Related papers: Rough differential equations with path-dependent c…
Existence and uniqueness for rough flows, transport and continuity equations driven by general geometric rough paths are established.
We give in this note a simple treatment of the non-explosion problem for rough differential equations driven by unbounded vector fields and weak geometric rough paths of arbitrary roughness.
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
In this paper we review and improve pathwise uniqueness results for some types of one-dimensional stochastic differential equations (SDE) involving the local time of the unknown process. The diffusion coefficient of the SDEs we consider is…
Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as…
This paper shows how to build a formal analytical solution for a differential equation of arbitrary order and with variable coefficients. It proofs that the most known approximated solutions for such a problem can be derived from the…
In this paper, the existence and pathwise uniqueness of strong solutions for jump-type stochastic differential equations are investigated under non-Lipschitz conditions. A sufficient condition is obtained for ensuring the non-confluent…
In this short note we are presenting a method of finding particular solutions of nonhomegeneous linear equations. This approach is different from methods of undetermined coefficients or variation of parameters presented in virtually every…
We establish a simultaneous generalization of It\^o's theory of stochastic and Lyons' theory of rough differential equations. The interest in such a unification comes from a variety of applications, including pathwise stochastic filtering,…
We explore the limit of stochastic differential equations driven by some random processes satisfying singularly perturbed second order stochastic differential equations. The main tool we employ is the universal limit theorem in rough path…
In this paper hyperbolic partial differential equations with random coefficients are discussed. Such random partial differential equations appear for instance in traffic flow problems as well as in many physical processes in random media.…
We study different possibilities to apply the principles of rough paths theory in a non-commutative probability setting. First, we extend previous results obtained by Capitaine, Donati-Martin and Victoir in Lyons' original formulation of…
We exploit the link between the transport equation and derivatives of expectations to construct efficient pathwise gradient estimators for multivariate distributions. We focus on two main threads. First, we use null solutions of the…
We consider anticipative Stratonovich stochastic differential equations driven by some stochastic process lifted to a rough path. Neither adaptedness of initial point and vector fields nor commuting conditions between vector field is…
We study controlled differential equations driven by a rough path (in the sense of T. Lyons) with an additional, possibly unbounded drift term. We show that the equation induces a solution flow if the drift grows at most linearly.…
We study a class of stochastic differential equations with non-Lipschitzian coefficients.A unique strong solution is obtained and a large deviation principle of Freidln-Wentzell type has been established.
We define and solve Volterra equations driven by an irregular signal, by means of a variant of the rough path theory allowing to handle generalized integrals weighted by an exponential coefficient. The results are applied to the fractional…
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…
We give an elementary proof that Davie's definition of a solution to a rough differential equation and the notion of solution given by Bailleul in (Flows driven by rough paths) coincide. This provides an alternative point on view on the…
In this paper, we study the existence of solutions for second-order non-instantaneous impulsive differential equations with a perturbation term. By variational approach, we obtain the problem has at least one solution under assumptions that…