differential equation by using the random DTM. Besides obtaining the series solution of the Riccati equation, we provide approximations of the main statistical functions of the stochastic solution process such as the mean and variance. These approximations are compared to those obtained by the Euler and Monte Carlo methods. It is shown that this method applied to the random Riccati differential equation is more efficient than the two above mentioned methods.     

Lifetime consumption-portfolio choice under trading constraints,recursive preferences,and nontradeable income

We analyze the lifetime consumption-portfolio problem in a competitive securities market with continuous price dynamics, possibly nontradeable income, and convex trading constraints. We define a class of “translation-invariant” recursive preferences, which includes additive exponential utility, but also nonadditive recursive and multiple-prior formulations, and allows for first and second-order source-dependent risk aversion. For this class, we show that the solution reduces to a single constrained backward stochastic differential equation, which for an interesting class of incomplete-market problems simplifies to a system of ordinary differential equations of the Riccati type.     

Riccati微分方程特解新求法的研究

通过对一般Riccati方程进行初等变换,使之变为特殊的Riccati方程,然后利用公式、观察实验,或利用二阶微分方程的特解,或利用一阶微分方程组的特解等方法,求得这些Riccati方程的特解.     

Characterization of kernel functions associated with operator algebraic Riccati equations for linear delay systems

In infinite time quadratic control and stochastic filtering problems for linear delay systems, operator algebraic Riccati equations play a very important role. However, since these are abstract operator equations, it is very useful, in analyzing their structure, to be able to characterize the kernel functions associated with the solutions of the operator Riccati equations. The kernel functions are given by the unique solution of a set of coupled differential equations. By comparing these kernel equations with similar ones available in the literature, it is shown that this characterization result is somewhat stronger than previously known results. Possible extentions to systems with control, observation, as well as state delays are also pointed out.     

一类新的线性二次随机最优控制器的设计

给出一类正倒向随机微分方程解的存在唯一性结果,应用这个结果研究了一类新的推广的随机线性二次最优控制器的设计问题,得到了由正倒向随机微分方程解所表示的唯一最优控制器的显式结构;在推广的Riccati方程系统基础上,得到最优控制器精确的线性反馈形式.最后,给出了随机线性二次最优控制器的设计算法.     

Quadratic Control for Stochastic Systems Defined by Evolution Operators and Square Integrable Martingales

The infinite dimensional version of the linear quadratic cost control problem is studied by Curtain and Pritchard [2], Gibson [5] by using Riccati integral equations, instead of differential equations. In the present paper the corresponding stochastic case over a finite horizon is considered. The stochastic perturbations are given by Hilbert valued square integrable martingales and it is shown that the deterministic optimal feedback control is also optimal in the stochastic case. Sufficient conditions are given for the convergence of approximate solutions of optimal control problems.     

Perturbations and projections of Kalman–Bucy semigroups

We analyse various perturbations and projections of Kalman–Bucy semigroups and Riccati equations. For example, covariance inflation-type perturbations and localisation methods (projections) are common in the ensemble Kalman filtering literature. In the limit of these ensemble methods, the regularised sample covariance tends toward a solution of a perturbed/projected Riccati equation. With this motivation, results are given characterising the error between the nominal and regularised Riccati flows and Kalman–Bucy filtering distributions. New projection-type models are also discussed; e.g. Bose–Mesner projections. These regularisation models are also of interest on their own, and in, e.g., differential games, control of stochastic/jump processes, and robust control.     

由Lévy过程驱动的一类特殊的高维的BSRDE解的存在唯一性

讨论由Brownian运动和Lévy过程共同驱动的线性随机系统的随机LQ问题,其中代价泛函是关于Lévy过程生成的σ-代数取条件期望.得到由Lévy过程驱动的新的多维的倒向随机Riccati方程,利用Bellman拟线性原理和单调收敛方法证明了此随机Riccati方程的解的存在性.     

伊藤-泊松型随机微分方程的线性二次控制

本文研究伊藤-泊松型随机微分方程的线性二次控制问题,利用动态规划方法、伊藤公式等技巧,通过解HJB方程,我们得到了随机Riccati方程及另外两个微分方程,求出控制变量,解决了线性二次最优控制最优问题.     

Stochastic Linear Quadratic Optimal Control Problems

This paper is concerned with the stochastic linear quadratic optimal control problem (LQ problem, for short) for which the coefficients are allowed to be random and the cost functional is allowed to have a negative weight on the square of the control variable. Some intrinsic relations among the LQ problem, the stochastic maximum principle, and the (linear) forward—backward stochastic differential equations are established. Some results involving Riccati equation are discussed as well. Accepted 15 May 2000. Online publication 1 December 2000     

A Finite Horizon Linear Quadratic Optimal Stochastic Control Problem Driven by Both Brownian Motion and L\'{e}vy Processes

We study the linear quadratic optimal stochastic control problem which is jointly driven by Brownian motion and L\'{e}vy processes. We prove that the new affine stochastic differential adjoint equation exists an inverse process by applying the profound section theorem. Applying for the Bellman's principle of quasilinearization and a monotone iterative convergence method, we prove the existence and uniqueness of the solution of the backward Riccati differential equation. Finally, we prove that the optimal feedback control exists, and the value function is composed of the initial value of the solution of the related backward Riccati differential equation and the related adjoint equation.     

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??We study the linear quadratic optimal stochastic control problem which is jointly driven by Brownian motion and L\'{e}vy processes. We prove that the new affine stochastic differential adjoint equation exists an inverse process by applying the profound section theorem. Applying for the Bellman's principle of quasilinearization and a monotone iterative convergence method, we prove the existence and uniqueness of the solution of the backward Riccati differential equation. Finally, we prove that the optimal feedback control exists, and the value function is composed of the initial value of the solution of the related backward Riccati differential equation and the related adjoint equation.