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## Deterministic interpretation of stochastic differential equation

In Paul Wilmott on Quantitative Finance Sec. Ed. in vol. 3 on p. 784 and p. 809 the following stochastic differential equation: $$dS=\mu\ S\ dt\ +\sigma \ S\ dX$$ is approximated in discrete time by $$\delta S=\mu\ S\ \delta t\ +\sigma \ S\ \phi \ (\delta t)^{1/2}$$ where $\phi$ is drawn from a standardized normal distribution.

This reasoning which seems to follow naturally from the definition of a Wiener process triggered some thoughts and questions I cannot solve. Think of the following general diffusion process: $$dS=a(t,S(t))\ dt\ +b(t,S(t))\ dX$$ Now transform the second term in a similar fashion as above and drop the stochastic component:$$dS=a(t,S(t))\ dt\ +b(t,S(t))\ (dt)^{1/2}$$

NB: The last term is no Riemann-Stieltjes integral (that would e.g. be $d(t)^{1/2}$)

My questions
(1) How would you interpret the last formula and how can the (now deterministic) differential equation be solved analytically? Will you get an additional term with a second derivative like in Ito's lemma?
(2) Is the last term a fractional integral of order 1/2 (which is a Semi-Integral)? Or is this a completely different concept?
(3) Will there be a different result in the construction of the limit like with Ito integrals (from left endpoint) and Stratonovich integrals (average of left and right endpoint)?

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mathoverflow.net/questions/9307/… – Steve Huntsman Apr 1 2010 at 18:24
You can't just drop the random term. It will explode in the limit dt -> 0. The random term has zero mean so there will be a lot of cancelation between positive and negative samples. Without it you are just adding lots of positive terms which are large relAtive to dt. – George Lowther Apr 1 2010 at 19:05
Thank you for the comment. I am not convinced: doesn't that behaviour depend on the function b which could e.g. be a trig function with zero mean too? – vonjd Apr 1 2010 at 19:26
@vonjd your trig function won't have a zero mean over an arbitrarily small interval, brownian motion does. – Mio Apr 2 2010 at 8:49