Motivated by the Brownian bridge on random interval considered by Bedini et al [4], we introduce and study Gaussian bridges with random length with special emphasis to the Markov property. We prove that if the starting process is Markov then this property was kept by the bridge with respect to the usual augmentation of its natural filtration. This leads us to conclude that the completed natural filtration of the bridge satisfies the usual conditions of right-continuity and completeness.

In this paper we first shall establish an existence and uniqueness result for the semilinear stochastic differential equations in Hilbert space dX ¼ (AX þ f(X ))dt þ g(X )dW under weaker conditions than the Lipschitz one by investigating the convergence of the successive approximations. Secondly we show, under the assumption of so-called pathwise uniqueness (PU ), the convergence of the Euler and Lie-Trotter schemes in L p , p . 2 and the continuous dependence of the solutions on the initial data and on the coefficients for such equation. Finally we study the existence of the solutions when the coefficients f and g are only defined on a subset of the state Hilbert space.

Let [Formula: see text] be a fractional Brownian sheet with Hurst parameters H, H′ ≤ 1/2. We prove the existence and uniqueness of a strong solution for a class of hyperbolic stochastic partial differential equations with additive fractional Brownian sheet of the form [Formula: see text], where b(ζ, x) is a Borel function satisfying some growth and monotonicity assumptions. We also prove the convergence of Euler's approximation scheme for this equation.