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Brownian Motion and Martingales in Analysis (The Wadsworth mathematics series, 1984)

Richard Durrett

posted on 21 December 2005

reviewed by Joseph L. McCauley

This, and not Durrett's later book "Stochastic Calculus and Applications" is the right book for econophysicists and finance theorists. It doesn't define Martingales, refers too much to other works for basic definitions, but has enough examples that one can use them to decipher the mathematese language the book is written in. The notation is terrible, simply awful, e.g. <...> doesn't mean an average but rather is Durrett's own special notation for squares of various processes. I'm not satisfied with the book, but at least one can figure much of it out. The parts on Girsanov's theorem and solutions of pdes via Markov processes (Cameron-Martin transformation) are far better than average, make the book worth studying. In the theorems on stochastic integration, it's not clear (for practical applications) what "x" is limited to. Are arbitrary Markov processes defined by stochastic differential equations included? This fuzziness in presentation is the flaw in typical, formal discussions of Girsanov's theorem: are arbitrary stochastic differential equations dx=R(x,t)dt+?D(x,t)dB(t) allowed, with arbitrary (x,t) dependent diffusion coefficients? Clarity via examples rather than abstract nonconstructive proofs from mathematicians would be greatly appreciated. Note to mathtematicians: showing how Girsanov's theorem works via application to (merely) Gaussian or lognormal statistics is totally inadequate for our purposes, nontrivial examples (with (x,t) dependent D(x,t)) are needed.

12/17/05: after finally understanding the awful notation on pp. 52-54 I was able to work through Girsanov's theorem. Most other texts state Girsanov's therem in terms of Wiener processes B(t) and speak of 'removing the drift', which limits the drift to t-dependence only (see Baxter and Martin or Steele, e.g.). Because Durrett uses Martingales x (which take on the form dx=?D(x,t)dB(t) for a Markov process) instead of restricting to Wiener processes, and developes stochastic integration in terms of Martingales x, his proof makes clear just how broadly applicable is Girsanov's theorem: it applies to (x,t) dependent drift R(x,t) and diffusion D(x,t) in a general Markov process dx=R(x,t)dt+?D(x,t)dB(t). Here's a translation of Durrett's notation for the reader (starting on pg. 52): for Martingales x and y (which I always take to be Markov processes for convenience), d =(dx)^2=D(x,t)dt, and d=dxdy=?D(x,t)?E(y,t)dt where dy=?E(y,t)dB(t). The solution to example 2 on pg. 85 (Ornsten-uhlenbeck process) is clearly wrong, and it would seem at first sight that this mistake was corrected on pg.204 of his (otherwise far more abstruse and opaque) text "Stochastic Calculus" written in 1996. However, that solution is also wrong. Here is the right answer, and you can use Ito's lemma to check that alpha( -a (integral vdt) + sigmaB(t)) really is a Martingale:

alpha=exp(-a integral v(t)dB(t) - (1/2)a^2 integral v^2(t)dt)

Before attempting to follow Durrett, I strongly advise the reader to learn the basics of Ito calculus (Ito's theorem and Ito's lemma) from Klaus Schulten's statistical physics notes published free on his Univ. of Illinois website.

Let me state as an afterthought that, with all its flaws, this is the only book that I know that treats Girsanov's theorem in the generality needed for application to the far from Gaussian, far from lognormal Markov processes that we construct from the empirical analysis of finance markets. In my opinion, every serious student of option pricing should own a copy.