By Ioannis Karatzas

This ebook is designed as a textual content for graduate classes in stochastic tactics. it really is written for readers conversant in measure-theoretic chance and discrete-time approaches who desire to discover stochastic methods in non-stop time. The car selected for this exposition is Brownian movement, that is provided because the canonical instance of either a martingale and a Markov method with non-stop paths. during this context, the idea of stochastic integration and stochastic calculus is constructed. the ability of this calculus is illustrated through effects touching on representations of martingales and alter of degree on Wiener area, and those in flip allow a presentation of modern advances in monetary economics (option pricing and consumption/investment optimization).

This e-book encompasses a distinct dialogue of vulnerable and powerful recommendations of stochastic differential equations and a research of neighborhood time for semimartingales, with distinct emphasis at the thought of Brownian neighborhood time. The textual content is complemented by means of a good number of difficulties and exercises.

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**Example text**

If A is an increasing process and {M,,~; 0 :s; t < oo} is a bounded, right-continuous martingale. 5) E(M,A,) = EJ (O,t] MsdA s· 24 1. 4)' PROOF. I] M s- dA s. Consider a partition 0 = {t o,t 1 , ... :s; = t, and define The martingale property of M yields E r M~dAs = E k;lf. ) = E[f. 5). 0 The following concept is a strengthening of the notion of uniform integrability for submartingales. 8 Definition. :s; a) = I for a given finite number a > 0). :s; t < oo} is said to be of class D, if the family {XThe S" is uniformly integrable; of class DL, if the family {XThey.

For each interval (ttl, t)~\), j = 0, I, ... , 2n - I we consider a right-continuous modification of the martingale ~~nJ = E[A tj(n) < t S /\ A,ln, I~], J" t)~\. 13. The resulting process g:nl; t sa} is right-continuous on (0, a) except possibly at the points of the partition, and dominates the increasing process {A. s. at the points t\nl, ... , t~~. t}~\] ~~~ dA s ; j = 0, I, ... ,! for any 0 S t S a. Now the process (n) _ rt, - {~~~ 0, - (A. 4. The Doob-Meyer Decomposition is an optional time of the right-continuous filtration {~}, hence a stopping time in Y:, (cf.

26). 17) 2-p E(Vf)::; --E(AH 1- P 0 < p< 1 holds for any stopping time T of {§;}. 5. Continuous, Square-Integrable Martingales In order to appreciate Brownian motion properly, one must understand the role it plays as the canonical example of various classes of processes. One such class is that of continuous, square-integrable martingales. 25). 1 Definition. Let X = {X" §;; 0 ::; t < oo} be a right-continuous martingale. We say that X is square-integrable if EX,2 < 00 for every t ~ O. , we write X EJt2 (or X EJt2, if X is also continuous).