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Stochastic Analysis and Diffusion Processes$
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Gopinath Kallianpur and P Sundar

Print publication date: 2014

Print ISBN-13: 9780199657063

Published to Oxford Scholarship Online: April 2014

DOI: 10.1093/acprof:oso/9780199657063.001.0001

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Probability Theory and Partial Differential Equations

Probability Theory and Partial Differential Equations

Chapter:
(p.202) 8 Probability Theory and Partial Differential Equations
Source:
Stochastic Analysis and Diffusion Processes
Author(s):

Gopinath Kallianpur

P. Sundar

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199657063.003.0008

The connection between stochastic differential equations and partial differential equations is discussed in this chapter. The Dirichlet problem is studied and several examples are presented. Next, the heat equation, and the Feynman-Kac formula are discussed. The Kolmogorov forward and backward equations are derived after proving the smoothness of solutions in the mean square sense. Applications to pricing of derivatives in finance theory are given.

Keywords:   Dirichlet problem, Feynman-Kac formula, Kolmogorov forward and backward equations, Black-Scholes equation

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