In this paper we develop a numerical method for efficiently approximating solutions of certain Zakai equations in high dimensions. The key idea is to transform given SPDE into PDE with random coefficients. We show that under suitable regularity assumptions on the coefficients equation, corresponding admits solution field which, almost all realizations coefficients, can be written as classical l...

We establish a probabilistic representation for a wide class of linear deterministic p.d.e.s with potential term, including the wave equation in spatial dimensions 1 to 3. Our representation applies to the heat equation, where it is related to the classical Feynman-Kac formula, as well as to the telegraph and beam equations. If the potential is a (random) spatially homogeneous Gaussian noise, t...

<p style='text-indent:20px;'>In this paper, we propose a model to price vulnerable European options where the dynamics of underlying asset value and counter-party's follow two jump-diffusion processes with fast mean-reverting stochastic volatility. First, derive an equivalent risk-neutral measure transfer pricing problem into solving partial differential equation (PDE) by Feynman-Kac form...

Motivated by the development of efficient Monte Carlo methods for PDE models in molecular dynamics, we establish a new probabilistic interpretation of a family of divergence form operators with discontinuous coefficients at the interface of two open subsets of R. This family of operators includes the case of the linearized Poisson-Boltzmann equation used to compute the electrostatic free energy...

In this paper, a hybrid approach for solving the Laplace equation in general threedimensional (3-D) domains is presented. The approach is based on a local method for the Dirichletto-Neumann (DtN) mapping of a Laplace equation by combining a deterministic (local) boundary integral equation (BIE) method and the probabilistic Feynman–Kac formula for solutions of elliptic partial differential equat...

Interacting particle methods are increasingly used to sample from complex high-dimensional distributions. They have found a wide range of applications in applied probability, Bayesian statistics and information engineering. Understanding rigorously these new Monte Carlo simulation tools leads to fascinating mathematics related to Feynman-Kac path integral theory and their interacting particle i...

In this article, we construct a Stratonovich solution for the stochastic wave equation in spatial dimension $$d \le 2$$ , with time-independent noise and linear term $$\sigma (u)=u$$ multiplying noise. The is spatially homogeneous its spectral measure satisfies an integrability condition which stronger than Dalang’s condition. We give probabilistic representation solution, similar to Feynman–Ka...

We consider stochastic differential equations driven by a general L\'evy processes (SDEs) with infinite activity and the related, via Feynman-Kac formula, Dirichlet problem for parabolic integro-differential equation (PIDE). approximate solution of PIDE using numerical method SDEs. The is based on three ingredients: (i) we small jumps diffusion; (ii) use restricted jump-adaptive time-stepping; ...