# How to understand the integral in the Girsanov theorem?

Let $W^P$ be a $d$-dimesional $P$-wiener procss. Define $L_t = > e^{\int_0^t \phi_s^T dW_s^P - \frac{1}{2} \int_0^t \| \phi_s\|^2 > ds}$.Assuming $E^PL_T = 1$, then the measure given by $dQ = L_TdP$ is a probability measy and $dW_t^P = \phi_tdt+dW_t^Q$ where $W_t^Q$ is a $W$-wiener process.

That's the Girsanov theorem, but what in the world is that first integral in the definition of $L_t$?

$\phi^T$ is a row vector ..... do we have multiple dimensional versions of stochastic integrals as well??

• As $\phi$ is a row vector, while $W^P$ is a column vector, you can multiply them out to obtain a sum of integrals on scalar processes.. – Gordon Jan 5 '17 at 18:36
• @Gordon yep. And this is an answer, not a comment. – SRKX Jan 6 '17 at 10:09
• @SRKX You are right. I wanted to write an answer but Gordon's comment is a short and good answer. – user16651 Jan 6 '17 at 10:26
• @SRKX and Behrouz Maleki, thank you both. – Gordon Jan 6 '17 at 13:43