Levy characterization, representation of martingale and change of probability

I am preparing for the final, so I write some notes for the course maths finance.

Levy characterization for Brownian motion: 
If $\phi_s^T \phi_s = Id$, then

$B_t = \int^t_0 \phi_s dW_s$

is a standard Brownian motion

This theorem is very useful and it describes the nature that after a con-formal transform, the BM keeps its properties.

Representation of martingale:

This theorem has different version. The most general version is that for a $\mathcal{F}_t$ adapted local martingale $M_t$, it can be written as 

$M_t = \mathbb{E}[M_t] + \int_0^t H_s dWs$

where $H_t \in \mathbb{H}^2_{loc}$.

This is a mathematical version of perfect duplication theorem. The proof starts from the case $L^2 \text{martingale} \rightarrow L^1 \text{martingale} \rightarrow \text{local martingale}$. It has many application in the stochastic calculus.

Change of probability: 

First we define $Z_T = \exp{(\int_0^T \phi_s dW_s - \frac{1}{2}\phi_s^2 ds)}$. Generally, it's only a local martingale and if it satisfies $\mathbb{E}[Z_T] = 1$, we can define a change of probability

$\frac{d\mathbb{Q}}{d\mathbb{P}} = Z_T$ 

then under the new probability $\mathbb{Q}$, we can define a new BM in the form

$\tilde{B}_t = B_t - \int_0^t \phi_s ds$

We remark that in the case $\phi$ is deterministic, then the there is no problem since in this case, $Z_T$ is well defined of expectation 1. Otherwise, the expectation is not so clear but there is a theorem Novikov, says that if $\exp{(\int_0^T \frac{1}{2}\phi_s^2 ds)} < \infty$, then all the condition is satisfied.

The change of probability  can simplify the formula and has important applications on Monte-Carlo algorithms.