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The objective of this article is to introduce and study Itô type stochastic integrals with respect to tempered fractional Brownian motion (TFBM) of Hurst index H ∈ (1 2 , 1) and tempering parameter λ > 0 , by using the Wick product. The main tools are fractional calculus and Malliavin calculus. The Itô formula for this stochastic integral is established for the Itô type processes driven by TFBM. Based on this new Itô formula, we analyze the stability of stochastic differential equations driven by TFBM in the sense of p -th moment. A numerical example is given to illustrate our stability results. [ABSTRACT FROM AUTHOR]

We consider the general diffusion equation driven by tempered fractional Brownian motion (TFBM) ∂ u (t , x) ∂ t = L u (t , x) + u (t , x) d B H , λ (t) d t , (t , x) ∈ R + × R d , where L is the infinitesimal generator of some time-homogeneous Markov process { X t } t ≥ 0 and { B H , λ (t) } t ∈ R is a tempered fractional Brownian motion with Hurst index H ∈ (0 , 1 2) ∪ (1 2 , 1) and tempering parameter λ > 0 which is independent of { X t } t ≥ 0. Based on approximating TFBM with a family of Gaussian processes possessing absolutely continuous sample paths, a unified framework of the Feynman-Kac formula u (t , x) = E x [ f (X t) ] e B H , λ (t) is established for the general stochastic diffusion equation driven by TFBM with the initial value f , Hurst index H ∈ (0 , 1 2) ∪ (1 2 , 1) and tempering parameter λ > 0. The difficulty is that the Hurst parameter H can be allowed to be less than 1/2. The idea is to explore the above simplest form, to utilize the techniques of Malliavin calculus. By using the properties of TFBM and especially that of the modified Bessel function of the second kind, we prove that the process defined by the Feynman-Kac formula is the mild and weak solutions of the general diffusion equation driven by TFBM. From the Feynman-Kac formula, we exhibit the smoothness of the density of the solution and the sharp Hölder regularity. We further show that lim t → ∞ ⁡ ln ⁡ ‖ u (t , ⋅) ‖ p g (t) = 0 in the almost surely sense where p ∈ R and the function g : R + → R + grows at least as fast as a linear function. The Feynman-Kac formula for the stochastic general diffusion equation in the Skorohod sense is also achieved by exploring the relationship between the Stratonovich and Skorohod integrals. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of backward doubly stochastic differential equations (BDSDEs for short) with general terminal value and general random generator. Those BDSDEs do not involve any forward diffusion processes. By using the techniques of Malliavin calculus, we are able to establish the L p -Hölder continuity of the solution pair. Then, an implicit numerical scheme for the BDSDE is proposed and the rate of convergence is obtained in the L p -sense. As a by-product, we obtain an explicit representation of the process Y in the solution pair to a linear BDSDE with random coefficients. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of stochastic differential equations with Markovian switching. The Malliavin calculus is used to study the smoothness of the density of the solution under a Hörmander type condition. Furthermore, we obtain a Bismut type formula which is used to establish the strong Feller property. [ABSTRACT FROM AUTHOR]

• Our generator is a weighted fractional space-time Brownian field. • And it's not a backward martingale. • The existence, uniqueness and explicit solution pair for Y and Z. • Some kind of sharp Hlder continuity for the solution is obtained. • The Feynman-Kac Formula is established. This paper is concerned with the backward stochastic differential equations whose generator is a weighted fractional Brownian field: Y t = ξ + ∫ t T Y s W (d s , B s) − ∫ t T Z s d B s , 0 ≤ t ≤ T , where W is a (d + 1) -parameter weighted fractional Brownian field of Hurst parameter H = (H 0 , H 1 , ⋯ , H d) , which provide probabilistic interpretations (Feynman-Kac formulas) for certain linear stochastic partial differential equations with colored space-time noise. Conditions on the Hurst parameter H and on the decay rate of the weight are given to ensure the existence and uniqueness of the solution pair. Moreover, the explicit expression for both components Y and Z of the solution pair is given. [ABSTRACT FROM AUTHOR]

This paper studies the existence and uniqueness of solution of Itô type stochastic differential equation , where B(t) is a fractional Brownian motion of Hurst parameter and is the Itô differential defined by using Wick product or divergence operator. The coefficients b and are random and anticipative. Using the relationship between the Itô and pathwise integrals we first write the equation as a stochastic differential equation involving pathwise integral plus a Malliavin derivative term. To handle this Malliavin derivative term the equation is then further reduced to a system of characteristic equations without Malliavin derivative, which is then solved by a careful analysis of Picard iteration, with a new technique to replace the Grönwall lemma which is no longer applicable. The solution of this system of characteristic equations is then applied to solve the original Itô stochastic differential equation up to a positive random time. In special linear and quasilinear cases the global solutions are proved to exist uniquely. [ABSTRACT FROM AUTHOR]

We prove a weak rate of convergence of a fully discrete scheme for stochastic Cahn–Hilliard equation with additive noise, where the spectral Galerkin method is used in space and the backward Euler method is used in time. Compared with the Allen–Cahn type stochastic partial differential equation, the error analysis here is much more sophisticated due to the presence of the unbounded operator in front of the nonlinear term. To address such issues, a novel and direct approach has been exploited which does not rely on a Kolmogorov equation but on the integration by parts formula from Malliavin calculus. To the best of our knowledge, the rates of weak convergence are revealed in the stochastic Cahn–Hilliard equation setting for the first time. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to establish the uniform convergence of the densities of a sequence of random variables, which are functionals of an underlying Gaussian process, to a normal density. Precise estimates for the uniform distance are derived by using the techniques of Malliavin calculus, combined with Steinʼs method for normal approximation. We need to assume some non-degeneracy conditions. First, the study is focused on random variables in a fixed Wiener chaos, and later, the results are extended to the uniform convergence of the derivatives of the densities and to the case of random vectors in some fixed chaos, which are uniformly non-degenerate in the sense of Malliavin calculus. Explicit upper bounds for the uniform norm are obtained for random variables in the second Wiener chaos, and an application to the convergence of densities of the least square estimator for the drift parameter in Ornstein–Uhlenbeck processes is discussed. [Copyright &y& Elsevier]

Multiple integrals with respect to several Hölder continuous functions of exponentare studied by using fractional calculus. They are applied to obtain the Volterra expansion (with remainder) for the solution of a differential system driven by rough paths. The results are applied to stochastic differential equations driven by fractional Brownian motions of Hurst parameter. For the solution of a stochastic differential equation driven by fractional Brownian motion, we obtain its chaos expansion, as well as the finite chaos expansion with remainder. To this end, we study the multiple Itô integral with random kernels. The Hu–Meyer formulas between multiple Itô and multiple pathwise integrals with random kernels are also obtained. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we apply Clark–Ocone formula to deduce an explicit integral representation for the renormalized self-intersection local time of the d-dimensional fractional Brownian motion with Hurst parameter . As a consequence, we derive the existence of some exponential moments for this random variable. [Copyright &y& Elsevier]