scholarly journals Backward Stochastic Differential Equations and Optimal Control of Marked Point Processes

2013 ◽  
Vol 51 (5) ◽  
pp. 3592-3623 ◽  
Author(s):  
Fulvia Confortola ◽  
Marco Fuhrman
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Point Processes ◽  
Marked Point ◽  
Backward Stochastic Differential Equations ◽  
Marked Point Processes

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.

Stochastic differential equations for ruin probabilities

1995 ◽  
Vol 32 (01) ◽  
pp. 74-89 ◽  
Author(s):  
Christian Max Møller
Keyword(s):  
Differential Equations ◽  
Point Processes ◽  
Diffusion Processes ◽  
Marked Point ◽  
Marked Point Processes ◽  
Ruin Probabilities ◽  
Probability Of Ruin ◽  
Time Of Ruin ◽  
Markov Structure ◽  
The Time Of Ruin

The present paper proposes a general approach for finding differential equations to evaluate probabilities of ruin in finite and infinite time. Attention is given to real-valued non-diffusion processes where a Markov structure is obtainable. Ruin is allowed to occur upon a jump or between the jumps. The key point is to define a process of conditional ruin probabilities and identify this process stopped at the time of ruin as a martingale. Using the theory of marked point processes together with the change-of-variable formula or the martingale representation theorem for point processes, we obtain differential equations for evaluating the probability of ruin. Numerical illustrations are given by solving a partial differential equation numerically to obtain the probability of ruin over a finite time horizon.

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