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Weak solutions of stochastic differential equations over the field of \(p\)-adic numbers. (English) Zbl 1136.60039

Summary: Study of stochastic differential equations on the field of \(p\)-adic numbers was initiated by the second author and has been developed by the first author, who proved several results for the \(p\)-adic case, similar to the theory of ordinary stochastic integrals with respect to Lévy processes on Euclidean spaces. In this article, we present an improved definition of a stochastic integral on the field and prove the joint (time and space) continuity of the local time for \(p\)-adic stable processes. Then we use the method of random time change to obtain sufficient conditions for the existence of a weak solution of a stochastic differential equation on the field, driven by the \(p\)-adic stable process, with a Borel measurable coefficient.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
60G52 Stable stochastic processes
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