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 str**on**g soluti**on**s of stochastic differentialequati**on**s.Introducti**on**For 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 functi**on** of S. A collecti**on**( Fc 0 5 t c w) of a-subalgebras of a o-algebra !F will becalled a filtrati**on** if Fs c 5 whenever 0 S s < t c w. Afiltrati**on** (7; : 0 5 t c -1 is said to be right-c**on**tinuous 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, ).C**on**sider a probability space (R, 5 P) and aftltrati**on** (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-c**on**tinuous **on** [O,w) andpossesses a finite left-hand limit at each t 6 (0.00). An**on**negative, extended real valued random variable T defined**on** (R, !E P) is said to be a stopping time (with respect to thefiltrati**on** (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 n**on**ckcnasing 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 variati**on** pnxcss if almost all sample paths of(Y(t): 0 5 t < -) are of fdte variati**on** **on** each **com**pactinterval 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 variati**on** 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 n**on**decnasing,rightc**on**tinuous functi**on** 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-c**on**tinuous 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 variati**on** 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 C**on**ference **on** Iqjormati**on** Sciences and Systems, March 20 - 22,1991, Baltimore, Maryland; to be published in the Proceedings of the Cortference.