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Guomin Liu

Publications and source records attributed to Guomin Liu.

13 recordsLinked to original sources
arXiv2025

Relationship between maximum principle and dynamic programming principle for recursive optimal control problem of stochastic evolution equations

This paper aims to study the relationship between the maximum principle and the dynamic programming principle for recursive optimal control problem of stochastic evolution equations, where the control domain is not necessarily convex and the value function may be nonsmooth. By making use of the notion of conditionally expected operator-valued backward stochastic integral equations, we establish a connection between the first and second-order adjoint processes in MP and the general derivatives of the value function. Under certain additional assumptions, the value function is shown to be $C^{1,1}$-regular. Furthermore, we discuss the smooth case and present several applications of our results.

math.OC↗
arXiv2025

Anticipated backward stochastic evolution equations and maximum principle for path-dependent systems in infinite dimensions

For a class of path-dependent stochastic evolution equations driven by cylindrical $Q$-Wiener process, we study the Pontryagin's maximum principle for the stochastic recursive optimal control problem. In this infinite-dimensional control system, the state process depends on its past trajectory, the control is delayed via an integral with respect to a general finite measure, and the final cost relies on the delayed state.To obtain the maximum principle, we introduce a functional adjoint operator for the non-anticipative path derivative and establish the well-posedness of an anticipated backward stochastic evolution equation in the path-dependent form, which serves as the adjoint equation.

math.OC↗
arXiv2024

Maximum principle for optimal control of stochastic evolution equations with recursive utilities

We consider the optimal control problem of stochastic evolution equations in a Hilbert space under a recursive utility, which is described as the solution of a backward stochastic differential equation (BSDE). A very general maximum principle is given for the optimal control, allowing the control domain not to be convex and the generator of the BSDE to vary with the second unknown variable $z$. The associated second-order adjoint process is characterized as a unique solution of a conditionally expected operator-valued backward stochastic integral equation.

math.OC↗
arXiv2024

Multi-dimensional reflected BSDEs driven by $G$-Brownian motion with diagonal generators

We consider the well-posedness problem of multi-dimensional reflected backward stochastic differential equations driven by $G$-Brownian motion ($G$-BSDEs) with diagonal generators. Two methods, i.e., the penalization method and the Picard iteration argument, are provided to prove the existence and uniqueness of solutions. We also study its connection with the obstacle problem of a system of fully nonlinear PDEs.

math.PR↗
arXiv2024

Maximum principle for recursive optimal control problem of stochastic delay evolution equations

For a class of stochastic delay evolution equations driven by cylindrical $Q$-Wiener process, we study the Pontryagin's maximum principle for the stochastic recursive optimal control problem. The delays are given as moving averages with respect to general finite measures and appear in all the coefficients of the control system. In particular, the final cost can contain the state delay. To derive the main result, we introduce a new form of anticipated backward stochastic evolution equations with terminals acting on an interval as the adjoint equations of the delayed state equations and deploy a proper dual analysis between them. Under certain convex assumption on the coefficient function and the Hamiltonian, we also show sufficiency of the maximum principle.

math.OC↗
arXiv2020

Exit times for semimartingales under nonlinear expectation

Let $\mathbb{\hat{E}}$ be the upper expectation of a weakly compact but non-dominated family $\mathcal{P}$ of probability measures. Assume that $Y$ is a $d$-dimensional $\mathcal{P}$-semimartingale under $\mathbb{\hat{E}}$. Given an open set $Q\subset\mathbb{R}^{d}$, the exit time of $Y$ from $Q$ is defined by \[ τ_{Q}:=\inf\{t\geq0:Y_{t}\in Q^{c}\}. \] The main objective of this paper is to study the quasi-continuity properties of $τ_{Q}$ under the nonlinear expectation $\mathbb{\hat{E}}$. Under some additional assumptions on the growth and regularity of $Y$, we prove that $τ_{Q}\wedge t$ is quasi-continuous if $Q$ satisfies the exterior ball condition. We also give the characterization of quasi-continuous processes and related properties on stopped processes. In particular, we get the quasi-continuity of exit times for multi-dimensional $G$-martingales, which nontrivially generalizes the previous one-dimensional result of Song.

math.PR↗
arXiv2020

Girsanov formula for $G$-Brownian motion: the degenerate case

In this paper, we prove the Girsanov formula for $G$-Brownian motion without the non-degenerate condition. The proof is based on the perturbation method in the nonlinear setting by constructing a product space of the $G$-expectation space and a linear space that contains a standard Brownian motion. The estimates for exponential martingale of $G$-Brownian motion are important for our arguments.

math.PR↗
arXiv2020

Multi-dimensional BSDEs driven by $G$-Brownian motion and related system of fully nonlinear PDEs

In this paper, we study the well-posedness of multi-dimensional backward stochastic differential equations driven by $G$-Brownian motion ($G$-BSDEs) with diagonal generators, the $z$ parts of whose $l$-th components only depend on the $l$-th columns. The existence and uniqueness of solutions are obtained via a contraction argument for $Y$ component and a backward iteration of local solutions. Furthermore, we show that, the solution of multi-dimensional $G$-BSDE in a Markovian framework provides a probabilistic formula for the viscosity solution of a system of nonlinear parabolic partial differential equations.

math.PR↗
arXiv2018

Local time and Tanaka formula of $G$-martingales

The objective of this paper is to study the local time and Tanaka formula of symmetric $G$-martingales. We introduce the local time of $G$-martingales and show that they belong to $G$-expectation space $L_{G}^{2}(Ω_{T})$. The bicontinuous modification of local time is obtained. We finally give the Tanaka formula for convex functions of $G$-martingales.

math.PR↗
arXiv2018

On the exit times of SDEs driven by $G$-Brownian motion

This paper is devoted to studying the properties of the exit times of stochastic differential equations driven by $G$-Brownian motion ($G$-SDEs). In particular, we prove that the exit times of $G$-SDEs has the quasi-continuity property. As an application, we give a probabilistic representation for a large class of fully nonlinear elliptic equations with Dirichlet boundary.

math.PR↗
arXiv2017

On the strong Markov property for stochastic differential equations driven by $G$-Brownian motion

In this paper we study the stochastic differential equations driven by $G$-Brownian motion ($G$-SDEs for short). We extend the notion of conditional $G$-expectation from deterministic time to the more general optional time situation. Then, via this conditional expectation, we develop the strong Markov property for $G$-SDEs. In particular, we obtain the strong Markov property for $G$-Brownian motion. Some applications including the reflection principle for $G$-Brownian motion are also provided.

math.PR↗