0.1 Stochastic Integration By Parts
Suppose \(f(s,\omega)\) is continuous and of bounded variation with respect to \(s \in[0,t]\), for almost all \(\omega\). Then \[ \int_{0}^t f(s)~dB_{s}=f(t)B_{t}-\int_{0}^tB_{s}df_{s}. \]
Example: We consider the integral \[ \int_{0}^tsdB_{s}. \] We consider the function \(g(t,x)=tx\) such that \[ Y_{t}=g(t,B_{t})=tB_{t}. \] By Itô Formula we have that \[ \begin{align} dY_{t} & =B_{t}dt+tdB_{t}+0 \\ d(tB_{t}) & = B_{t}dt+tdB_{t} \\ tB_{t} & =\int_{0}^t B_{s}ds + \int_{0}^t sdB_{s} \\ \implies \int_{0}^t sdB_{s} & = tB_{t}-\int_{0}^t B_{s}ds, \end{align} \] the same result as stated by integration by parts.