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By Doran R.S. (ed.)

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DiPaola moments and cumulants are related to each other by means of non-linear algebraic recurrence relationship (with argument omitted), ml = kl =Is Is ~ +kl ml ~ = + 2 ml + kl ~ m 4 = k4 + 3 m 1 + 3 k 2 ~ + k 1 ~ Is Is (6) where the coefficients in equation (6) are those of the Tartaglia triangle. From equations (1), (2) and (4), it is evident that the complete probabilistic description of the stochastic process X(t) at a fixed time t can be obtained indifferently by means of the knowledge of its probability density function or of its characteristic function or of its moments and/or cumulants of every order.

1 -+oj=1 n [Z(t ) + Z(t s 2 s -1 ) , t ] s-1 (B(t ) - B(t s . s-1 )) (82) 49 Stochastic Differential Calculus where (S) before the integral stands for Stratonovich. It can be shown that (S) { B(t) dB(t) = H B2(t)- B2(t, l] (83) from this equation we recognize that the Stratonovich integral obeys to the classical rules of integration. However for this integral the very useful relationships (78-81) are not valid. In general no relationship between the Ito integral and the Stratonovich integral exists.

It is interesting to note that expansion (110), neglecting higher order infinitesimal, can be also written in the form a

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