Tohoku Mathematical Journal
2007
December
SECOND SERIES VOL. 59, NO. 4

Tohoku Math. J.
59 (2007), 547-564

Title WEAK SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS OVER THE FIELD OF $p$-ADIC NUMBERS

Author Hiroshi Kaneko and Anatoly N. Kochubei

(Received March 20, 2006, revised July 30, 2007)
Abstract. 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 integral 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.

2000 Mathematics Subject Classification. Primary 60H10; Secondaly 11S80, 60G52.

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