A Sample-Wise Data Driven Control Solver for the Stochastic Optimal Control Problem with Unknown Model Parameters

In this work,an efficient sample-wise data driven control solver will be developed to solve the stochastic optimal control problem with unknown model parameters.A direct filter method will be applied as an online parameter estimation method that dynamically estimates the target model parameters upon receiving the data,and a sample-wise optimal control solver will be provided to efficiently search for the optimal control.Then,an effective overarching algorithm will be introduced to combine the parameter estimator and the optimal control solver.Numerical experiments will be carried out to demonstrate the effectiveness and the efficiency of the sample-wise data driven control method.

Erratum to:Turnpike Properties for Stochastic Linear-Quadratic Optimal Control Problems

Erratum to:https://doi.0rg/10.1007/s11401-022-0374-x 1.The third line of page 999,the first author's name Jiangrui SUN should be changed to JingruiSUN.2.The seventh lineofpage 6 of file totalcontents-43B6.pdf,JiangruiSUN should be changed to Jingrui SUN.

FOUR STEP SCHEME FOR GENERAL MARKOVIAN FORWARD-BACKWARD SDES

This paper studies a class of forward-backward stochastic differential equations (FBSDE)in a general Markovian framework.The forward SDE represents a large class of strong Markov semimartingales,and the backward generator requires only mild regularity assumptions.The authors showthat the Four Step Scheme introduced by Ma,et al.(1994) is still effective in this case.Namely,the authors show that the adapted solution of the FBSDE exists and is unique over any prescribedtime duration;and the backward components can be determined explicitly by the forward componentvia the classical solution to a system of parabolic integro-partial differential equations.An importantconsequence the authors would like to draw from this fact is that,contrary to the general belief,in aMarkovian set-up the martingale representation theorem is no longer the reason for the well-posednessof the FBSDE,but rather a consequence of the existence of the solution of the decoupling integralpartialdifferential equation.Finally,the authors briefly discuss the possibility of reducing the regularityrequirements of the coefficients by using a scheme proposed by F.Delarue (2002) to the current case.

Turnpike Properties for Stochastic Linear-Quadratic Optimal Control Problems

This paper analyzes the limiting behavior of stochastic linear-quadratic optimal control problems in finite time-horizon[0,T]as T→∞.The so-called turnpike properties are established for such problems,under stabilizability condition which is weaker than the controllability,normally imposed in the similar problem for ordinary differential systems.In dealing with the turnpike problem,a crucial issue is to determine the corresponding static optimization problem.Intuitively mimicking the deterministic situations,it seems to be natural to include both the drift and the diffusion expressions of the state equation to be zero as constraints in the static optimization problem.However,this would lead us to a wrong direction.It is found that the correct static problem should contain the diffusion as a part of the objective function,which reveals a deep feature of the stochastic turnpike problem.

Heat Equation with Memory in Anisotropic and Non-Homogeneous Media

A modified Fourier＇s law in an anisotropic and non-homogeneous media results in a heat equation with memory, for which the memory kernel is matrix-valued and spatially dependent. Different conditions on the memory kernel lead to the equation being either a parabolic type or a hyperbolic type. Well-posedness of such a heat equation is established under some general and reasonable conditions. It is shown that the propagation speed for heat pulses could be either infinite or finite, depending on the different types of the memory kernels. Our analysis indicates that, in the framework of linear theory, heat equation with hyperbolic kernel is a more realistic model for the heat conduction, which might be of some interest in physics.

Mean-field stochastic linear quadratic optimal control problems: closed-loop solvability

An optimal control problem is studied for a linear mean-field stochastic differential equation with a quadratic cost functional.The coefficients and the weighting matrices in the cost functional are all assumed to be deterministic.Closedloop strategies are introduced,which require to be independent of initial states;and such a nature makes it very useful and convenient in applications.In this paper,the existence of an optimal closed-loop strategy for the system(also called the closedloop solvability of the problem)is characterized by the existence of a regular solution to the coupled two(generalized)Riccati equations,together with some constraints on the adapted solution to a linear backward stochastic differential equation and a linear terminal value problem of an ordinary differential equation.

Backward stochastic differential equations and backward stochastic Volterra integral equations with anticipating generators

For a backward stochastic differential equation(BSDE,for short),when the generator is not progressively measurable,it might not admit adapted solutions,shown by an example.However,for backward stochastic Volterra integral equations(BSVIEs,for short),the generators are allowed to be anticipating.This gives,among other things,an essential difference between BSDEs and BSVIEs.Under some proper conditions,the well-posedness of such BSVIEs is established.Further,the results are extended to path-dependent BSVIEs,in which the generators can depend on the future paths of unknown processes.An additional finding is that for path-dependent BSVIEs,in general,the situation of anticipating generators is not avoidable,and the adaptedness condition similar to that imposed for anticipated BSDEs by Peng−Yang[22]is not necessary.