By Ioannis Karatzas

This ebook is designed as a textual content for graduate classes in stochastic techniques. it's written for readers conversant in measure-theoretic chance and discrete-time procedures who desire to discover stochastic procedures in non-stop time. The automobile selected for this exposition is Brownian movement, that's awarded because the canonical instance of either a martingale and a Markov procedure with non-stop paths. during this context, the idea of stochastic integration and stochastic calculus is constructed. the facility of this calculus is illustrated by means of effects bearing on representations of martingales and alter of degree on Wiener area, and those in flip let a presentation of contemporary advances in monetary economics (option pricing and consumption/investment optimization).

This publication includes a certain dialogue of vulnerable and robust strategies of stochastic differential equations and a research of neighborhood time for semimartingales, with detailed emphasis at the idea of Brownian neighborhood time. The textual content is complemented by means of a number of difficulties and exercises.

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**Sample text**

16 1. 8 that the submartingale X has rightcontinuous sample paths. It is of interest to investigate conditions under which we may assume this to be the case. 13 Theorem. Let X = {X" ~; 0 :s; t < oo} be a submartingale, and assume the filtration {~} satisfies the usual conditions. Then the process X has a rightcontinuous modification if and only if the function t f--+ EX, from [0,00) to IR is right-continuous. If this right-continuous modification exists, it can be chosen so as to be RCLL and adapted to {~}, hence a submartingale with respect to {~}.

4. F, P). Let us consider a process A = {A,; Os; t < oo} adapted to {~}. 4 Definition. e. have WE 0 we (a) Ao(w) = 0 (b) t 1-+ A,(w) is a nondecreasing, right-continuous function, and E(A,) < 00 holds for every t E [0, 00). An increasing process is called integrable if E(A oo ) < 00, where Aoo = lim,_oo A,. 5 Definition. ,) M s- dA s' for every 0 < t < 00. 6 Remarks. (i) If A is an increasing and X a measurable process, then with WE 0 fixed, the sample path {X,(w); 0 s; t < oo} is a measurable function from [0, 00) into IR.

5. PROOF. 13). This shows existence of A. To prove uniqueness, suppose there exists another process B satisfying the conditions on A. Then M ~ (XY - A) - (XY - B) =B - A is a continuous martingale with finite first variation. If we define 36 1. Martingales, Stopping Times, and Filtrations T" = inf{t 2 0: IMtl = n}, then {Mfn) ~ M t /\ Tn' ~; 0 ~ t < oo} is a continuous, bounded (hence squareintegrable) martingale, with finite first variation on every interval [0, t]. s. P. 14 Problem. Show that for X, Y E vIt~ and [0, t], m lim IIllll-O I.