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Register a new userStrong approximation of particular one-dimensional diffusions
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Authors
Madalina Deaconu
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This paper develops a new technique for the path approximation of one-dimensional stochastic processes, more precisely the Brownian motion and families of stochastic differential equations sharply linked to the Brownian motion (usually known as L and G-classes). We are interested here in the $epsilon$-strong approximation. We propose an explicit and easy to implement procedure that constructs jointly, the sequences of exit times and corresponding exit positions of some well chosen domains. The main results control the number of steps to cover a fixed time interval and the convergence theorems for our scheme. We combine results on Brownian exit times from time-depending domains (one-dimensional heat balls) and classical renewal theory. Numerical examples and issues are also described in order to complete the theoretical results.
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We consider small perturbations of a dynamical system on the one-dimensional torus. We derive sharp estimates for the pre-factor of the stationary state, we examine the asymptotic behavior of the solutions of the Hamilton-Jacobi equation for the pre-factor, we compute the capacities between disjoint sets, and we prove the metastable behavior of the process among the deepest wells following the martingale approach. We also present a bound for the probability that a Markov process hits a set before some fixed time in terms of the capacity of an enlarged process.
We consider sequences of additive functionals of difference approximations for uniformly non-degenerate multidimensional diffusions. The conditions are given, sufficient for such a sequence to converge weakly to a W-functional of the limiting process. The class of the W-functionals, that can be obtained as the limiting ones, is completely described in the terms of the associated W-measures, and coincides with the class of the functionals that are regular w.r.t. the phase variable.
Let $X$ be the branching particle diffusion corresponding to the operator $Lu+beta (u^{2}-u)$ on $Dsubseteq mathbb{R}^{d}$ (where $beta geq 0$ and $beta otequiv 0$). Let $lambda_{c}$ denote the generalized principal eigenvalue for the operator $L+beta $ on $D$ and assume that it is finite. When $lambda_{c}>0$ and $L+beta-lambda_{c}$ satisfies certain spectral theoretical conditions, we prove that the random measure $exp {-lambda_{c}t}X_{t}$ converges almost surely in the vague topology as $t$ tends to infinity. This result is motivated by a cluster of articles due to Asmussen and Hering dating from the mid-seventies as well as the more recent work concerning analogous results for superdiffusions of cite{ET,EW}. We extend significantly the results in cite{AH76,AH77} and include some key examples of the branching process literature. As far as the proofs are concerned, we appeal to modern techniques concerning martingales and `spine decompositions or `immortal particle pictures.
In order to approximate the exit time of a one-dimensional diffusion process, we propose an algorithm based on a random walk. Such an algorithm was already introduced in both the Brownian context and in the Ornstein-Uhlenbeck context. Here the aim is therefore to generalize this efficient numerical approach in order to obtain an approximation of both the exit time and position for either a general linear diffusion or a growth diffusion. The efficiency of the method is described with particular care through theoretical results and numerical examples.
For a stochastic differential equation(SDE) driven by a fractional Brownian motion(fBm) with Hurst parameter $H>frac{1}{2}$, it is known that the existing (naive) Euler scheme has the rate of convergence $n^{1-2H}$. Since the limit $Hrightarrowfrac{1}{2}$ of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler scheme is the extension of the classical Euler scheme for It^{o} SDEs for $H=frac{1}{2}$, the convergence rate of the naive Euler scheme deteriorates for $Hrightarrowfrac{1}{2}$. In this paper we introduce a new (modified Euler) approximation scheme which is closer to the classical Euler scheme for Stratonovich SDEs for $H=frac{1}{2}$, and it has the rate of convergence $gamma_n^{-1}$, where $gamma_n=n^{2H-{1}/2}$ when $H<frac{3}{4}$, $gamma_n=n/sqrt{log n}$ when $H=frac{3}{4}$ and $gamma_n=n$ if $H>frac{3}{4}$. Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if ${X_t,0le tle T}$ is the solution of a SDE driven by a fBm and if ${X_t^n,0le tle T}$ is its approximation obtained by the new modified Euler scheme, then we prove that $gamma_n(X^n-X)$ converges stably to the solution of a linear SDE driven by a matrix-valued Brownian motion, when $Hin(frac{1}{2},frac{3}{4}]$. In the case $H>frac{3}{4}$, we show the $L^p$ convergence of $n(X^n_t-X_t)$, and the limiting process is identified as the solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this scheme. We also apply our approach to the naive Euler scheme.
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