Numerical methods for the 2nd moment of stochastic ODEs
Andreev, Roman; Kirchner, Kristin (2016), Numerical methods for the 2nd moment of stochastic ODEs. https://basepub.dauphine.fr/handle/123456789/17410
TypeDocument de travail / Working paper
Series titlecahier de recherche CEREMADE- Paris-Dauphine
MetadataShow full item record
Laboratoire Jacques-Louis Lions [LJLL]
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)Numerical methods for stochastic ordinary differential equations typically estimate moments of the solution from sampled paths. Instead, in this paper we directly target the deterministic equation satisfied by the first and second moments. For the canonical examples with additive noise (Ornstein-Uhlenbeck process) or multiplicative noise (geometric Brownian motion) we derive these deterministic equations in variational form and discuss their well-posedness in detail. Notably, the second moment equation in the multiplicative case is naturally posed on projective-injective tensor products as trial-test spaces. We propose Petrov-Galerkin discretizations based on tensor product piecewise polynomials and analyze their stability and convergence in the natural norms.
Subjects / Keywordsprojective and injective tensor product; Hilbert tensor product; variational problems; multiplicative noise; additive noise; Stochastic ordinary differential equations; Petrov-Galerkin discretizations
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