On a nonlinear stochastic pseudo-differential equation driven by fractional noise

Abstract

In this paper, we study the existence, uniqueness and Hölder regularity of the solution to a class of nonlinear stochastic pseudo-differential equation of the following form [Formula: see text] where [Formula: see text] is a pseudo-differential operator with negative definite symbol of variable order which generates a stable-like process with transition density, the coefficient [Formula: see text] is a measurable function, and [Formula: see text] is a double-parameter fractional noise. In addition, the existence and Gaussian type estimates for the density of the mild solution are proved via Malliavin calculus.

Weak convergence of SFDEs driven by fractional Brownian motion with irregular coefficients

“…[13] obtained existence and uniqueness of solutions to SDEs driven by fBMs with Hurst parameter H ∈ ( 1 2 , 1) by using Young integrals (see [30]) and p-variation estimate; [3] derived the existence and uniqueness result for H ∈ ( 1 4 , 1 2 ) through the same rough-type arguments in [13]; [25] studied SDEs driven by fBMs by using fractional calculus developed in [31]. For more results on existence and uniqueness of solutions to SDEs driven by fBMs, we refer to [2,8,9,12,17,24] for instance. Stochastic functional differential equations (SFDEs) are also used to characterise stochastic systems with memory effects.…”

Weak convergence of path-dependent SDEs driven by fractional Brownian motion with irregular coefficients

“…[11] obtained existence and uniqueness of solutions to SDEs driven by fBMs with Hurst parameter H ∈ ( 1 2 , 1) by using Young integrals (see [27]) and p-variation estimate; [3] derived the existence and uniqueness result for H ∈ ( 1 4 , 1 2 ) through the same rough-type arguments in [11]; [22] studied SDEs driven by fBMs by using fractional calculus developed in [28]. For more results on existence and uniqueness of solutions to SDEs driven by fBMs, we refere to [2,8,9,10,15,21] for instance. Stochastic functional differential equations (SFDEs) are also used to characterise stochastic systems with memory effects.…”

Moment estimates and applications for SDEs driven by fractional Brownian motion with irregular drifts

“…For other results on the existence and uniqueness, the reader may consult e.g. [5,13,16,19,22,28] and the references therein. In [6,7,16,26,30], by rough path techniques or Malliavin calculus the authors handled the existence of densities of the solutions under regularity assumptions.…”