Telegraph Processes and Option Pricing by Alexander D. Kolesnik & Nikita Ratanov

Author:Alexander D. Kolesnik & Nikita Ratanov
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

(3.2.2)

Let be the telegraph process which starts from point .

Theorem 3.5.

The functions

(3.2.3)

(3.2.4)

defined by the conditional distributions (given the initial direction ) for all such that satisfy Cauchy problem (3.2.2).

Proof

The initial conditions of the Cauchy problem (3.2.2) follow immediately from (3.2.3)–(3.2.4). To prove that equations in (3.2.2) are valid for we first obtain the system of integral equations.

Denote by and the conditional probability densities for the particle that currently moves forward (), given the initial direction and respectively,

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