Stochastic Integrals by H. P. McKean (PDF)

Description

Stochastic Integrals discusses one area of diffusion processes: the differential and integral calculus based upon the Brownian motion. The book reviews Gaussian families, construction of the Brownian motion, the simplest properties of the Brownian motion, Martingale inequality, and the law of the iterated logarithm. It also discusses the definition of the stochastic integral by Wiener and by Ito, the simplest properties of the stochastic integral according to Ito, and the solution of the simplest stochastic differential equation. The book explains diffusion, Lamperti’s method, forward equation, Feller’s test for the explosions, Cameron-Martin’s formula, the Brownian local time, and the solution of dx=e(x) db + f(x) dt for coefficients with bounded slope. It also tackles Weyl’s lemma, diffusions on a manifold, Hasminski’s test for explosions, covering Brownian motions, Brownian motions on a Lie group, and Brownian motion of symmetric matrices. The book gives as example of a diffusion on a manifold with boundary the Brownian motion with oblique reflection on the closed unit disk of R squared. The text is suitable for economists, scientists, or researchers involved in probabilistic models and applied mathematics.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is an amazing book. It begins with Wiener vs. Ito definitions of the stochastic integral and then takes on through stochastic calculus on manifolds, including stochastic calculus in the complex plane and on lie groups. Dyson had written earlier on the latter topic, and one of his papers is referenced. Here are some half-baked notes:1. The Markov property is not mentioned, it’s asserted that ‘Brownian motion starts afresh at stopping times’ (pg. 10, 50).2. Time changes make it clear that only t-dependent difusion coefficients are treated in that case (pg. 31, 36, 41, 68 (Levy’s theorem)).3. the connection of Hermite polynomials to the exponential martingale is treated thoroughly (pg. 37).4. Ito’s proof of uniqueness and the condition for lack of finite time singularities is given on pg. 52.5. Finite time blow up is treated further (this explains why the stopping time is an important topic), pp. 54, 65, 796. Kolmogorov’s backward diffusion eqn. has the wrong sign for the time derivative (pg. 64). McKean probably asumed time independent drift and diffusion and wrote ‘t-s’ without informing the reader.7.Problem 2 for stopping times on pg 27 is easy and instructive.8. Sdes on tangent space are introduced via metric defined by elliptic operator, pg. 82 90, with patching together of solutions treated formally with no example (typical of discussions in differential geometry, the real work is left for the reader).9. Cameron-Martin-Girsanov on pp. 67.10. Brownian motion in the complex plane (!) via stereographic projection from the 2-sphere, pg. 106.11. Brownian motion on Lie groups, pg. 115.I’ve only scanned the book superficially to se what’s in it, working easy exercises (with Friedman and Durrett as basis), topics 8, 10 and 11 are interesting and advanced and will require a lot of hard work.

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