Some Comments on Stochastic Calculus - MathGeek.com
SOME COMMENTS ON STOCHASTIC CALCULUSEric B. HallDepartment of Electrical EngineeringSouthern Methodist UniversityDallas, Texas 75275Gary L. WiseDepamncnt of Electrical andComputer EngineeringandDcpcaanent of MathematicsThe University of Texas at AustinAustin, Texas 787 12AbstractThis paper surveys some unappreciatedcounterexamples related to the uniqueness and existence ofweak and strong solutions of stochastic differentialequations.IntroductionFor a topological space T we will let denote thefamily of Bore1 subsets of T, and for a subset S we will letIS( ') denote the indicator function of S. A collection( Fc 0 5 t c w) of a-subalgebras of a o-algebra !F will becalled a filtration if Fs c 5 whenever 0 S s < t c w. Afiltration (7; : 0 5 t c -1 is said to be right-continuous ifjFt = n jFt+, whenever 0 5 t c -. For a positive integer k,s>Owe will let Ak denote the k x k identity matrix. For arandom process (X(t, ): 0 5 t c -) we will let X(t) denoteX(t, ).Consider a probability space (R, 5 P) and aftltration (Fc 0 5 t c -) of F A random process(X(t): 0 5 t c -0) defined on (0, % P) is said to be 5-adapted if X(t) is &-measurable for each t E [O, w). Arandom process (X(t): 0 S t c so) defined on a probabilityspace (R, 5 P) is said to be cadlag if for each fixed wg thesample path X(t, 0) is right-continuous on [O,w) andpossesses a finite left-hand limit at each t 6 (0.00). Anonnegative, extended real valued random variable T definedon (R, !E P) is said to be a stopping time (with respect to thefiltration (yt : t E [0, w))) if(o E R: T(o) 5 t) E Ft for each t E [O, w). A local martingaledefined on (R, 5 P) is an 7;-adapted, as. cadlagprocess (X(t): 0 5 t c 00) such that X(0) = 0 a.s. and suchthat thenexists a nonckcnasing sequence of stopping times(Tnln (with respect to (Ft: 0 5 t c w)) such that(X(min(t, T,,)), Ft: 0 S t < -1 is a martingale for eachpositive integer n and such that lim Tn = 00 a.s. Ann -+ 00%-adapted, as. cadlag random process (Y(t): 0 I t c -1 iscalled a finite variation pnxcss if almost all sample paths of(Y(t): 0 5 t < -) are of fdte variation on each compactinterval of [0, -), A semimartingale defined on (R, 71 P) isan %-adapted, a.s. cadlag random process (M(t): 0 5 t c -)such that M(t) = M(0) + N(t) + B(t) where N(0) = 0,B(0) = 0, (N(t): 0 5 t < -) is a local martingale and(B(t): 0 S t c w) is a finite variation process. An Ft-adaptedrandom process (X(t): 0 S t c -) is said to be an increasingprocess if X(0) = 0 a.s., if X(t) is a.s. a nondecnasing,rightcontinuous function oft, and if E[X(t)] is finite foreach t E [O, w). A random process (X(t): 0 5 t c w) is saidto be a natural random process if it is increasing, and iffor every bounded, right-continuous martingale(M(t), Ft: 0 St c w) and for every t E (0, w). A martingale(M(t), 7;: 0 5 t c w) is said to be square integrable ifE[W] is finite for each t 2 0. The quadratic variation of asquare integrable martingale (M(t),%: 0 I t c -) is denotedby ((M(t)), %: 0 5 t < 0) and is the unique (up toindistinguishability), 7;-adapted, natural random processsuch that (M(0)) = 0 a.s. and((M(t))2 - (M(t)), 7;: 0 St c -) is a martingale.Presented at the 1991 Conference on Iqjormation Sciences and Systems, March 20 - 22,1991, Baltimore, Maryland; to be published in the Proceedings of the Cortference.