In this paper we show the existence and uniqueness of a solution for a stochastic differential equation driven by an additive noise which is the sum of two fractional Brownian motions with different Hurst parameters. The proofs are based on the techniques of fractional calculus and Girsanov theorem. In particular, we show that the regularization effect of the fractional Brownian motion with the smaller Hurst index dominates.
We consider the stochastic continuity equation perturbed by a fractional Brownian motion and the drift is allowed to be discontinuous. We show that for almost all paths of the fractional Brownian motion there exists a solution to the equation.
We consider two related linear PDE's perturbed by a fractional Brownian motion. We allow the drift to be discontinuous, in which case the corresponding deterministic equation is ill-posed. However, the noise will be shown to have a regularizing effect on the equations in the sense that we can prove existence of solutions for almost all paths of the fractional Brownian motion.
We study a regularization by noise phenomenon for the continuous parabolic Anderson model with a potential shifted along paths of fractional Brownian motion. We demonstrate that provided the Hurst parameter is chosen sufficiently small, this shift allows to establish well-posedness and stability to the corresponding problem - without the need of renormalization - in any dimension. We moreover provide a robustified Feynman-Kac type formula for the unique solution to the regularized problem building upon regularity estimates for the local time of fractional Brownian motion as well as non-linear …
In this paper we study the controllability of fractional neutral stochastic functional differential equations with infinite delay driven by fractional Brownian motion in a real separable Hilbert space. The controllability results are obtained by using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is provided to demonstrate the effectiveness of the theoretical result.
We first state a special type of Itô formula involving stochastic integrals of both standard and fractional Brownian motions. Then we use Doss-Sussman transformation to establish the link between backward doubly stochastic differential equations, driven by both standard and fractional Brownian motions, and backward stochastic differential equations, driven only by standard Brownian motions. Following the same technique, we further study associated nonlinear stochastic partial differential equations driven by fractional Brownian motions and partial differential equations with stochastic coeffic…
This paper defines a new class of fractional differential operators alongside a family of random variables whose density functions solve fractional differential equations equipped with these operators. These equations can be further used to construct fractional integro-differential equations for the ruin probabilities in collective renewal risk models, with inter-arrival time distributions from the aforementioned family. Gamma-time risk models and fractional Poisson risk models are two specific cases among them, whose ruin probabilities have explicit solutions, when claim sizes distributions e…
In this paper we study a stochastic differential equation driven by a fractional Brownian motion with a discontinuous coefficient. We also give an approximation to the solution of the equation. This is a first step to define a fractional version of the skew Brownian motion.
By using Malliavin calculus, Bismut derivative formulae are established for a class of stochastic (functional) differential equations driven by fractional Brownian motions. As applications, Harnack type inequalities and strong Feller property are presented.
In this paper, we consider a stochastic differential equation driven by a fractional Brownian motion (fBm) and a Wiener process and having jumps. We prove that this equation has a unique solution and show that all its moments are finite.
In the paper, Harnack inequalities are established for stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H<1/2$. As applications, strong Feller property, log-Harnack inequality and entropy-cost inequality are given.
We study some functional inequalities satisfied by the distribution of the solution of a stochastic differential equation driven by fractional Brownian motions. Such functional inequalities are obtained through new integration by parts formulas on the path space of a fractional Brownian motion.
In this work we study rough differential equations driven by a fractional Brownian motion with Hurst parameter H>1/4 and establish Varadhan's small time estimates for the density of solutions of such equations under Hormander's type conditions.
We study the existence of a unique solution to semilinear fractional backward doubly stochastic differential equation driven by a Brownian motion and a fractional Brownian motion with Hurst parameter less than 1/2. Here the stochastic integral with respect to the fractional Brownian motion is the extended divergence operator and the one with respect to Brownian motion is Itô's backward integral. For this we use the technique developed by R.Buckdahn to analyze stochastic differential equations on the Wiener space, which is based on the Girsanov theorem and the Malliavin calculus, and we reduce …
We consider a mixed stochastic differential equation driven by possibly dependent fractional Brownian motion and Brownian motion. Under mild regularity assumptions on the coefficients, it is proved that the equation has a unique solution.
In this work we study the smoothing effect of rough differential equations driven by a fractional Brownian motion with parameter $H>1/4$. The regularization estimates we obtain generalize to the fractional Brownian motion previous results by Kusuoka and Stroock and can be seen as a quantitative version of the existence of smooth densities under Hörmander's type conditions.
In this paper, we study the existence and (Hölder) regularity of local times of stochastic differential equations driven by fractional Brownian motions. In particular, we show that in one dimension and in the rough case H<1/2, the Hölder exponent (in t) of the local time is 1-H, where H is the Hurst parameter of the driving fractional Brownian motion.
In this paper, we study a class of stochastic partial differential equations (SPDEs) driven by space-time fractional noises. Our method consists in studying first the nonlocal SPDEs and showing then the convergence of the family of these equations and the limit gives the solution to the SPDE.
In this paper we show that under some assumptions, for a $d$-dimensional fractional Brownian motion with Hurst parameter $H>1/2$, the density of solution of stochastic differential equation driven by it has a short-time expansion similar to that in the Brownian motion case.
We study pathwise regularization by noise for equations on the plane in the spirit of the framework outlined by Catellier and Gubinelli (Stochastic Process. Appl., 2016). To this end, we extend the notion of non-linear Young equations to a two dimensional domain and prove existence and uniqueness of such equations. This concept is then used in order to prove regularization by noise for stochastic equations on the plane. The statement of regularization by noise is formulated in terms of the regularity of the local time associated to the perturbing stochastic field. For this, we provide two quan…
In this note we prove that a fractional stochastic delay differential equation which satisfies natural regularity conditions generates a continuous random dynamical system on a subspace of a Hölder space which is separable.
The work in this paper is four-fold. Firstly, we introduce an alternative approach to solve fractional ordinary differential equations as an expected value of a random time process. Using the latter, we present an interesting numerical approach based on Monte Carlo integration to simulate solutions of fractional ordinary and partial differential equations. Thirdly, we show that this approach allows us to find the fundamental solutions for fractional partial differential equations (PDEs), in which the fractional derivative in time is in the Caputo sense and the fractional in space one is in the…
In this paper we develop a method to solve evolution equations on Gelfand triples with time-fractional derivative based on monotonicity techniques. Applications include deterministic and stochastic quasi-linear partial differential equations with time-fractional derivatives, including time-fractional (stochastic) porous media equations (including the case where the Laplace operator is also fractional) and $p$-Laplace equations as special cases.
We discuss maximum likelihood estimation of parameters for models governed by a stochastic differential equation driven by a mixed fractional Brownian motion with random effects.
