Consider the Wonham optimal ltering problem for a nite state ergodic Markov process in both discrete and continuous time, and let be the noise intensity for the observation. We examine the sensitivity of the solution with respect to the lter's initial conditions in terms of the gap between the rst two Lyapunov exponents of the Zakai equation for the unnormalized conditional probability. This ga...

Neural networks are increasingly used to construct numerical solution methods for partial differential equations. In this expository review, we introduce and contrast three important recent approaches attractive in their simplicity suitability high-dimensional problems: physics-informed neural networks, based on the Feynman–Kac formula of backward stochastic The article is accompanied by a suit...

The article develops and investigates a new model for the formation of interest rates on consumer loans based an analysis commercial interests logic behavior banks. assumes that borrowers’ incomes are described by geometric Brownian motion. Commercial banks assess default risk borrowers. According to Feynman–Kac formula, assessment is reduced solving boundary value problem partial differential ...

We derive a local-time path-integral representation for a generic one-dimensional time-independent system. In particular, we show how to rephrase the matrix elements of the Bloch density matrix as a path integral over x-dependent local-time profiles. The latter quantify the time that the sample paths x(t) in the Feynman path integral spend in the vicinity of an arbitrary point x. Generalization...

A simulation based method for the numerical solution of PDEs with random coefficients is presented. By the Feynman-Kac formula, the solution can be represented as conditional expectation of a functional of a corresponding stochastic differential equation driven by independent noise. A time discretization of the SDE for a set of points in the domain and a subsequent Monte Carlo regression lead t...

Consider the Wonham optimal filtering problem for a finite state ergodic Markov process in both discrete and continuous time, and let σ be the noise intensity for the observation. We examine the sensitivity of the solution with respect to the filter’s initial conditions in terms of the gap between the first two Lyapunov exponents of the Zakai equation for the unnormalized conditional probabilit...

We study Cauchy’s problem for a second-order linear parabolic stochastic partial differential equation (SPDE) driven by a cylindrical Brownian motion. Existence and uniqueness of a generalized (soft) solution is established in Sobolev, Hölder, and Lipschitz classes. We make only minimal assumptions, virtually identical to those common to similar deterministic problems. A stochastic Feynman–Kac ...

On the grounds of a Feynman-Kac–type formula for Hamiltonian lattice systems we derive analytical expressions for the matrix elements of the evolution operator. These expressions are valid at long times when a central limit theorem applies. As a remarkable result we find that the ground-state energy as well as all the correlation functions in the ground state are determined semi-analytically by...

A Lower bound for the density of a Bose gas in Z in the grand canonical ensemble is calculated for general dimension d, and for l(Z) potential interactions. The rigorous formalism of the measure on continuous time random walks, analogous to the Wiener measure, is briefly explained, along with justification of the Feynman-Kac formula using this. These are then used to bound the density, although...

In [3], the authors proved an existence result for BSDEs with quadratic generators with respect to the variable z and with unbounded terminal conditions. However, no uniqueness result was stated in that work. The main goal of this paper is to fill this gap. In order to obtain a comparison theorem for this kind of BSDEs, we assume that the generator is convex with respect to the variable z. Unde...