Vol. 25, No. 2, 1968

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Expected values of functionals with respect to the Ito distribution

Michael Schilder

Vol. 25 (1968), No. 2, 371–380

Stochastic differential equations of the type (written symbolically)

x(n)(t) +m (t,x (n−1)(t),x(n− 2)(t),⋅⋅⋅ ,x′(t),x(t),t) = z′(t)
′             (n− 1)
x(S) = a0,x (S) = a1,⋅⋅⋅ ,x  (S) = an−1 S ≦ t ≦ T
where z(t) is Brownian motion, arise in physics and engineering and are also the object of study of pure mathematicians. In this paper it will be shown that the integral associated with the distribution of the function x() may be expressed in terms of a Wiener integral with a weighting functional (the Radon-Nikodym derivative). Thus the expected values of functionals with respect to the distribution of x() can be easily and concisely expressed. Also it will be shown that certain partial differential equations of physics naturally have their solutions associated with this integral.

Mathematical Subject Classification
Primary: 60.62
Received: 24 February 1967
Published: 1 May 1968
Michael Schilder