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a Stochastic Differential Equation |
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Consider a stochastic differential equation of the form
where t is time and dz is a random variable with a normal distribution of mean zero and standard deviation of unity. The solution of such an equation might be represented in integral form as
This representation of the solution is based upon the assumption that the integral exist. There is no problem with the existence of the first integral but the second integral is a different matter. The function z(t) is not well behaved, in particular it may not be of bounded variation which is required for the existence of the integral. Even the graphical depiction of such a function is a problem because there is a lack of coherence between z(t) and z(t+h) even when h is vanishingly small. When one tries to present the graph of such a function, as is shown in the figure below, the result looks like dust. Actually the graph below shows only the values of the function for discrete values of t. One cannot physically display z(t) for all values of t in an interval.
In order to visualize the function better lines have been drawn between the successive sampled values in the following graph. Actually this is for another realization of the stochastic processes because of the use of a random number generator in creating the function values.
Eugene Wong and Moshe Zakai discovered a remarkable thing. Suppose one takes a particular stochastic function z(t) and considers approximations to z(t) created by taking the values of z(t) for discrete values of t. For convenience assume t ranges over the interval [0,1] and the discrete values of t are {0, 1/n, 2/n, ..., n/n}. If one uses the values of z(t) at the discrete values and linear interpolations of these values of z for the values of t between the discrete values one gets an ordinary differential equation. This differential equation can be easily solved by evaluating the corresponding integral. As the discrete values of t are chosen closer and closer together by choosing larger values of n the solutions to the differential equation will approach a limit solution, say S'(t). The remarkable thing is that this limit solution S'(t) is not the solution to the original stochasic differential equation. Instead the limit solution is the solution of a related stochastic differential equation, one that can be derived from the original stochastic differential equation by making some simple adjustment. For the above stochastic differential equation the limit solution is a solution to:
(To be continued)
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