In this paper, we study spatial averages for the parabolic Anderson model in Skorohod sense driven by rough Gaussian noise, which is colored space and time. We include case of a fractional noise with Hurst parameters $H_0$ time $H_1$ space, satisfying $H_0 \in (1/2,1)$, $H_1\in (0,1/2)$ + H_1 > 3/4$. Our main result functional central limit theorem averages. As an important ingredient our analy...

Abstract This article addresses the problem of pricing European options when underlying asset is not perfectly liquid. A liquidity discounting factor as a function market-wide governed by mean-reverting stochastic process and sensitivity price to firstly introduced, so that impact on can be captured option model. The characteristic analytically worked out using Feynman–Kac theorem closed-form f...

We study the Feynman-Kac semigroup generated by the Schrödinger operator based on the fractional Laplacian −(−∆)−q in R, for q ≥ 0, α ∈ (0, 2). We obtain sharp estimates of the first eigenfunction φ1 of the Schrödinger operator and conditions equivalent to intrinsic ultracontractivity of the Feynman-Kac semigroup. For potentials q such that lim|x|→∞ q(x) = ∞ and comparable on unit balls we obta...

Kac integral [1, 2, 3] appears as a path-wise presentation of Brownian motion and shortly becomes, with Feynman approach [4], a powerful tool to study different processes described by the wave-type or diffusion-type equations. In the basic papers [1, 4], the paths distribution was based on averaging over the Wiener measure. It is worthwhile to mention the Kac comment that the Wiener measure can...

The Feynman-Kac formula provides a way to understand solutions elliptic partial differential equations in terms of expectations continuous time Markov processes. This connection allows for the creation numerical schemes based on samples these processes which have advantages over traditional methods some cases. However, naïve implementations suffer from issues related statistical bias and sampli...

We find Feynman–Kac type representation theorems for generalized diffusions. To do this we need to establish existence, uniqueness and regularity results for equations with measure-valued coefficients.