Stochastic integral representation of solutions to Hodge theoretic Poisson's equations on Graphs, and cooperative value allocation of Shapley and Nash
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The fundamental connection between stochastic differential equations (SDEs) and partial differential equations (PDEs) has found numerous applications in diverse fields. We explore a similar link between stochastic calculus and combinatorial PDEs on graphs with Hodge structure, by showing that the solution to the Hodge-theoretic Poisson's equation on graphs allows for a stochastic integral representation driven by a canonical time-reversible Markov chain. When the underlying graph has a hypercube structure, we further show that the solution to the Poisson's equation can be fully characterized by five properties, which can be thought of as a completion of the Lloyd Shapley's four axioms.
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