We first generalize, in an abstract framework, results on the order of convergence of a semi-discretization in time by an implicit Euler scheme of a stochastic parabolic equation. In this part, all the coefficients are globally Lipchitz. The case when the nonlinearity is only locally Lipchitz is then treated. For the sake of simplicity, we restrict our attention to the Burgers equation. We are not able in this case to compute a pathwise order of the approximation, we introduce the weaker notion...

We first generalize, in an abstract framework, results on the order of convergence of a semi-discretization in time by an implicit Euler scheme of a stochastic parabolic equation. In this part, all the coefficients are globally Lipchitz. The case when the nonlinearity is only locally Lipchitz is then treated. For the sake of simplicity, we restrict our attention to the Burgers equation. We are not able in this case to compute a pathwise order of the approximation, we introduce the weaker notion...

In this work a family of stochastic differential equations whose solutions are multidimensional diffusion-type (non necessarily markovian) processes is considered, and the estimation of a parametric vector θ which relates the coefficients is studied. The conditions for the existence of the likelihood function are proved and the estimator is obtained by continuously observing the process. An application for Diffusion Branching Processes is given. This problem has been studied in some special cases...

This paper considers stochastic differential equations with solutions which are multidimensional diffusion processes with drift coefficient depending on a parametric vector θ. By considering a trajectory observed up to a stopping time, the maximum likelihood estimator for θ has been obtained and its consistency and asymptotic normality have been proved.

1. Introduction Random Integral Equations play a significant role in characterizing of many biological and engineering problems [4,5,6,7]. We present here new existence theorems for a class of integral equations with advancing argument. Our method is based on the notion of a measure of noncompactness in Banach spaces and the fixed point theorem of Darbo type. We shall deal with random integral equation with advancing argument $x(t,\omega )=h(t,\omega )+{\int }^{t+\delta \left(t\right)}₀k(t,\tau ,\omega )f(\tau ,x_{\tau }\left(\omega \right))d\tau$, (t,ω) ∈ R⁺ × Ω, (1) where (i) (Ω,A,P) is a complete probability space, (ii)...

In this paper we use the Schauder fixed point theorem and methods of integral inequalities in order to prove a result on the existence, uniqueness and parametric dependence on the coefficients of the solution processes in McShane stochastic integral equations.

Necessary and sufficient conditions are found for the exponential Orlicz norm (generated by ${\psi }_p\left(x\right)={exp\left(\right|x|}^p)-1$ with 0 < p ≤ 2) of $ma{x}_{0\le t\le \tau }\left|B_t\right|$ or $|B_{\tau }|$ to be finite, where $B={\left(B_t\right)}_{t\ge 0}$ is a standard Brownian motion and τ is a stopping time for B. The conditions are in terms of the moments of the stopping time τ. For instance, we find that $\parallel ma{x}_{0\le t\le \tau }\left|B_t\right|{\parallel }_{{\psi }_1}<\infty$ as soon as $E\left({\tau }^k\right)=O\left(C^k{k}^k\right)$ for some constant C > 0 as k → ∞ (or equivalently $\parallel \tau {\parallel }_{{\psi }_1}<\infty$). In particular, if τ ∼ Exp(λ) or $\left|N\right(0,{\sigma }^2\left)\right|$ then the last condition is satisfied, and we obtain $\parallel ma{x}_{0\le t\le \tau }\left|B_t\right|{\parallel }_{{\psi }_1}\le K\surd E\left(\tau \right)$ with some universal constant K > 0....

Consider a stochastic heat equation ∂tu=κ  ∂xx2u+σ(u)ẇ for a space–time white noise ẇ and a constant κ&gt;0. Under some suitable conditions on the initial function u0 and σ, we show that the quantities lim sup t→∞t−1sup x∈Rln El(|ut(x)|2) and lim sup t→∞t−1ln E(sup x∈R|ut(x)|2) are equal, as well as bounded away from zero and infinity by explicit multiples of 1/κ. Our proof works by demonstrating quantitatively that the peaks of the stochastic process x↦ut(x) are highly concentrated...

This paper is devoted to the helices processes, i.e. the solutions H : ℝ × Ω → ℝd, (t, ω) ↦ H(t, ω) of the helix equation $\begin{array}{ccc}\mathit{H}\mathrm{\left(}\mathrm{0}\mathit{,\omega }\mathrm{\right)}\mathrm{=}\mathrm{0}\mathrm{;} \mathit{H}\mathrm{(}\mathit{s}\mathrm{+}\mathit{t,\omega }\mathrm{)}\mathrm{=}\mathit{H}\mathrm{\left(}\mathit{s,}\mathrm{\Phi }\mathrm{\right(}\mathit{t,\omega }\mathrm{\left)}\mathrm{\right)}\mathrm{+}\mathit{H}\mathrm{\left(}\mathit{t,\omega }\mathrm{\right)}& & \end{array}$ where Φ : ℝ × Ω → Ω, (t, ω) ↦ Φ(t, ω) is a dynamical system on a measurable space (Ω, ℱ).More precisely, we investigate dominated solutions and non differentiable solutions of the helix equation. For the last case, the Wiener helix plays a fundamental role. Moreover, some relations with the cocycle equation defined...

In this paper we solve the basic fractional analogue of the classical infinite time horizon linear-quadratic gaussian regulator problem. For a completely observable controlled linear system driven by a fractional brownian motion, we describe explicitely the optimal control policy which minimizes an asymptotic quadratic performance criterion.

In this paper we solve the basic fractional analogue of the classical infinite time horizon linear-quadratic Gaussian regulator problem. For a completely observable controlled linear system driven by a fractional Brownian motion, we describe explicitely the optimal control policy which minimizes an asymptotic quadratic performance criterion.

The Kaczmarz algorithm of successive projections suggests the following concept. A sequence $\left(e_k\right)$ of unit vectors in a Hilbert space is said to be effective if for each vector x in the space the sequence (xₙ) converges to x where (xₙ) is defined inductively: x₀ = 0 and $xₙ=x_{n-1}+\alpha ₙeₙ$, where $\alpha ₙ=⟨x-x_{n-1},eₙ⟩$. We prove the effectivity of some sequences in Hilbert spaces. We generalize the concept of effectivity to sequences of vectors in Banach spaces and we prove some results for this more general concept.