Related papers: Feynman-Kac Formulas for Solutions to Degenerate E…
We prove existence and uniqueness of stochastic representations for solutions to elliptic and parabolic boundary value and obstacle problems associated with a degenerate Markov diffusion process. In particular, our article focuses on the…
We prove and implement stochastic solution (or Feynman-Kac) formulas for boundary value problems involving the spectral fractional Laplacian with nonzero Dirichlet boundary condition. The main tools used in the proofs are the abstract…
The Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance, is a paradigm for a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square…
The Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance, is a paradigm for a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square…
This paper establishes a Feynman-Kac formula to represent the solution to general time inhomogeneous stochastic parabolic partial differential equations driven by multiplicative fractional Gaussian noises in bounded domain where L_t is a…
We study the stochastic solution to a Cauchy problem for a degenerate parabolic equation arising from option pricing. When the diffusion coefficient of the underlying price process is locally H\"older continuous with exponent $\delta\in (0,…
We present a computational alternative to probabilistic simulations for non-smooth stochastic dynamical systems that are prevalent in engineering mechanics. As examples, we target (1) stochastic elasto-plastic problems, which involve…
A representation formula for solutions of stochastic partial differential equations with Dirichlet boundary conditions is proved. The scope of our setting is wide enough to cover the general situation when the backward characteristics that…
The Heston stochastic volatility process is a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this process…
Motivated by entropic optimal transport, we investigate an extended notion of solution to the parabolic equation $( \partial_t + b\cdot \nabla + \Delta _{ a}/2 +V)g =0$ with a final boundary condition. It is well-known that the viscosity…
We propose a novel non-compact, positivity-preserving scheme for linear non-divergence form elliptic equations. Based on the Feynman--Kac formula, the solution is represented as a conditional expectation associated with a diffusion…
The classical Feynman-Kac identity represents solutions of linear partial differential equations in terms of stochastic differential euqations. This representation has been generalized to nonlinear partial differential equations on the one…
The Feynman-Kac formula provides a way to understand solutions to elliptic partial differential equations in terms of expectations of continuous time Markov processes. This connection allows for the creation of numerical schemes for…
Probabilistic solutions of the so called Schr\"{o}dinger boundary data problem provide for a unique Markovian interpolation between any two strictly positive probability densities designed to form the input-output statistics data for the…
It is known that Markovian forward-backward stochastic differential equations provide nonlinear Feynman-Kac representation formulae for semilinear parabolic PDEs. We show that non-Markovian forward-backward stochastic differential equations…
We prove existence of solutions to boundary value problems and obstacle problems for degenerate-elliptic, linear, second-order partial differential operators with partial Dirichlet boundary conditions using a new version of the Perron…
The challenge to fruitfully merge state-of-the-art techniques from mathematical finance and numerical analysis has inspired researchers to develop fast deterministic option pricing methods. As a result, highly efficient algorithms to…
This article is devoted to the stochastic anticipating equations with the extended stochastic integral with respect to the Gaussian processes of a special type. In the particular cases the solutions of such an equations are the well-known…
We prove a stochastic representation formula for the viscosity solution of Dirichlet terminal-boundary value problem for a degenerate Hamilton-Jacobi-Bellman integro-partial differential equation in a bounded domain. We show that the unique…
Let $D\subset R^d$ be a bounded domain and denote by $\mathcal P(D)$ the space of probability measures on $D$. Let \begin{equation*} L=\frac12\nabla\cdot a\nabla +b\nabla \end{equation*} be a second order elliptic operator. Let…