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On two dimensional Markov processes with branching property

1969; American Mathematical Society; Volume: 136; Linguagem: Inglês

10.1090/s0002-9947-1969-0234531-1

1088-6850

Shinzo Watanabe,

Stochastic processes and financial applications

Introduction.A continuous state branching Markov process (C.B.P.) was introduced by Jirina [8] and recently Lamperti [10] determined all such processes on the half line.(A quite similar result was obtained independently by the author.)This class of Markov processes contains as a special case the diffusion processes (which we shall call Feller's diffusions) studied by Feller [2].The main objective of the present paper is to extend Lamperti's result to multi-dimensional case.For simplicity we shall consider the case of 2-dimensions though many arguments can be carried over to the case of higher dimensions(2).In Theorem 2 below we shall characterize all C.B.P.'s in the first quadrant of a plane and construct them.Our construction is in an analytic way, by a similar construction given in Ikeda, Nagasawa and Watanabe [5], through backward equations (or in the terminology of [5] through S-equations) for a simpler case and then in the general case by a limiting procedure.A special attention will be paid to the case of diffusions.We shall show that these diffusions can be obtained as a unique solution of a stochastic equation of Ito (Theorem 3).This fact may be of some interest since the solutions of a stochastic equation with coefficients Holder continuous of exponent 1/2 (which is our case) are not known to be unique in general.Next we shall examine the behavior of sample functions near the boundaries (xj-axis or x2-axis).We shall explain, for instance, the case of xraxis.There are two completely different types of behaviors.In the first case Xj-axis acts as a pure exit boundary: when a sample function reaches the Xj-axis then it remains on it moving as a one-dimensional Feller diffusion up to the time when it hits the origin and then it is stopped.In the second case, there is a point x0 on x^-axis such that 22 = (0, *o) acts as a reflecting boundary and Ex= (x0, oo) acts as a pure entrance boundary (Theorem 4 and Corollaries).1. Definitions and the main theorem.Let D° = {x = (x1, x2) : xx>0, x2>0}, D = {x = (x1, x2) : xr ä 0, x2 ^ 0} and D = D u {A} be the one-point compactification of D.