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The Malliavin differentiability of a SDE plays a crucial role in the study of density smoothness and ergodicity among others. For Gaussian driven SDEs the differentiability issue is solved essentially in Cass et al., (2013). In this paper, we consider the Malliavin differentiability for the Euler scheme of such SDEs. We will focus on SDEs driven by fractional Brownian motions (fBm), which is a very natural class of Gaussian processes. We derive a uniform (in the step size n) path-wise upper-bound estimate for the Euler scheme for stochastic differential equations driven by fBm with Hurst parameter H > 1 / 3 and its Malliavin derivatives. [ABSTRACT FROM AUTHOR]

In this paper, we use the techniques of fractional calculus to study the existence of a unique solution to semilinear fractional differential equation driven by a γ -Hölder continuous function 𝜃 with γ ∈ (2 3 , 1). Here, the initial condition is a function that may not be defined at zero and the involved integral with respect to 𝜃 is the extension of the Young integral [An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936) 251–282] given by Zähle [Integration with respect to fractal functions and stochastic calculus I, Probab. Theory Related Fields 111 (1998) 333–374]. [ABSTRACT FROM AUTHOR]

We propose a local linearization scheme to approximate the solutions of non-autonomous stochastic differential equations driven by fractional Brownian motion with Hurst parameter 1 / 2 < H < 1. Toward this end, we approximate the drift and diffusion terms by means of a first-order Taylor expansion. This becomes the original equation into a local fractional linear stochastic differential equation, whose solution can be figured out explicitly. As in the Brownian motion case (i.e., H = 1/2), the rate of convergence, in our case, is twice the one of the Euler scheme. Numerical examples are given to demonstrate the performance of the method. [ABSTRACT FROM AUTHOR]

Using Malliavin Calculus techniques, we derive closed-form expressions for the at-the-money behaviour of the forward implied volatility, its skew and its curvature, in general Markovian stochastic volatility models with continuous paths. [ABSTRACT FROM AUTHOR]

In this article, we study a class of stochastic differential equations driven by a fractional Brownian motion withH> 1/2 and a discontinuous coefficient in the diffusion. We prove existence and uniqueness for the solution of these equations. This is a first step to define a fractional version of the skew Brownian motion. [ABSTRACT FROM AUTHOR]

We study the uniqueness in the path-by-path sense (i.e. ω -by- ω ) of solutions to stochastic differential equations with additive noise and non-Lipschitz autonomous drift. The notion of path-by-path solution involves considering a collection of ordinary differential equations and is, in principle, weaker than that of a strong solution, since no adaptability condition is required. We use results and ideas from the classical theory of ode's, together with probabilistic tools like Girsanov's theorem, to establish the uniqueness property for some classes of noises, including Brownian motion, and some drift functions not necessarily bounded nor continuous. [ABSTRACT FROM AUTHOR]

By means of Itô's formula and a comparison theorem for integral equations, we study the blow up in finite time of semilinear stochastic differential equations of the form Here, is non-negative and non-decreasing by components, is a predictable and continuous process, is an -Brownian motion and is an -measurable random variable. The results of this paper can be seen as extensions of the Feller and Osgood criteria. [ABSTRACT FROM AUTHOR]

The Malliavin derivative operator for the Poisson process introduced by Carlen and Pardoux [Differential calculus and integration by parts on a Poisson space, inStochastics, Algebra and Analysis in Classical and Quantum Dynamics, S. Albeverio et al. (eds), Kluwer, Dordrecht, 1990, pp. 63–73] is extended to Lévy processes. It is a true derivative operator (in the sense that it satisfies the chain rule), and we deduce a sufficient condition for the absolute continuity of functionals of the Lévy process. As an application, we analyse the absolute continuity of the law of the solution of some stochastic differential equations with jumps. [ABSTRACT FROM AUTHOR]

Abstract: 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 Itôʼs backward integral. For this we use the technique developed by R. Buckdahn (1994) to analyze stochastic differential equations on the Wiener space, which is based on the Girsanov theorem and the Malliavin calculus, and we reduce the backward doubly stochastic differential equation to a backward stochastic differential equation driven by the Brownian motion. We also prove that the solution of semilinear fractional backward doubly stochastic differential equation defines the unique stochastic viscosity solution of a semilinear stochastic partial differential equation driven by a fractional Brownian motion. [Copyright &y& Elsevier]

We prove the existence of a unique solution for a one-dimensional stochastic parabolic partial differential equation with random and adapted coefficients perturbed by a two-parameter white noise. The proof is based on a maximal inequality for the Skorohod integral deduced from Itô's formula for this anticipating stochastic integral. [ABSTRACT FROM AUTHOR]

We study the existence and uniqueness of pathwise solutions to backward and forward stochastic differential equations on the Poisson space. We obtain the structure of these pathwise solutions to give the relationship between them. Also, in the bilinear case, we calculate the explicit form of their chaos decompositions. [ABSTRACT FROM AUTHOR]