Related papers: On the strong convergence of multiple ordinary int…
In this paper we consider the Stratonovich reflected stochastic differential equation $dX_t=\sigma(X_t)\circ dW_t+b(X_t)dt+dL_t$ in a bounded domain $\O$ which satisfies conditions, introduced by Lions and Sznitman, which are specified…
We consider anticipative Stratonovich stochastic differential equations driven by some stochastic process (not necessarily a semi-martingale). No adaptedness of initial point or vector fields is assumed. Under a simple condition on the…
We prove that a sequence of semi-discrete approximations converges to a multiplicative functional for reflected Brownian motion, which intuitively represents the Lyapunov exponent for the corresponding stochastic flow. The method of proof…
The Levy transform of a Brownian motion B is the Brownian motion B't, the integral over (O,t) of sign of Bs with respect to dBs. Call T the corresponding transformation on the Wiener space W. We establish that a.s. the orbit of w in W under…
Stochastic integration w.r.t. fractional Brownian motion (fBm) has raised strong interest in recent years, motivated in particular by applications in finance and Internet traffic modelling. Since fBm is not a semi-martingale, stochastic…
We revisit the Markov approximation necessary to derive ordinary Brownian motion from a model widely adopted in literature for this specific purpose. We show that this leads to internal inconsistencies, thereby implying that further search…
Let X be the solution of the multidimensional stochastic differential equationdX(t) = b(t, X(t)) dt + sigma(t, X(t)) dW(t)\, with X(0)=x where W is a standard Brownian motion. We show that when b is measurable and sigma is in an appropriate…
Sub-fractional Brownian motion is a process analogous to fractional Brownian motion but without stationary increments. In \cite{GGL1} we proved a strong uniform approximation with a rate of convergence for fractional Brownian motion by…
In this article, an uniform discretization of stochastic integrals $\int_{0}^{1} f'_-(B_t)\ud B_t$, with respect to fractional Brownian motion with Hurst parameter $H \in (1/2,1)$, for a large class of convex functions $f$ is considered. In…
The article is devoted to the expansions of iterated Stratonovich stochastic integrals on the basis of the method of generalized multiple Fourier series that converge in the sense of norm in Hilbert space $L_2([t, T]^k),$ $k\in\mathbb{N}.$…
We give a new estimate on Stieltjes integrals of H\"older continuous functions and use it to prove an existence-uniqueness theorem for solutions of ordinary differential equations with H\"older continuous forcing. We construct stochastic…
We discuss Meyers-Serrin's type results for smooth approximations of functions $b=b(t,x):\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^m$, with convergence of an energy of the form \[ \int_{\mathbb{R}}\int_{\mathbb{R}^n} w(t,x)…
The purpose of this paper is to establish the convergence in law of the sequence of "midpoint" Riemann sums for a stochastic process of the form f'(W), where W is a Gaussian process whose covariance function satisfies some technical…
We prove a general functional limit theorem for multiparameter fractional Brownian motion. The functional law of the iterated logarithm, functional L\'{e}vy's modulus of continuity and many other results are its particular cases.…
We study pathwise approximation of scalar stochastic differential equations at a single point. We provide the exact rate of convergence of the minimal errors that can be achieved by arbitrary numerical methods that are based (in a…
We examine the numerical approximation of a quasilinear stochastic differential equation (SDE) with multiplicative fractional Brownian motion. The stochastic integral is interpreted in the Wick-It\^o-Skorohod (WIS) sense that is well…
We generalize the results of Ambrosio [Invent. Math. 158 (2004), 227--260] on the existence, uniqueness and stability of regular Lagrangian flows of ordinary differential equations to Stratonovich stochastic differential equations with BV…
The weak Stratonovich integral is defined as the limit, in law, of Stratonovich-type symmetric Riemann sums. We derive an explicit expression for the weak Stratonovich integral of $f(B)$ with respect to $g(B)$, where $B$ is a fractional…
We establish global universal approximation theorems on spaces of piecewise linear paths, stating that linear functionals of the corresponding signatures are dense with respect to $L^p$- and weighted norms, under an integrability condition…
Stochastic calculus with respect to fractional Brownian motion (fBm) has attracted a lot of interest in recent years, motivated in particular by applications in finance and Internet traffic modeling. Multifractional Brownian motion (mBm) is…