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Stochastic Differential Equations

Avner Friedman

posted on 05 March 2007

reviewed by Joseph L. McCauley

First, regardless of what mathematicians may believe or say, this is not a book for beginners. Second, if you understand stochastic processes at the level of Gnedenko, Stratonovich, and Wax and want to understand what is possible beyond that, then this is a valuable reference. From a modern standpoint the index is terrible, there's far more in the book than the index and table of contents indicate. I review the book to the chapter before the Cameron-Martin-Girsanov theorem, for that topic I recommend Durrett's 1984 book. Let me indicate briefly what you can find in Friedman.

The claim (without proof) that the Chapman-Kolmogorov eqn. defines a semi-group even in the absence of time translational invariance (pg. 23). Time translational invariance means that the drift R(x,t) and diffusion D(x,t) coefficients are t-independent, but the process is not stationary unless the 1-point density f(x,t) approaches a statistical equilibrium density f(x), and this is typically impossible when R and D depend on x (see L. Arnold, e.g.). Proving a semi-group for the case of time translational invariance is trivial. Would have been nice to see the proof for the general case.

The condition for a Wiener process (pg. 36) emphasizes time-translational invariance. This is misleading. The condition for a Wiener process is stationary increments combined with variance linear in time. This yields time translational invariance. Generalizing time translational invariance leads to stationary processes, this includes the Zwanzig-Mori memory processes (these are not Ito processes)) popular in physics in the 1970s. Combining stationary increments with variance nonlinear in time leads to processes with long time increment autocorrlelations like fractional Brownian motion (fBm), also not an Ito process. But, one can easily generalize from Wiener processes to Martingales, processes with uncorrelated and generally nonstationary increments x(t,T)=x(t+T)-x(t), and from there to general Ito processes with drift. Increments and increment autocorrelations are not discussed in Friedman, but see our 2005-2007 papers posted on

Hölder continuity, nondifferentiability and infinite length of Brownian paths are discussed on pp. 39-40.

On pg. 46, eqns. (5.1,2), "Levy's Theorem", a word of warning: for an arbitrary martingale x(t), x^2(t)-t is not a martingale. An arbitrary martingale is topologically inequivalent to a Wiener process. The assumption that martingales are generally equivalent to Wiener processes is the source of much too confusion in financial math texts.

The derivation of both Kolmogorov's backward time pde (K1) and the Fokker-Planck pde (K2) from Ito's lemma (pp. 139-150). The proof that the Green functions (transition densities) of K1 and K2 are adjoints of each other implies the Chapman-Kolmogorov eqn. (assigned as problem 11, pg. 151)! This is the reverse of the usual derivation, where one derives K2 from a Chapman-Kolmogorov eqn., and is extremely enlightening. Friedman's assumption is that we have a Markov process, and that is guaranteed iff. the drift and diffusion coefficients R and D are history-independent, depend on (x,t) alone and on no earlier states. Because Friedman's emphasis is on solving elliptic and parabolic pdes, rather than on transition densities for Markov processes, he first writes the Fokker-Planck pde (Kolmogorov's second pde, or K2) in standard elliptic form (pg. 142). For the same reason, he could have started with K1 directly on pg. 139 but instead delays introducing K1 until pg. 142. I.e., he first discusses how to solve elliptic and parabolic pdes by 'running an Ito process'.

The 'Feynman-Katz' formula is found on pp. 147-8, and is applied to Black-Scholes type pdes over 20 yeas before Duffie published and popularized the same solution in the financial economics literature.

Friedman's very general treatment shows that not only Kolmogorov's two pdes but a more general class of Black-Scholes like pdes and adjoints satisfy the Chapman-Kolmogorov eqn. Feller's example of a discrete nonMarkov system satisfying the C-K eqn. was known in that era, maybe not to Friedman. Friedman provides us with the general formalism of nonMarkov systems continuous in both x and t that satisfy C-K.

A word of warning: Friedman (mis-)defines a Markov process by the Chapman-Kolmogorov eqn. As Feller showed, systems containing memory also satisfy that eqn. Friedman apparently did not realize this. Friedman's theoretical development is very general and applies to Ito processes with finite memory, which is a big plus. See cond-mat/0702517 on