ON THE RELATION BETWEEN ORDINARY AND STOCHASTIC DIFFERENTIAL EQUATIONS,
CALIFORNIA UNIV BERKELEY ELECTRONICS RESEARCH LAB
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The following problem is considered in this paper het x sub t be a solution to the stochastic differential equation dx sub t mx sub t, tdt Sx sub t, t dy sub t where y sub t is the Brownian motion process. het the nth derivative of x sub t be the solution to the ordinary differential equation which is obtained from the stochastic differential equation by replacing y sub t with the nth derivative of y sub t where this derivative is a continuous piecewise linear approximation to the Brownian motion and converges to y sub t as n approaches infinity. If x sub t is the solution to the stochastic differential equation in the sense of Ito does the sequence of the solutions converge to x sub t. It is shown that the answer is in general negative it is, however, shown that the nth derivative of x sub t converges in the mean to the solution of another stochastic differential equation which is given.