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In this paper, we study a representation for the solutions to sticky stochastic differential equations driven by a continuous process. The involved stochastic integral is interpreted in three different ways. Namely, we deal with Young integral defined by the fractional calculus, and the forward and symmetric integrals in the Russo and Vallois sense. The representation obtained in this paper depends on the amount of time spent by the solution at zero. Hence, we obtain the uniqueness for the solution among the processes that spend zero time at 0. [ABSTRACT FROM AUTHOR]

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]

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]

In this note we give several methods to construct nontrivial solutions to the equation d y t = σ ( y t ) d x t , where x is a γ -Hölder R d -valued signal with γ ∈ ( 1 / 2 , 1 ) and σ is a function behaving like a power function | ξ | κ , with κ ∈ ( 0 , 1 ) . In this situation, classical Young integration techniques allow to get existence and uniqueness results whenever γ ( κ + 1 ) > 1 , while we focus on cases where γ ( κ + 1 ) ≤ 1 . Our analysis then relies on Zähle’s extension (Zähle, 1998) of Young’s integral allowing to cover the situation at hand. [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]

In this paper, we introduce a stochastic integral with respect to the solution X of the fractional heat equation on [0,1], interpreted as a divergence operator. This allows to use the techniques of the Malliavin calculus in order to establish an Ito-type formula for the process X. [ABSTRACT FROM AUTHOR]

In this article, we use the chaos decomposition approach to establish the existence of a unique continuous solution to linear fractional differential equations of the Skorohod type. Here, the coefficients are deterministic, the initial condition is anticipating and the underlying fractional Brownian motion has Hurst parameter less than 1/2. We provide an explicit expression for the chaos decomposition of the solution in order to show our results. [ABSTRACT FROM AUTHOR]

In this paper we combine the techniques of the Malliavin and the fractional calculus to establish a Clark-Ocone integral representation for fractional Brownian motion with Hurst parameter . The uniqueness of this adapted representation is proved. [ABSTRACT FROM AUTHOR]

Here, we review some results of fractional volatility models, where the volatility is driven by fractional Brownian motion (fBm). In these models, the future average volatility is not a process adapted to the underlying filtration, and fBm is not a semimartingale in general. So, we cannot use the classical Itô's calculus to explain how the memory properties of fBm allow us to describe some empirical findings of the implied volatility surface through Hull and White type formulas. Thus, Malliavin calculus provides a natural approach to deal with the implied volatility without assuming any particular structure of the volatility. The aim of this paper is to provides the basic tools of Malliavin calculus for the study of fractional volatility models. That is, we explain how the long and short memory of fBm improves the description of the implied volatility. In particular, we consider in detail a model that combines the long and short memory properties of fBm as an example of the approach introduced in this paper. The theoretical results are tested with numerical experiments. [ABSTRACT FROM AUTHOR]