arXiv subjects

Oleg Reichmann

Publications and source records attributed to Oleg Reichmann.

3 recordsLinked to original sources

arXiv2019

Existence and uniqueness results for time-inhomogeneous time-change equations and Fokker--Planck equations

We prove existence and uniqueness of solutions to Fokker--Planck equations associated to Markov operators multiplicatively perturbed by degenerate time-inhomogeneous coefficients. Precise conditions on the time-inhomogeneous coefficients are given. In particular, we do not necessarily require the coefficients to be neither globally bounded nor bounded away from zero. The approach is based on constructing random time-changes and studying related martingale problems for Markov processes with values in locally compact, complete and separable metric spaces.

math.PR↗

arXiv2019

On Skorokhod Embeddings and Poisson Equations

The classical Skorokhod embedding problem for a Brownian motion $W$ asks to find a stopping time $τ$ so that $W_τ$ is distributed according to a prescribed probability distribution $μ$. Many solutions have been proposed during the past 50 years and applications in different fields emerged. This article deals with a generalized Skorokhod embedding problem (SEP): Let $X$ be a Markov process with initial marginal distribution $μ_0$ and let $μ_1$ be a probability measure. The task is to find a stopping time $τ$ such that $X_τ$ is distributed according to $μ_1$. More precisely, we study the question of deciding if a finite mean solution to the SEP can exist for given $μ_0, μ_1$ and the task of giving a solution which is as explicit as possible. If $μ_0$ and $μ_1$ have positive densities $h_0$ and $h_1$ and the generator $\mathcal A$ of $X$ has a formal adjoint operator $\mathcal A^*$, then we propose necessary and sufficient conditions for the existence of an embedding in terms of the Poisson equation $\mathcal A^* H=h_1-h_0$ and give a fairly explicit construction of the stopping time using the solution of the Poisson equation. For the class of Lévy processes we carry out the procedure and extend a result of Bertoin and Le Jan to Lévy processes without local times.

math.PR↗

arXiv2018

Dirichlet Forms and Finite Element Methods for the SABR Model

We propose a deterministic numerical method for pricing vanilla options under the SABR stochastic volatility model, based on a finite element discretization of the Kolmogorov pricing equations via non-symmetric Dirichlet forms. Our pricing method is valid under mild assumptions on parameter configurations of the process both in moderate interest rate environments and in near-zero interest rate regimes such as the currently prevalent ones. The parabolic Kolmogorov pricing equations for the SABR model are degenerate at the origin, yielding non-standard partial differential equations, for which conventional pricing methods ---designed for non-degenerate parabolic equations--- potentially break down. We derive here the appropriate analytic setup to handle the degeneracy of the model at the origin. That is, we construct an evolution triple of suitably chosen Sobolev spaces with singular weights, consisting of the domain of the SABR-Dirichlet form, its dual space, and the pivotal Hilbert space. In particular, we show well-posedness of the variational formulation of the SABR-pricing equations for vanilla and barrier options on this triple. Furthermore, we present a finite element discretization scheme based on a (weighted) multiresolution wavelet approximation in space and a $θ$-scheme in time and provide an error analysis for this discretization.

q-fin.MF↗