scholarly journals Nonlinear Kolmogorov equations in infinite dimensional spaces: the backward stochastic differential equations approach and applications to optimal control

2002 ◽  
Vol 30 (3) ◽  
pp. 1397-1465 ◽  
Author(s):  
Marco Fuhrman
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Backward Stochastic Differential Equations ◽  
Kolmogorov Equations ◽  
Infinite Dimensional

A Monte Carlo method for backward stochastic differential equations with Hermite martingales

2019 ◽  
Vol 25 (1) ◽  
pp. 37-60
Author(s):  
Antoon Pelsser ◽  
Kossi Gnameho
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Approximation ◽  
Mathematical Finance ◽  
Backward Stochastic Differential Equations ◽  
Terminal Condition ◽  
Dimensional System ◽  
Infinite Dimensional ◽  
The Family ◽  
Infinite Dimensional System

Abstract Backward stochastic differential equations (BSDEs) appear in many problems in stochastic optimal control theory, mathematical finance, insurance and economics. This work deals with the numerical approximation of the class of Markovian BSDEs where the terminal condition is a functional of a Brownian motion. Using Hermite martingales, we show that the problem of solving a BSDE is identical to solving a countable infinite-dimensional system of ordinary differential equations (ODEs). The family of ODEs belongs to the class of stiff ODEs, where the associated functional is one-sided Lipschitz. On this basis, we derive a numerical scheme and provide numerical applications.

scholarly journals Fully Coupled Mean-Field Forward-Backward Stochastic Differential Equations and Stochastic Maximum Principle

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Hui Min ◽  
Ying Peng ◽  
Yongli Qin
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Optimal Control Problems ◽  
Mean Field ◽  
Backward Stochastic Differential Equations ◽  
Maximum Principles ◽  
Control Problems ◽  
Linear Quadratic Optimal Control ◽  
Fully Coupled

We discuss a new type of fully coupled forward-backward stochastic differential equations (FBSDEs) whose coefficients depend on the states of the solution processes as well as their expected values, and we call them fully coupled mean-field forward-backward stochastic differential equations (mean-field FBSDEs). We first prove the existence and the uniqueness theorem of such mean-field FBSDEs under some certain monotonicity conditions and show the continuity property of the solutions with respect to the parameters. Then we discuss the stochastic optimal control problems of mean-field FBSDEs. The stochastic maximum principles are derived and the related mean-field linear quadratic optimal control problems are also discussed.

scholarly journals Delayed Stochastic Linear-Quadratic Control Problem and Related Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Li Chen ◽  
Zhen Wu ◽  
Zhiyong Yu
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Stochastic Differential Equations ◽  
Backward Stochastic Differential Equations ◽  
Delay System ◽  
Linear Quadratic ◽  
Stochastic Linear System ◽  
And Control

We discuss a quadratic criterion optimal control problem for stochastic linear system with delay in both state and control variables. This problem will lead to a kind of generalized forward-backward stochastic differential equations (FBSDEs) with Itô’s stochastic delay equations as forward equations and anticipated backward stochastic differential equations as backward equations. Especially, we present the optimal feedback regulator for the time delay system via a new type of Riccati equations and also apply to a population optimal control problem.

scholarly journals Stochastic Optimization Theory of Backward Stochastic Differential Equations Driven by G-Brownian Motion

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Zhonghao Zheng ◽  
Xiuchun Bi ◽  
Shuguang Zhang
Keyword(s):  
Optimal Control ◽  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Optimal Control ◽  
Optimal Control Problems ◽  
Backward Stochastic Differential Equations ◽  
Dynamic Programming Principle ◽  
Control Problems ◽  
Order Partial Differential Equation

We consider the stochastic optimal control problems under G-expectation. Based on the theory of backward stochastic differential equations driven by G-Brownian motion, which was introduced in Hu et al. (2012), we can investigate the more general stochastic optimal control problems under G-expectation than that were constructed in Zhang (2011). Then we obtain a generalized dynamic programming principle, and the value function is proved to be a viscosity solution of a fully nonlinear second-order partial differential equation.

scholarly journals Stochastic differential equations in infinite dimensional Hilbert space and its optimal control problem with Lévy processes

2021 ◽  
Vol 7 (2) ◽  
pp. 2427-2455
Author(s):  
Meijiao Wang ◽  
◽  
Qiuhong Shi ◽  
Maoning Tang ◽  
Qingxin Meng ◽  
...  
Keyword(s):  
Optimal Control ◽  
Hilbert Space ◽  
Differential Equations ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Stochastic Differential Equations ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
Teugels Martingales

<abstract><p>The paper is concerned with a class of stochastic differential equations in infinite dimensional Hilbert space with random coefficients driven by Teugels martingales which are more general processes and the corresponding optimal control problems. Here Teugels martingales are a family of pairwise strongly orthonormal martingales associated with Lévy processes (see Nualart and Schoutens <sup>[<xref ref-type="bibr" rid="b21">21</xref>]</sup>). There are three major ingredients. The first is to prove the existence and uniqueness of the solutions by continuous dependence theorem of solutions combining with the parameter extension method. The second is to establish the stochastic maximum principle and verification theorem for our optimal control problem by the classic convex variation method and dual techniques. The third is to represent an example of a Cauchy problem for a controlled stochastic partial differential equation driven by Teugels martingales which our theoretical results can solve.</p></abstract>

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