具有一般终端时刻和非一致线性增长生成元的倒向随机微分方程的$L^p$解

In this paper, we establish the existence of the minimal L~p(p 1) solution of backward stochastic differential equations(BSDEs) where the time horizon may be finite or infinite and the generators have a non-uniformly linear growth with respect to t. The main idea is to construct a sequence of solutions {(Y~n, Z~n)} which is a Cauchy sequence in S~p× M~p space, and finally we prove {(Y~n, Z~n)} converges to the L~p(p 1) solution of BSDEs.     

L~P SOLUTION OF GENERAL MEAN-FIELD BSDES WITH CONTINUOUS COEFFICIENTS

In this paper we consider one dimensional mean-field backward stochastic differential equations (BSDEs) under weak assumptions on the coefficient.Unlike[3],the generator of our mean-field BSDEs depends not only on the solution (Y,Z) but also on the law P_(Y )of Y.The first part of the paper is devoted to the existence and uniqueness of solutions in L~p,1p≤2,where the monotonicity conditions are satisfied.Next,we show that if the generator f is uniformly continuous in (μ,y,z),uniformly with respect to (t,ω),and if the terminal valueξbelongs to L~p(?,F,P) with 1p≤2,the mean-field BSDE has a unique L~psolution.     

FULLY COUPLED FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH GENERAL MARTINGALE

The article first studies the fully coupled Forward-Backward Stochastic Differential Equations (FBSDEs) with the continuous local martingale. The article is mainly divided into two parts. In the first part, it considers Backward Stochastic Differential Equations (BSDEs) with the continuous local martingale. Then, on the basis of it, in the second part it considers the fully coupled FBSDEs with the continuous local martingale. It is proved that their solutions exist and are unique under the monotonicity conditions.     

Properties of Solutions of BSDEs with Integrable Parameters

The main result of this study is to obtain,using the localization method in Briand et al.Levi,Fatou and Lebesgue type theorems for the solutions of certain one-dimensional backward stochastic differentialequation(BSDEs)with integrable parameters with respect to the terminal condition.     

GLOBAL STABILITY OF SOLUTIONS WITH DISCONTINUOUS INITIAL DATA CONTAINING VACUUM STATES FOR THE RELATIVISTIC EULER EQUATIONS

The global stability of Lipschitz continuous solutions with discontinuous initial data for the relativistic Euler equations is established in a broad class of entropy solutions in L∞containing vacuum states. As a corollary, the uniqueness of Lipschitz solutions with discontinuous initial data is obtained in the broad class of entropy solutions in     

Uniform invariance principle of discontinuous systems with parameter variation

An extension of the invariance principle for a class of discontinuous righthand sides systems with parameter variation in the Filippov sense is proposed. This extension allows the derivative of an auxiliary function V, also called a Lyapunov-like function, along the solutions of the discontinuous system to be positive on some sets. The uniform estimates of attractors and basin of attractions with respect to parameters are also obtained. To this end, we use locally Lipschitz continuous and regular Lyapunov functions, as well as Filippov theory. The obtained results settled in the general context of differential inclusions, and through a uniform version of the LaSalle invariance principle. An illustrative example shows the potential of the theoretical results in providing information on the asymptotic behavior of discontinuous systems.     

Constrained LQ Problem with a Random Jump and Application to Portfolio Selection

This paper deals with a constrained stochastic linear-quadratic(LQ for short)optimal control problem where the control is constrained in a closed cone. The state process is governed by a controlled SDE with random coefficients. Moreover, there is a random jump of the state process. In mathematical finance, the random jump often represents the default of a counter party. Thanks to the It-Tanaka formula, optimal control and optimal value can be obtained by solutions of a system of backward stochastic differential equations(BSDEs for short). The solvability of the BSDEs is obtained by solving a recursive system of BSDEs driven by the Brownian motions. The author also applies the result to the mean variance portfolio selection problem in which the stock price can be affected by the default of a counterparty.     

Macroscopic regularity for the Boltzmann equation

The regularity of solutions to the Boltzmann equation is a fundamental problem in the kinetic theory. In this paper, the case with angular cut-off is investigated. It is shown that the macroscopic parts of solutions to the Boltzmann equation, i.e., the density, momentum and total energy are continuous functions of(x, t) in the region R3×(0, +∞). More precisely, these macroscopic quantities immediately become continuous in any positive time even though they are initially discontinuous and the discontinuities of solutions propagate only in the microscopic level. It should be noted that such kind of phenomenon can not happen for the compressible Navier-Stokes equations in which the initial discontinuities of the density never vanish in any finite time, see [22]. This hints that the Boltzmann equation has better regularity effect in the macroscopic level than compressible Navier-Stokes equations.     

Solutions to BSDEs driven by both fractional Brownian motions and the underlying standard Brownian Motions

The local existence and uniqueness of the solutions to backward stochastic differential equations(BSDEs, in short) driven by both fractional Brownian motions with Hurst parameter H ∈(1/2, 1) and the underlying standard Brownian motions are studied. The generalization of the It formula involving the fractional and standard Brownian motions is provided. By theory of Malliavin calculus and contraction mapping principle, the local existence and uniqueness of the solutions to BSDEs driven by both fractional Brownian motions and the underlying standard Brownian motions are obtained.     

DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR A FORWARD-BACKWARD HEAT EQUATION

A space-time finite element method,discontinuous in time but continuous in space, is studied to solve the nonlinear forward-backward heat equation. A linearized technique is introduced in order to obtain the error estimates of the approximate solutions. And the numerical simulations are given.