Graph-Theoretic Approach to Stochastic Integrals with Clifford Algebras |
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Authors: | George Stacey Staples |
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Institution: | (1) Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1653, USA |
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Abstract: | Given a fixed probability space (Ω,ℱ,ℙ) and m≥1, let X(t) be an L2(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 L2(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 L2(Ω) processes, Hermite and Poisson-Charlier polynomials are recovered in this manner. |
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Keywords: | Algebraic probability Combinatorial probability Geometric algebra |
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