[944] Differentiating Integrals

[Gautam Iyer]

On Fri, Nov 17, 2017 at 07:09:33AM -0500, Richard Rosenbaum wrote:

> When differentiating integrals, is there a difference between Ito
> integrals and regular Riemann integrals?

Yes there's a big difference. If

    F(t) = integral from 0 to t of f(s) ds

then the derivative of F(t) = f(t), in the honest usual way. But if

    M(t) = integral from 0 to t of g(s) dW(s)

then M is not differentiable with respect to time. "dM", technically has
no meaning, like the regular derivative does and

    dM = g(s) dW(s)

is only shorthand notation for

    M(t) - M(0) = integral from 0 to t of g(s) dW(s)

(Note M(0) = 0, so the above is consistent.)

See the text following remark 4.7 and remark 5.2 in the notes for a
latexed version of the above.

GI

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