📂Stochastic Differential Equations

Integration by Parts

Theorem ¹

If the bounded continuous function $f(s,\omega) = f(s)$ in $[0,t]$ depends only on $s$, then
$$\int_{0}^{t} f(s) d W_{s} = f (t) W_{t} - \int_{0}^{t} W_{s} d f (s)$$

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• $W_{t}$ is a Wiener process.

Description

It’s a theorem about Itô integration, not much different from the integration by parts we commonly know. It’s important to note that the integrand has changed. The derivation is also similar to that of the standard integration by parts method.

$$\begin{align*} & {{ d } \over { ds }} f(s) W_{s} = {{ d } \over { ds }} f(s) \cdot W_{s} + f(s) {{ d } \over { ds }} W_{s} \\ \implies& \int_{0}^{t} d f(s) W_{s} = \int_{0}^{t} W_{s} d f(s) + \int_{0}^{t} f(s) d W_{s} \\ \implies& f (t) W_{t} = \int_{0}^{t} f(s) d W_{s} + \int_{0}^{t} W_{s} d f (s) \end{align*}$$