Related papers: Pathwise solvability of stochastic integral equati…
For any real-valued stochastic process $X$ with c\'rdl\'rg paths we define non-empty family of processes which have locally finite total variation, have jumps of the same order as the process $X$ and uniformly approximate its paths on…
We derive consistency and asymptotic normality results for quasi-maximum likelihood methods for drift parameters of ergodic stochastic processes observed in discrete time in an underlying continuous-time setting. The special feature of our…
We study a numerical method to compute probability density functions of solutions of stochastic differential equations. The method is sometimes called the numerical path integration method and has been shown to be fast and accurate in…
Unique existence of analytically strong solutions to stochastic partial differential equations (SPDE) with drift given by the subdifferential of a quasi-convex function and with general multiplicative noise is proven. The proof applies a…
We provide a general framework for the stability of solutions to stochastic partial differential equations with respect to perturbations of the drift. More precisely, we consider stochastic partial differential equations with drift given as…
We prove the solvability of It\^o stochastic equations with uniformly nondegenerate, bounded, measurable diffusion and drift in $L_{d+1}(\mathbb{R}^{d+1})$. Actually, the powers of summability of the drift in $x$ and $t$ could be different.…
In this paper we study general nonlinear stochastic differential equations, where the usual Brownian motion is replaced by a L\'evy process. We also suppose that the coefficient multiplying the increments of this process is merely Lipschitz…
We consider It\^o SDE $\d X_t=\sum_{j=1}^m A_j(X_t) \d w_t^j + A_0(X_t) \d t$ on $\R^d$. The diffusion coefficients $A_1,..., A_m$ are supposed to be in the Sobolev space $W_\text{loc}^{1,p} (\R^d)$ with $p>d$, and to have linear growth;…
In this paper we study the randomized non-autonomous complete linear differential equation. The diffusion coefficient and the source term in the differential equation are assumed to be stochastic processes and the initial condition is…
We study the Cauchy problem for Schr\"odinger type stochastic partial differential equations with uniformly bounded coefficients on a curved space. We give conditions on the coefficients, on the drift and diffusion terms, on the Cauchy…
We study quasi-linear stochastic partial differential equations with discontinuous drift coefficients. Existence and uniqueness of a solution is already known under weaker conditions on the drift, but we are interested in the regularity of…
We present a path integral formulation of Darcy's equation in one dimension with random permeability described by a correlated multi-variate lognormal distribution. This path integral is evaluated with the Markov chain Monte Carlo method to…
Stochastic diffusion equations are crucial for modeling a range of physical phenomena influenced by uncertainties. We introduce the generalized finite difference method for solving these equations. Then, we examine its consistency,…
We present a detailed analysis of non-degenerate time-homogeneous It\^o-stochastic differential equations with low local regularity assumptions on the coefficients. In particular the drift coefficient may only satisfy a local integrability…
The classical result by It\^o on the existence of strong solutions of stochastic differential equations (SDEs) with Lipschitz coefficients can be extended to the case where the drift is only measurable and bounded. These generalizations are…
We examine the relation between a stochastic version of the rough path integral with the symmetric-Stratonovich integral in the sense of regularization. Under mild regularity conditions in the sense of Malliavin calculus, we establish…
In this article we study the existence of pathwise Stieltjes integrals of the form $\int f(X_t)\, dY_t$ for nonrandom, possibly discontinuous, evaluation functions $f$ and H\"older continuous random processes $X$ and $Y$. We discuss a…
Motivated by the recent advances in the theory of stochastic partial differential equations involving nonlinear functions of distributions, like the Kardar-Parisi-Zhang (KPZ) equation, we reconsider the unique solvability of one-dimensional…
We consider one-dimensional stochastic differential equations with generalized drift which involve the local time $L^X$ of the solution process: X_t = X_0 + \int_0^t b(X_s) dB_s + \int_\mathbb{R} L^X(t,y) \nu(dy), where b is a measurable…
In this work we prove the existence of a smooth density for the solution to an SDE with locally Lipschitz and semimonotone drift, and will derive an exponential decay for this density and all of its derivatives as well. Our main tool in…