Continuous Transformations and Stochastic Differential Equations ( )
نویسنده
چکیده
where (x(£),x(0) = 0,0fktfíl\ is a Brownian motion process. Equation (0.1) has been studied by S. Bernstein [l], J. L. Doob [5] and others [2], [ 10]. In general, the solution given here is different from that given by these authors. Equation (0.1) is almost purely formal since the derivative dx/dt fails to exist with probability one. In [2], [5], [ 10], the stochastic integral of K. Ito [7], [8] is used to define an integrated form of (0.1), which is solved as in [8] to obtain a transformation of sample functions. The present work involves a transformation of sample functions but the integral used is the functional integral of R. Cameron and R. F agen [4]. E. B. Dynkin [6] has given a different method of attacking similar problems. Note that /(y|f) is a functional. 1. Comparison with previous work. Let m, re, and v be constants and specialize (0.1) to (1.1) dyit) = miyit) + v)dt + re(y«) + v)dx(t).
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