Multidimensional stochastic differential equations with distributional drift

This paper investigates a time-dependent multidimensional stochastic differential equation with drift being a distribution in a suitable class of Sobolev spaces with negative derivation order. This is done through a careful analysis of the corresponding Kolmogorov equation whose coefficient is a distribution

Forward-backward SDEs with distributional coefficients

Forward-backward stochastic differential equations (FBSDEs) have attracted significant attention since they were introduced almost 30 years ago, due to their wide range of applications, from solving non-linear PDEs to pricing American-type options. Here, we consider two new classes of multidimensional FBSDEs with distributional coefficients (elements of a Sobolev space with negative order). We introduce a suitable notion of a solution, show existence and uniqueness of a strong solution of the first FBSDE, and weak existence for the second. We establish a link with PDE theory via a nonlinear Feynman-Kac representation formula. The associated semi-linear second order parabolic PDE is the same for both FBSDEs, also involves distributional coefficients and has not previously been investigated; our analysis uses mild solutions, Sobolev spaces and semigroup theory.Comment: 40 pages, no figures - new improved version with shorter proof of Thm 18, extended results in Thm 25 and Thm 27. Other minor clarifications adde

Fractional Brownian motions ruled by nonlinear equations

In this note we consider generalized diffusion equations in which the diffusivity coefficient is not necessarily constant in time, but instead it solves a nonlinear fractional differential equation involving fractional Riemann-Liouville time-derivative. Our main contribution is to highlight the link between these generalised equations and fractional Brownian motion (fBm). In particular, we investigate the governing equation of fBm and show that its diffusion coefficient must satisfy an additive evolutive fractional equation. We derive in a similar way the governing equation of the iterated fractional Brownian motion.Comment: 7 page

Blow-up for a nonlinear PDE with fractional Laplacian and singular quadratic nonlinearity

We consider a parabolic-type PDE with a diffusion given by a fractional Laplacian operator and with a quadratic nonlinearity of the 'gradient' of the solution, convoluted with a singular term b. Our first result is the well-posedness for this problem: We show existence and uniqueness of a (local in time) mild solution. The main result is about blow-up of said solution, and in particular we find sufficient conditions on the initial datum and on the term b to ensure blow-up of the solution in finite time

Stochastic partial differential equations with fractal noise: two different approaches

This thesis deals with stochastic partial differential equations driven by fractional noises. In this work, problems related to this topics are tackled and solved from two fairly different points of view. On one side we prove existence, uniqueness and regularity for mild solutions to a parabolic transport diffusion type equation that involves a non-smooth coefficient. We investigate related Cauchy problems on bounded smooth domains with Dirichlet boundary conditions by means of semigroup theory and fixed point arguments. Main ingredients are the definition of a product of a function and a (not too irregular) distribution as well as a corresponding norm estimate. As an application, transport stochastic partial differential equations driven by fractional Brownian noises are considered in the pathwise sense. On the other side we deal with stochastic differential equations driven by fractal noises in Banach spaces. More precisely, we deal with abstract Cauchy problems driven by fractional Brownian processes in Banach spaces and look for weak and mild solutions. To this aim, a fractional Brownian motion in separable Banach spaces is introduced by means of cylindrical processes. The related stochastic integral is then defined as cylindrical stochastic process and its properties are investigated. When the Banach space is a function space then the equation becomes a stochastic partial differential equation driven by a fractional noise

A numerical scheme for stochastic differential equations with distributional drift

In this paper we present a scheme for the numerical solution of stochastic differential equations (SDEs) with distributional drift. The approximating process, obtained by the scheme, converges in law to the (virtual) solution of the SDE in a general multi-dimensional setting. When we restrict our attention to the case of a one-dimensional SDE we also obtain a rate of convergence in a suitable $L^1$-norm. Moreover, we implement our method in the one-dimensional case, when the drift is obtained as the distributional derivative of a sample path of a fractional Brownian motion. To the best of our knowledge this is the first paper to study (and implement) numerical solutions of SDEs whose drift cannot be expressed as a function of the state.Comment: 35 pages, 8 figure

Convergence rate of numerical scheme for SDEs with a distributional drift in Besov space

This paper is concerned with numerical solutions of one-dimensional SDEs with the drift being a generalised function, in particular belonging to the Holder-Zygmund space $C^{-\gamma}$ of negative order $-\gamma<0$ in the spacial variable. We design an Euler-Maruyama numerical scheme and prove its convergence, obtaining an upper bound for the strong $L^1$ convergence rate. We finally implement the scheme and discuss the results obtained.Comment: 20 pages, 3 figure