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Topics in the Theory of Random Noise

R. Stratonovich

posted on 19 February 2004

reviewed by Joe McCauley

The first volume of this two volume set provides a first rate, readable and fairly complete introduction to stochastic processes by one of the main experts in the field. Notable, however, is that 'Stratonovich calculus' is not presented, which is just as well because Ito calculus is used nearly universally in finance theory now.

Implicit conditions for an approach to equilibrium are presented via eigenfunction expansions, but the reader should take care to notice that when the spectrum becomes continuous then there is no stationary limit, in spite of the fact that the stationary distribution still appears in the expansion with vanishing eigenvalue. Nonstationary processes are discussed later in the text, but the reader should pay attention that the integrals (4.67) required for the existence of the spectral density (4.66) do not exist (are infinite) whenever the process is nonstationary, which includes most cases of practical interest (for a Wiener process, e.g.). I emphasize this because financial math texts often toss bones to the idea of 'equilibrium conditions' and then discuss processes that are only nonstationary with no stationary limit (the Osborne-Black-Scholes lognormal model, e.g.).

This book is highly recommended for physics and engineering students, as well as for students of economics and finance.