Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Defined a standard Wiener process $W = (W_t , \mathcal F_t)_{t≥0}$ and a deterministic, continuously differentiable function $f : [0, ∞) → \mathbb R$. Prove that $$f(t)W_t=\int_0^tW_sf'(s)ds+\int_0^tf(s)dW_s$$

share|cite|improve this question
up vote 2 down vote accepted

If you know Ito Lemma, then you just consider $F(t,x) = f_t\cdot x$ and the differential of $F$ is $$ \mathrm dF(t,W_t) = f'_tW_t\mathrm dt + f_t\mathrm dW_t + \mathrm d[f,W]_t $$ and since $f$ is deterministic and continuous, the latter term is zero. Hence $$ \mathrm dF(t,W_t) = f'_tW_t\mathrm dt + f_t\mathrm dW_t \quad\Leftrightarrow \quad F(T,W_T) = F(0,W_0) + \int_0^T f'_tW_t\mathrm dt + \int_0^Tf_t\mathrm dW_t. $$

share|cite|improve this answer
Thank you very much! – user79133 May 29 '13 at 11:20
@user79133: welcome! – Ilya May 29 '13 at 11:27

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.