Semi-linear degenerate backward stochastic partial differential equations and associated forward–backward stochastic differential equations

In this paper, we consider the Cauchy problem of semi-linear degenerate backward stochastic partial differential equations (BSPDEs) under general settings without technical assumptions on the coefficients. For the solution of semi-linear degenerate BSPDE, we first give a proof for its existence and uniqueness, as well as regularity. Then the connection between semi-linear degenerate BSPDEs and forward–backward stochastic differential equations (FBSDEs) is established, which can be regarded as an extension of the Feynman–Kac formula to the non-Markovian framework.     

Strong solution of Itô type set-valued stochastic differential equation

In this paper, we shall firstly illustrate why we should introduce an It5 type set-valued stochastic differential equation and why we should notice the almost everywhere problem. Secondly we shall give a clear definition of Aumann type Lebesgue integral and prove the measurability of the Lebesgue integral of set-valued stochastic processes with respect to time t. Then we shall present some new properties, especially prove an important inequality of set-valued Lebesgue integrals. Finally we shall prove the existence and the uniqueness of a strong solution to the It5 type set-valued stochastic differential equation.     

Anticipated mean-field backward stochastic differential equations with jumps∗

Lithuanian Mathematical Journal - In this paper, we prove an existence and uniqueness theorem and a comparison theorem for a class of anticipated mean-field backward stochastic differential...     

Solvability of forward–backward stochastic partial differential equations

In this paper we study the solvability of a class of fully-coupled forward–backward stochastic partial differential equations (FBSPDEs). These FBSPDEs cannot be put into the framework of stochastic evolution equations in general, and the usual decoupling methods for the Markovian forward–backward SDEs are difficult to apply. We prove the well-posedness of the FBSPDEs, under various conditions on the coefficients, by using either the method of contraction mapping or the method of continuation. These conditions, especially in the higher dimensional case, are novel in the literature.     

Limit theorem and uniqueness theorem of backward stochastic differential equations

This paper establishes a limit theorem for solutions of backward stochastic differential equations (BSDEs). By this limit theorem, this paper proves that, under the standard assumption g(t,y,0) = 0, the generator g of a BSDE can be uniquely determined by the corresponding g-expectationεg;this paper also proves that if a filtration consistent expectation S can be represented as a g-expectationεg, then the corresponding generator g must be unique.     

Reflected forward–backward stochastic differential equations with continuous monotone coefficients

In this work, we prove that there exists at least one solution for the reflected forward–backward stochastic differential equations satisfying the obstacle constraint with continuous monotone coefficients. The distinct character of our result is that the coefficient of the forward SDEs contains the solution variable of the reflected BSDEs.     

Stability of linear impulsive Itô differential equations with bounded delays

We use the method of “model” equations to study the exponential p-stability (2 ≤ p < ∞) of the trivial solution with respect to the initial function for a linear impulsive system of Itô differential equations with bounded delays. The specific form of the equation and the method used permit one to analyze the stability of solutions starting from an arbitrary point of the half-line [0,∞) and obtain constructive sufficient conditions in terms of the parameters of the equations to be studied.     

New explicit stabilized stochastic Runge-Kutta methods with weak second order for stiff Itô stochastic differential equations

Numerical Algorithms - This paper introduces a new class of weak second-order explicit stabilized stochastic Runge-Kutta methods for stiff Itô stochastic differential equations. The...     

On representations of solutions of 1–dimensional stochastic differential equations with reflecting boundary conditions

In this paper we give representations of the solution of 1–dimensional stochastic differential equation (SDE for short) with reflecting barrieres. To this we construct the solution of deterministic Skorohod equation with two reflecting boundaries and show which can be expressed by the operator “sup inf”. Then the solution of given SDE can be represented by a form that depend on a reflecting Brownian motion determined by solving the deterministic Skorohod eqyation     

A class of new Magnus-type methods for semi-linear non-commutative Itô stochastic differential equations

Numerical Algorithms - In this paper, a class of new Magnus-type methods is proposed for non-commutative Itô stochastic differential equations (SDEs) with semi-linear drift term and...     

Pathwise uniqueness of multi-dimensional stochastic differential equations with H?lder diffusion coefficients

We extend Yamada-Watababe’s criterion [J. Math. Kyoto Univ., 1971, 11: 553–563] on the pathwise uniqueness of one-dimensional stochastic differential equations to a special class of multi-dimensional stochastic differential equations.     

A Legendre-based computational method for solving a class of Itô stochastic delay differential equations

Numerical Algorithms - This paper provides a numerical method for solving a class of Itô stochastic delay differential equations (SDDEs). The method’s novelty is its use of the spectral...     

Carathéodory approximate solutions for a class of perturbed stochastic differential equations with reflecting boundary

Abstract

In this work, we study the Carathéodory approximate solution for a class of one-dimensional perturbed stochastic differential equations with reflecting boundary (PSDERB). Based on the Carathéodory approximation procedure, we prove that PSDERB have a unique solution and show that the Carathéodory approximate solution converges to the solution of PSDERB whose both drift and diffusion coefficients are non-Lipschitz. After that, we establish an explicit rate of convergence in the case of PSDERB with Lipschitz coefficients.     

Mean-square A-stable diagonally drift-implicit integrators of weak second order for stiff Itô stochastic differential equations

We introduce two drift-diagonally-implicit and derivative-free integrators for stiff systems of Itô stochastic differential equations with general non-commutative noise which have weak order 2 and deterministic order 2, 3, respectively. The methods are shown to be mean-square A-stable for the usual complex scalar linear test problem with multiplicative noise and improve significantly the stability properties of the drift-diagonally-implicit methods previously introduced (Debrabant and Rößler, Appl. Numer. Math. 59(3–4):595–607, 2009).     

Reflected forward–backward stochastic differential equations and related PDEs

In this article, we study a type of coupled reflected forward–backward stochastic differential equations (reflected FBSDEs, for short) with continuous coefficients, including the existence and the uniqueness of the solution of our reflected FBSDEs as well as the comparison theorem. We prove that the solution of our reflected FBSDEs gives a probabilistic interpretation for the viscosity solution of an obstacle problem for a quasilinear parabolic partial differential equation.     

Strong convergence rates for backward Euler–Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients

In this work, we generalize the current theory of strong convergence rates for the backward Euler–Maruyama scheme for highly non-linear stochastic differential equations, which appear in both mathematical finance and bio-mathematics. More precisely, we show that under a dissipative condition on the drift coefficient and super-linear growth condition on the diffusion coefficient the BEM scheme converges with strong order of a half. This type of convergence gives theoretical foundations for efficient variance reduction techniques for Monte Carlo simulations. We support our theoretical results with relevant examples, such as stochastic population models and stochastic volatility models.     

Stochastic optimization theory of backward stochastic differential equations with jumps and viscosity solutions of Hamilton–Jacobi–Bellman equations

In this paper we study stochastic optimal control problems with jumps with the help of the theory of Backward Stochastic Differential Equations (BSDEs) with jumps. We generalize the results of Peng [S. Peng, BSDE and stochastic optimizations, in: J. Yan, S. Peng, S. Fang, L. Wu, Topics in Stochastic Analysis, Science Press, Beijing, 1997 (Chapter 2) (in Chinese)] by considering cost functionals defined by controlled BSDEs with jumps. The application of BSDE methods, in particular, the use of the notion of stochastic backward semigroups introduced by Peng in the above-mentioned work allows a straightforward proof of a dynamic programming principle for value functions associated with stochastic optimal control problems with jumps. We prove that the value functions are the viscosity solutions of the associated generalized Hamilton–Jacobi–Bellman equations with integral-differential operators. For this proof, we adapt Peng’s BSDE approach, given in the above-mentioned reference, developed in the framework of stochastic control problems driven by Brownian motion to that of stochastic control problems driven by Brownian motion and Poisson random measure.