Related papers: Rough paths and symmetric-Stratonovich integrals d…
Stochastic approximation is a framework unifying many random iterative algorithms occurring in a diverse range of applications. The stability of the process is often difficult to verify in practical applications and the process may even be…
We establish the existence of solutions to path-dependent rough differential equations with non-anticipative coefficients. Regularity assumptions on the coefficients are formulated in terms of horizontal and vertical derivatives.
The stochastic partial differential equation analyzed in this work is the Cahn-Hilliard equation perturbed by an additive fractional white noise (fractional in time and white in space). We work in the case of one spatial dimension and apply…
The present paper is devoted to the study of sample paths of G-Brownian motion and stochastic differential equations (SDEs) driven by G-Brownian motion from the view of rough path theory. As the starting point, we show that quasi-surely,…
We consider the stochastic continuity equation perturbed by a fractional Brownian motion and the drift is allowed to be discontinuous. We show that for almost all paths of the fractional Brownian motion there exists a solution to the…
We develop the connection between large deviation theory and more applied approaches to stochastic hybrid systems by highlighting a common underlying Hamiltonian structure. A stochastic hybrid system involves the coupling between a…
In this paper, we consider a stochastic system described by a differential equation admitting a spatially varying random coefficient. The differential equation has been employed to model various static physics systems such as elastic…
We demonstrate how path integrals often used in problems of theoretical physics can be adapted to provide a machinery for performing Bayesian inference in function spaces. Such inference comes about naturally in the study of inverse…
We investigate the stochastic modified equation which plays an important role in the stochastic backward error analysis for explaining the mathematical mechanism of a numerical method. The contribution of this paper is threefold. First, we…
T. Lyons' rough path theory is something like a deterministic version of K. Ito's theory of stochastic differential equations, combined with ideas from K. T. Chen's theory of iterated path integrals. In this article we survey rough path…
A new paradigm recently emerged in financial modelling: rough (stochastic) volatility, first observed by Gatheral et al. in high-frequency data, subsequently derived within market microstructure models, also turned out to capture…
We consider a Stratonovich heat equation in $(0,1)$ with a nonlinear multiplicative noise driven by a trace-class Wiener process. First, the equation is shown to have a unique mild solution. Secondly, convolutional rough paths techniques…
Langevin equation with a multiplicative stochastic force is considered. That force is uncorrelated, it has the L\'evy distribution and the power-law intensity. The Fokker-Planck equations, which correspond both to the It\^o and Stratonovich…
We consider a differential equation driven by a Brownian motion as well as a rough path. We prove a Girsanov-type result for this equation to construct a weak solution in the probabilistic sense.
In this paper, we first prove that the local time associated with symmetric $\alpha$-stable processes is of bounded $p$-variation for any $p>\frac{2}{\alpha-1}$ partly based on Barlow's estimation of the modulus of the local time of such…
We use a path integral approach for solving the stochastic equations underlying the financial markets, and we show the equivalence between the path integral and the usual SDE and PDE methods. We analyze both the one-dimensional and the…
We discuss stochastic differential equations with a stiff linear part and their approximation by stochastic exponential integrators. Representing the exact and approximate solutions using B-series and rooted trees, we derive the order…
We prove the existence and uniqueness of solutions to a class of stochastic scalar conservation laws with joint space-time transport noise and affine-linear noise driven by a geometric p-rough path. In particular, stability of the solutions…
In the setting of stochastic Volterra equations, and in particular rough volatility models, we show that conditional expectations are the unique classical solutions to path-dependent PDEs. The latter arise from the functional It\^o formula…
The aim of the paper is to show the probabilistically strong well-posedness of rough differential equations with distributional drifts driven by the Gaussian rough path lift of fractional Brownian motion with Hurst parameter…