Taylor development to show that for Stratonovich stochastic calculus the chain regulation takes the kind of the timeless one

How can I show with a heuristic argument based upon a Taylor development that for Stratonovich. stochastic calculus the chain regulation takes the kind of the timeless (Newtonian ) one?

Worrying Ito calculus the reality that dX ^ 2 = dt results using a Taylor development in Ito's lemma - this reality needs to remain the very same with Stratonovich yet it need to in some way negate in there - I simply do not recognize just how ...

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2019-05-07 13:04:08
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Answers: 1

In the adhering to, $W_t$ is a Wiener procedure with increments having mean absolutely no and also typical inconsistency 1.

Primarily, the Stratonovich formula does the adhering to in regards to the Itô formula :

$$f(W_t) \circ dW_t = \frac{f(W_t+dW_t)+f(W_t)}{2}dW_t \; ,$$

which can be revised as

$$f(W_t) \circ dW_t = \frac{f(W_t+dW_t)-f(W_t)}{2}dW_t + f(W_t)dW_t \; .$$

Using Taylor development on the first term and also Itô calculus regulations, this can be streamlined to

$$f(W_t) \circ dW_t = \frac{f'(W_t)}{2}dt + f(W_t)dW_t \; .$$

Now, we can change $f$ with $f'$ in the formula to offer

$$f'(W_t) \circ dW_t = \frac{f''(W_t)}{2}dt + f'(W_t)dW_t \; .$$

But you can conveniently examine that the right-hand man side is just $df(W_t)$ according to Itô calculus regulations, consequently

$$df(W_t) = f'(W_t) \circ dW_t \; .$$

Which is simply a grandfather clause of the chain regulation. I presume from below on out you can generalise the argument.

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2019-12-03 05:42:15
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