Sharp regularity near an absorbing boundary for solutions to second order SPDEs in a half-line with constant coefficients

Sean Ledger

Research output: Contribution to journalArticle (Academic Journal)peer-review

Abstract

We prove that the weak version of the SPDE problem \begin{align*} dV_{t}(x) & = [-\mu V_{t}'(x) + \frac{1}{2} (\sigma_{M}^{2} + \sigma_{I}^{2})V_{t}''(x)]dt - \sigma_{M} V_{t}'(x)dW^{M}_{t}, \quad x > 0, \\ V_{t}(0) &= 0 \end{align*} with a specified bounded initial density, $V_{0}$, and $W$ a standard Brownian motion, has a unique solution in the class of finite-measure valued processes. The solution has a smooth density process which has a probabilistic representation and shows degeneracy near the absorbing boundary. In the language of weighted Sobolev spaces, we describe the precise order of integrability of the density and its derivatives near the origin, and we relate this behaviour to a two-dimensional Brownian motion in a wedge whose angle is a function of the ratio $\sigma_{M}/\sigma_{I}$. Our results are sharp: we demonstrate that better regularity is unattainable.
Original languageEnglish
Pages (from-to)1
Number of pages26
JournalStochastic Partial Differential Equations: Analysis and Computations
Volume2
Issue number1
DOIs
Publication statusPublished - 1 Mar 2014

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