# Integration on Wiener Process

How can I show that below equation holds ?

$$\int\limits_{0}^{t} f \left( s \right)W_s ds = W_t \int\limits_{0}^{t}f \left( s \right)ds - \int\limits_{0}^{t}\int\limits_{0}^{s} f\left( u \right)dudW_s$$

$$W_t$$ is regular Wiener process.

## 2 Answers

Let

$$g_s = \int_0^s f_u du$$

By Ito-Leibniz product rule:

$$d(W_sg_s) = W_sdg_s+ g_sdW_s +d[g,W]_s$$

Assuming $$f_s$$ is deterministic, $$d[g,W]_s = 0$$ and we get:

$$d(W_sg_s) = W_sdg_s + g_sdW_s$$ In integral form, this is:

$$W_tg_t = \int_0^t W_sdg_s + \int_0^t g_sdW_s$$

Getting back to $$f$$:

$$W_t \int_0^t f_u du = \int_0^t W_sf_sds + \int_0^t \int_0^s f_u du dW_s$$

• Should be $d g_s$ after the Ito leibniz rule. – oliversm Aug 2 '20 at 13:04
• @oliversm fixed it. thank you. – ir7 Aug 2 '20 at 13:14

We can use Stochastic Integration by Parts to show this.

Taking the corollary from the link above \begin{align} X_t Y_t = X_0 Y_0 + \int_0^t X_s dY_s + \int_0 ^t Y_{s-} dX_s \end{align}

We set $$X_t$$ and $$Y_t$$ equal to the following: \begin{align} X_t &\to \int_0^t f(u) du\\ Y_t &\to W_t \end{align}

then \begin{align} W_t \int_0^t f(u) du &= W_0 \int_0^0 f(u) du + \int_0^t \Bigl( \int_0^s f(u) du \Bigr) dW_s + \int_0^t W_s d\Bigl( \int_0^s f(u) du \Bigr)\\ \end{align}

The second term is $$0$$ (since the integration range is $$0$$ and $$W_0 = 0$$). The fourth term simplifies via the Fundamental Theorem of Calculus which says $$d\Bigl( \int_0^s f(u) du \Bigr) = f(s)ds\\\\$$, so: \begin{align} W_t \int_0^t f(u) du &= \int_0^t \int_0^s f(u) du dW_s + \int_0^t W_s f(s)ds\\ \int_0^t W_s f(s)ds &= W_t \int_0^t f(u) du - \int_0^t \Bigl( \int_0^s f(u) du \Bigr) dW_s \end{align}

which is the expression in your question.