Related papers: It\^{o}'s formula for linear fractional PDEs

We consider random variables of the form $F=f(V_1,...,V_n)$, where $f$ is a smooth function and $V_i,i\in\mathbb{N}$, are random variables with absolutely continuous law $p_i(y) dy$. We assume that $p_i$, $i=1,...,n$, are piecewise…

In general, adding a stochastic perturbation to a differential equation possessing an invariant manifold destroys the invariance as far as the It\^o formalism is used. In this article, we propose an invariantization method for perturbations…

In this paper, a Feynman-Kac formula is established for stochastic partial differential equation driven by Gaussian noise which is, with respect to time, a fractional Brownian motion with Hurst parameter $H<1/2$. To establish such a…

For the fundamental solutions of heat-type equations of order $n$ we give a general stochastic representation in terms of damped oscillations with generalized gamma distributed parameters. By composing the pseudo-process $X_n$ related to…

We give a probabilistic numerical method for solving a partial differential equation with fractional diffusion and nonlinear drift. The probabilistic interpretation of this equation uses a system of particles driven by L\'evy alpha-stable…

A peculiar feature of It\^o's calculus is that it is an integral calculus that gives no explicit derivative with a systematic differentiation theory counterpart, as in elementary calculus. So, can we define a pathwise stochastic derivative…

In this paper we obtain a Feynman-Kac formula for the solution of a fractional stochastic heat equation driven by fractional noise. One of the main difficulties is to show the exponential integrability of some singular nonlinear functionals…

In the present work we discuss higher order multi-term partial differential equation (PDE) with the Caputo-Fabrizio fractional derivative in time. We investigate a boundary value problem for fractional heat equation involving higher order…

A computational method for numeric resolution of a PDEs system, based on a Finite Differences schema integrated by interpolations of partial results, and an estimate of the error of its solution respect to the normal FD solution.

The paper suggests a way of stochastic integration of random integrands with respect to fractional Brownian motion with the Hurst parameter H> 1/2. The integral is defined initially on the processes that are "piecewise" predictable on a…

In this paper we give a concrete application of the spectral theory based on the notion of $S$-spectrum to fractional diffusion process. Precisely, we consider the Fourier law for the propagation of the heat in non homogeneous materials,…

The applicability of stochastic differential equations to thermodynamics is considered and a new form, different from the classical Ito and Stratonovich forms, is introduced. It is shown that the new presentation is more appropriate for the…

In this paper we obtain an It\^o differential representation for a class of singular stochastic Volterra integral equations. As an application, we investigate the rate of convergence in the small time central limit theorem for the solution.

The object of this paper is to present a computable solution of a fractional partial differential equation associated with a Riemann-Liouville derivative of fractional order as the time-derivative and Riesz-Feller fractional derivative as…

Given a multi-dimensional It\^{o} process whose drift and diffusion terms are adapted processes, we construct a weak solution to a stochastic differential equation that matches the distribution of the It\^{o} process at each fixed time.…

We present the stochastic solution to a generalized fractional partial differential equation involving a regularized operator related to the so-called Prabhakar operator and admitting, amongst others, as specific cases the fractional…

Fractional calculus has been used to describe physical systems with complexity. Here, we show that a fractional calculus approach can restore or include complexity in any physical systems that can be described by partial differential…

Let $X$ be a (two-sided) fractional Brownian motion of Hurst parameter $H\in (0,1)$ and let $Y$ be a standard Brownian motion independent of $X$. Fractional Brownian motion in Brownian motion time (of index $H$), recently studied in…

The Malliavin integration-by-parts formula is a key ingredient to develop stochastic analysis on the Wiener space. In this article we show that a suitable integration-by-parts formula also characterizes a wide class of Gaussian processes,…

In the framework of fractional stochastic calculus, we study the existence and the uniqueness of the solution for a backward stochastic differential equation, formally written as: [{[c]{l}% -dY(t)= f(t,\eta(t),Y(t),Z(t))dt-Z(t)\delta…