On Stochastic Differential Equations
Cód:
491_9781406742176
De: R$ 0,00Por: R$ 191,30ou X de
MEMOIRS O F T H i-AMERICAN 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 FCs-A jjs E. W. Feller has discussed the case in which it has the following form 3 F s-A 2 , JJS A E 1-p s, I yA 2 G s-A 2 , j js A E yA 2 p s, j P s, 3 , E o yA 2 , where G s-Ag, 5 s A, j, E is a probability distribution as a function of E and satisfies 5 T- T f 1 2 J -j h-jl f 6 2 J, l-J G s-A 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 Mx-V E-3 , 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 law-distance 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 logarithmic characteristic function Received by the editors March 29, 5 KIYOSI I TO V , L S of L f is given by 7 z, L ib s, j z - a s, j z p s, 5 f 03 e iu2 - 1 P s, J , du J . 6 8 j 7 - 00 A problem of stochastic differential equations is to construct a Markoff process whose stochastic differential coefficient L. - is given as a function of t, . 9 W. Feller has deduced the following integro-differential equation from 3, 4, 5 and 6 F t, J s, E - P t, j F t, J s, E p t, f F t, 7 s,
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