In this paper we prove that a Gaussian white noise on the d-dimensional torus has paths in the Besov spaces $B_{p,\infty }^{-d/2}{(}^d)$ with p ∈ [1,∞). This result is shown to be optimal in several ways. We also show that Gaussian white noise on the d-dimensional torus has paths in the Fourier-Besov space $b{̂}_{p,\infty }^{-d/p}{(}^d)$. This is shown to be optimal as well.

We prove that the exit times of diffusion processes from a bounded open set Ω almost surely belong to the Besov space $B_{p,q}^{\alpha }\left(\Omega \right)$ provided that pα < 1 and 1 ≤ q < ∞.

We show that the effective diffusivity of a random diffusion with a drift is a continuous function of the drift coefficient. In fact, in the case of a homogeneous and isotropic random environment the function is $C^{\infty }$ smooth outside the origin. We provide a one-dimensional example which shows that the diffusivity coefficient need not be differentiable at 0.

A directed-edge-reinforced random walk on graphs is considered. Criteria for the walk to end up in a limit cycle are given. Asymptotic stability of some neural networks is shown.

We describe a new model of multiple reinsurance. The main idea is that the reinsurance premium is paid conditionally. It is motivated by some analysis of the ultimate price of the reinsurance contract. For simplicity we assume that the underlying risk pricing functional is the L₂-norm. An unexpected relation to the general theory of sample regularity of stochastic processes is given.

2000 Mathematics Subject Classification: Primary 60G51, secondary 60G70, 60F17.We discuss weak limit theorems for a uniformly negligible triangular array (u.n.t.a.) in Z = [0, ∞) × [0, ∞)^d as well as for the associated with it sum and extremal processes on an open subset S . The complement of S turns out to be the explosion area of the limit Poisson point process. In order to prove our criterion for weak convergence of the sum processes we introduce and study sum processes over explosion area....

Let $C[0,t]$ denote a generalized Wiener space, the space of real-valued continuous functions on the interval $[0,t]$, and define a random vector $Z_n:C[0,t]\to {ℝ}^{n+1}$ by $$Z_n\left(x\right)=\left(x\left(0\right)+a\left(0\right),{\int }_0^{t_1}h\left(s\right)\mathrm{d}x\left(s\right)+x\left(0\right)+a\left(t_1\right),\cdots ,{\int }_0^{t_n}h\left(s\right)\mathrm{d}x\left(s\right)+x\left(0\right)+a\left(t_n\right)\right),$$ where $a\in C[0,t]$, $h\in L_2[0,t]$, and $0<t_1<\cdots <t_n\le t$ is a partition of $[0,t]$. Using simple formulas for generalized conditional Wiener integrals, given $Z_n$ we will evaluate the generalized analytic conditional Wiener and Feynman integrals of the functions $F$ in a Banach algebra which corresponds to Cameron-Storvick’s Banach algebra $𝒮$. Finally, we express the generalized analytic conditional Feynman...