arXiv subjects

D. Farazakis

Publications and source records attributed to D. Farazakis.

3 recordsLinked to original sources

arXiv2024

Malliavin Calculus for the stochastic heat equation and results on the density

We study the one-dimensional stochastic heat equation with unbounded, nonlinear,Lipschitz coefficients with Dirichlet boundary conditions. Using Malliavin calculus, we construct a piecewise approximation of the solution u and establish regularity results. This approximation enables us to provide a new proof of the existence of a density for the random variable u(t, x) at any fixed t, x. Unlike existing proofs, which rely on comparison principles ([10], [12]), our approach is based purely on a localization argument, which allows us to handle the unbounded coefficients.

math.AP↗

arXiv2023

Existence of maximal solutions for the financial stochastic Stefan problem of a volatile asset with spread

In this work, we consider the outer Stefan problem for the short-time prediction of the spread of a volatile asset traded in a financial market. The stochastic equation for the evolution of the density of sell and buy orders is the Heat Equation with a non-smooth noise in the sense of Walsh, posed in a moving boundary domain with velocity given by the Stefan condition. This condition determines the dynamics of the spread, and the solid phase $[s^-(t),s^+(t)]$ defines the bid-ask spread area wherein the transactions vanish. We introduce a reflection measure and prove existence and uniqueness of maximal solutions up to stopping times in which the spread $s^+(t)-s^-(t)$ stays a.s. non-negative and bounded. For this, we use a Picard approximation scheme and some of the estimates of \cite{BH} for the Green's function and the associated to the reflection measure obstacle problem. Analogous results are obtained for the equation without reflection corresponding to a signed density. Additionally, we apply some formal asymptotics when the noise depends only on time to derive that the spread is given by the integral of the solution of a linear diffusion stochastic equation.

math.PR↗

arXiv2018

Malliavin calculus for the stochastic Cahn-Hilliard / Allen Cahn equation with unbounded noise diffusion

The stochastic partial differential equation analyzed in this work, is motivated by a simplified mesoscopic physical model for phase separation. It describes pattern formation due to adsorption and desorption mechanisms involved in surface processes, in the presence of a stochastic driving force. This equation is a combination of Cahn-Hilliard and Allen-Cahn type operators with a multiplicative, white, space-time noise of unbounded diffusion. We apply Malliavin calculus, in order to investigate the existence of a density for the stochastic solution $u$. In dimension one, according to the regularity result in \cite{AKM}, $u$ admits continuous paths a.s. Using this property, and inspired by a method proposed in \cite{CW1}, we construct a modified approximating sequence for $u$, which properly treats the new second order Allen-Cahn operator. Under a localization argument, we prove that the Malliavin derivative of $u$ exists locally, and that the law of $u$ is absolutely continuous, establishing thus that a density exists.

math.PR↗