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Let $$B = { (B_t^{1}, ..., B_t^{d} ),t \geq 0}$$ be a d-dimensional fractional Brownian motion with Hurst parameter H and let $$R_{t} = \sqrt {(B_t^1 )^2 + ... + (B_t^{d} )^{2} }$$ be the fractional Bessel process. Itô’s formula for the fractional Brownian motion leads to the equation $$R_t = \sum_{i = 1}^d ,\int_0^{t} \frac{B_s^{i} }{R_{s} }\ {d} B_s^i + H(d -1)\int_0^{t} \frac{s^{2H - 1}} {R_s }\ {d} s$$ . In the Brownian motion case $$(H=1/2), X_t = \sum\nolimits_{i = 1}^d {\int_0^t {\frac{{B_s^i }} {{R_s }}} } \d B_s^i $$ is a Brownian motion. In this paper it is shown that Xt is not an $${\cal F}^{B}$$ -fractional Brownian motion if H ≠ 1/2. We will study some other properties of this stochastic process as well. [ABSTRACT FROM AUTHOR]

This paper is devoted to study backward stochastic differential equations in the plane driven by a Brownian sheet, where the value of the solution at the corner ( s0, t0) is fixed. The existence and uniqueness of a solution is obtained by means of Picard's approximation scheme and a suitable two-parameter Gronwall's type lemma. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of the quadratic covariation [f( X), X], where f is a locally square integrable function and Xt = ∫ t 0 us dWs is a smooth nondegenerate Brownian martingale. This result is based on some moment estimates for Riemann sums which are established by means of the techniques of the Malliavin calculus. [ABSTRACT FROM AUTHOR]

We establish the existence and uniqueness of the solution to a multidimensional linear Skorohod stochastic differential equation with deterministic diffusion matrix, using the notions of Wick product and Stransform. If the diffusion matrix is constant and has real eigenvalues, the solution is a stochastic process with moments of all orders, provided that the initial condition is differentiable up to a suitable order. The case of a diffusion matrix in the first Wiener chaos is discussed in the last section. [ABSTRACT FROM AUTHOR]

We study reflected solutions of the heat equation on the spatial interval [0, 1] with Dirichlet boundary conditions, driven by an additive space-time white noise. Roughly speaking, at any point ( x, t) where the solution u(x, t) is strictly positive it obeys the equation, and at a point ( x, t) where u(x, t) is zero we add a force in order to prevent it from becoming negative. This can be viewed as an extension both of one-dimensional SDEs reflected at 0, and of deterministic variational inequalities. An existence and uniqueness result is proved, which relies heavily on new results for a deterministic variational inequality. [ABSTRACT FROM AUTHOR]

In this paper we study the conditional independence of sigma-fields on the Wiener space using the tools of the Stochastic Calculus of Variations. Particular emphasis is given to the relation between the splitting of the (random) tangent spaces associated to the sigma-fields and the conditional independence. [ABSTRACT FROM AUTHOR]

In this work we study sigma fields and their tangent spaces on the Wiener space which are invariant in some sense with respect to the basic operators of the Malliavin Calculus. [ABSTRACT FROM AUTHOR]

In this article we deal with stochastic differential equations driven by an infinite dimensional Brownian motion. Under some non-degeneracy conditions, the existence and smoothness of the density for the law of the solution is proved. [ABSTRACT FROM AUTHOR]

We study the stochastic integral defined by Skorohod in [24] of a possibly anticipating integrand, as a function of its upper limit, and establish an extended Itô formula. We also introduce an extension of Stratonovich's integral, and establish the associated chain rule. In all the results, the adaptedness of the integrand is replaced by a certain smoothness requirement. [ABSTRACT FROM AUTHOR]

In this paper we show that the local time of the Brownian motion belongs to the Sobolev space $$\mathbb{D}^{\alpha {\text{,}}p}$$ for any p≥2 and 0<α<1/ p. In order to prove this result we first discuss the smoothness and integrability properties of the composition of the Dirac function δ with a Wiener integral W(h), and we show that this composition belongs to $$\mathbb{D}^{ - \alpha {\text{,}}p}$$ , for any α>0 and p>1 such that α+1/ p>1. [ABSTRACT FROM AUTHOR]

In this paper we apply the Malliavin Calculus to deduce the existence and smoothness of density for the solution of stochastic differential equations with respect to a multidimensional two-parameter Wiener process. [ABSTRACT FROM AUTHOR]