MEMOIRS O F T H iAMERICAN MATHEMATICAL SOCIETY NLMBKR 4 ON STOCHASTIC DlFFliRL. NT. lAL LUAUONS KFYOSl 1TO PUBLISHED BY THh AMERICAN MATHEMATFCAL SCXJF1T 531 West 116th St., New York City ON STOCHASTIC DIFFERENTIAL EQUATIONS By KIYOSI ITO Let Xj. be a simple Markoff process with a continuous parameter t, and F t, s, E be the transition probability law of the process D F t, ,s, E Prfx E X.3, where the right side means the probability of x a E under the condition x. f Hie differential of x. at t s is given by the transition probability law of x in an infinitesimal neighborhood of t s 2 FCsA jjs E. W. Feller has discussed the case in which it has the following form 3 F sA 2 , JJS A E 1p s, I yA 2 G sA 2 , j js A E yA 2 p s, j P s, 3 , E o yA 2 , where G sAg, 5 s A, j, E is a probability distribution as a function of E and satisfies 5 T T f 1 2 J j hjl f 6 2 J, lJ G sA 2 , J js dn b t, J, for A A and p s, J and P s, J , E is a probability distribution in E. The special case of M p s, J O 11 has already been treated by A, Kolmogoroff and S. Bernstein. 3 We shall introduce a somewhat general definition of the differential of the process x. Cf. 85. Let P A denote the conditional probability law L 8,5 ,, 2 MxV E3 , A V A 2 0. If the 1 A times convolution of P fl A tends to a probability law L with regard to Levys lawdistance as A A 0, then L is called the I d S, J stochastic differential coefficient at s. L is clearly an infinitely divisible law. In the above Fellers case the...

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