NDLI: Graph-Theoretic Approach to Stochastic Integrals with Clifford Algebras

Content Provider	Springer Nature Link
Author	Staples, George Stacey
Copyright Year	2007
Abstract	Given a fixed probability space (Ω,ℱ,ℙ) and m≥1, let X(t) be an L$^{2}$(Ω) process satisfying necessary regularity conditions for existence of the mth iterated stochastic integral. For real-valued processes, these existence conditions are known from the work of D. Engel. Engel’s work is extended here to L$^{2}$(Ω) processes defined on Clifford algebras of arbitrary signature (p,q), which reduce to the real case when p=q=0. These include as special cases processes on the complex numbers, quaternion algebra, finite fermion algebras, fermion Fock spaces, space-time algebra, the algebra of physical space, and the hypercube. Next, a graph-theoretic approach to stochastic integrals is developed in which the mth iterated stochastic integral corresponds to the limit in mean of a collection of weighted closed m-step walks on a growing sequence of graphs. Combinatorial properties of the Clifford geometric product are then used to create adjacency matrices for these graphs in which the appropriate weighted walks are recovered naturally from traces of matrix powers. Given real-valued L$^{2}$(Ω) processes, Hermite and Poisson-Charlier polynomials are recovered in this manner.
Starting Page	257
Ending Page	274
Page Count	18
File Format	PDF
ISSN	08949840
Journal	Journal of Theoretical Probability
Volume Number	20
Issue Number	2
e-ISSN	15729230
Language	English
Publisher	Kluwer Academic Publishers-Plenum Publishers
Publisher Date	2007-04-14
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Algebraic probability Combinatorial probability Geometric algebra Statistics Probability Theory and Stochastic Processes
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Statistics, Probability and Uncertainty

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