Related papers: A mild Ito formula for SPDEs
In this paper, we propose a semi-implicit Euler scheme to discretize the stochastic nonlinear Maxwell equations with multiplicative Ito noise, which is implicit in the drift term and explicit in the diffusion term of the equations, in order…
We study a family of stationary increment Gaussian processes, indexed by time. These processes are determined by certain measures sigma (generalized spectral measures), and our focus here is on the case when the measure sigma is a singular…
Correlation and smoothness are terms used to describe a wide variety of random quantities. In time, space, and many other domains, they both imply the same idea: quantities that occur closer together are more similar than those further…
Stochastic differential equations (SDE) often exhibit large random transitions. This property, which we denote as pathwise stiffness, causes transient bursts of stiffness which limit the allowed step size for common fixed time step explicit…
In this article, we derive a Stratonovich and Skorohod type change of variables formula for a multidimensional Gaussian process with low H\"older regularity (typically lower than 1/4). To this aim, we combine tools from rough paths theory…
Dynamical systems are essential to model various phenomena in physics, finance, economics, and are also of current interest in machine learning. A central modeling task is investigating parameter sensitivity, whether tuning atmospheric…
We construct and derive uniform stochastic estimates on the renormalised model for a class of fourth-order conservative quasilinear singular SPDEs in arbitrary dimension $d\geq 1$ and in the full subcritical regime of noise regularity. The…
In this paper, we are interested in the analytical study of a nonlinear Stochastic Partial Differential Equation (SPDE) arising as a model of phytoplankton aggregation. This SPDE consists in a diffusion equation with a chemotaxis term…
In the paper stochastic Volterra equations of nonscalar type in Hilbert space are studied. The aim of the paper is to provide some results on stochastic convolution and mild solutions to those Volterra equations. The motivation of the paper…
We prove the existence and uniqueness of mild solutions for a specific class of time-fractional $\psi$-Caputo evolution systems with a derivative order ranging from 1 to 2 in Banach spaces. By using the properties of cosine and sine family…
This paper presents an efficient approach to image segmentation that approximates the piecewise-smooth (PS) functional in [12] with explicit solutions. By rendering some rational constraints on the initial conditions and the final solutions…
The Poisson process is the most elementary continuous-time stochastic process that models a stream of repeating events. It is uniquely characterised by a single parameter called the rate. Instead of a single value for this rate, we here…
The main objective consists in generalizing a well-known It{\^o} formula of J. Jacod and A. Shiryaev: given a c{\`a}dl{\`a}g process S, there is an equivalence between the fact that S is a semimartingale with given characteristics (B^k , C,…
These are course notes on the application of SDEs to options pricing. The author was partially supported by NSF grant DMS-0739195.
In order to approximate solutions of stochastic partial differential equations (SPDEs) that do not possess commutative noise, one has to simulate the involved iterated stochastic integrals. Recently, two approximation methods for iterated…
Mounting empirical evidence suggests that the observed extreme prices within a trading period can provide valuable information about the volatility of the process within that period. In this paper we define a class of stochastic volatility…
A stochastic representation for the solutions of the Poisson-Vlasov equation, with several charged species, is obtained. The representation involves both an exponential and a branching process and it provides an intuitive characterization…
We provide existence and uniqueness of global (and local) mild solutions for a general class of semilinear stochastic partial differential equations driven by Wiener processes and Poisson random measures under local Lipschitz and linear…
We prove the existence and uniqueness of a mild solution for a class of non-autonomous parabolic mixed stochastic partial differential equations defined on a bounded open subset $D \subset \mathbb{R}^d$ and involving standard and fractional…
The article is devoted to the expansions of iterated Ito stochastic integrals based on generalized multiple Fourier series converging in the sense of norm in the space $L_2([t, T]^k),$ $k\in\mathbb{N}.$ The method of generalized multiple…