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A large deviation principle is established for a two-scale stochastic system in which the slow component is a continuous process given by a small noise finite dimensional It\^{o} stochastic differential equation, and the fast component is a…

The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order $\beta \in (0,1)$. The fundamental solution for the Cauchy problem is…

The present work considers diffusive shock acceleration at non-relativistic shocks using a system of stochastic differential equations (SDE) equivalent to the Fokker-Planck equation. We compute approximate solutions of the transport of…

Consider jump-type stochastic differential equations with the drift, diffusion and jump terms. Logarithmic derivatives of densities for the solution process are studied, and the Bismut-Elworthy-Li type formulae can be obtained under the…

Existence and uniqueness results of fully coupled forward stochastic differential equations without drifts and backward stochastic differential equations in a degenerate case are obtained for an arbitrarily large time duration.

We introduce a guided stochastic sampling method that augments sampling from diffusion models with physics-based guidance derived from partial differential equation (PDE) residuals and observational constraints, ensuring generated samples…

This work concerns generalized backward stochastic differential equations, which are coupled with a family of reflecting diffusion processes. First of all, we establish the large deviation principle for forward stochastic differential…

This article introduces and solves a general class of fully coupled forward-backward stochastic dynamics by investigating the associated system of functional differential equations. As a consequence, we are able to solve many different…

Diffusion models have emerged as powerful generative tools with applications in computer vision and scientific machine learning (SciML), where they have been used to solve large-scale probabilistic inverse problems. Traditionally, these…

We propose a new, unified approach to solving jump-diffusion partial integro-differential equations (PIDEs) that often appear in mathematical finance. Our method consists of the following steps. First, a second-order operator splitting on…

The inverse problem of backward diffusion is known to be ill-posed and highly unstable. Backward diffusion processes appear naturally in image enhancement and deblurring applications. It is therefore greatly desirable to establish a…

We obtain the variational equations for backward stochastic differential equations in recursive stochastic optimal control problems, and then get the maximum principle which is novel. The control domain need not be convex, and the generator…

Stemmed from the derivation of the optimal control to a stochastic linear-quadratic control problem with Markov jumps, we study one kind of backward stochastic differential equations (BSDEs) that the generator f is affected by a Markovian…

In this paper, we study the solvability of a class of multi-dimensional forward backward stochastic differential equations (FBSDEs) with oblique reflection and unbounded stopping time. Under some mild assumptions on the coefficients in such…

In this paper, we study the diffusion approximation for slow-fast stochastic differential equations with state-dependent switching, where the slow component $X^{\varepsilon}$ is the solution of a stochastic differential equation with…

We derive novel algorithms for optimization problems constrained by partial differential equations describing multiscale particle dynamics, including non-local integral terms representing interactions between particles. In particular, we…

We investigate evolution equations for anomalous diffusion employing fractional derivatives in space and time. Linkage between the space-time variables leads to a new type of fractional derivative operator. Fractional diffusion equations…

In this paper we establish a comparison theorem for stochastic differential delay equations with jumps. An example is constructed to demonstrate that the comparison theorem need not hold whenever the diffusion term contains a delay function…

In this paper, we study both the direct and inverse random source problems associated with the multi-term time-fractional diffusion-wave equation driven by a fractional Brownian motion. Regarding the direct problem, the well-posedness is…

This paper presents a novel approach to numerically solve stochastic differential games for nonlinear systems. The proposed approach relies on the nonlinear Feynman-Kac theorem that establishes a connection between parabolic deterministic…