Let $(X_t)$ be an Ito process of the form
$$ dX_t = u(\omega, t) dt + dB_t, \ t \in [0, T]$$
where $T \in [0, \infty)$ is given and $(B_t)$ is a standard Brownian motion (for simplicity $1$ dimensional) with respect to the law $\mathbb{P}$ given a probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
Define $\mathcal{F}_t$ to be the $\sigma$-algebra generated by the random variables $B_s$, $s \leq t$.
I pressume that you wanted to assume the coefficient $u$ is bounded. To have it a bit more general we could have assumed the Novikov's condition, that is,
$$ \mathbb{E}_{\mathbb{P}}\left[ \exp\left\{ \frac{1}{2}\int_0^T u^2(\omega, s) ds\right\} \right] < \infty$$
The Novikov condition yields that $(L_t)$ is a martingale w.r.t $(\mathcal{F}_t)$ and $\mathbb{P}$. This martingale is know as an exponential martingale and it can be shown that
$$ dL_t = - L_t u(\omega, t)dB_t.$$
Usually in the literature your coefficient $u(\omega, t)$ is replaced by $a(\omega, t)=-u(\omega, t)$ and so the corresponding SDE becomes
$$dL_t = L_ta(\omega, t)dB_t$$
probably this is the reason (you missed the minus) why you couldn't reduce the drift term.
You wish to show that $(Z_t:=X_tL_t)$ is a martingale w.r.t $(\mathcal{F}_t)$ and $\mathbb{P}$. We proceed by applying Ito's product formula, namely
$$dZ_t = X_t dL_t + L_t dX_t + dX_t dL_t$$
thus we obtain
$$dZ_t = -X_t L_t u(\omega, t)dB_t + L_t \left(u(\omega, t) dt + dB_t\right) -L_tu(\omega, t)dt$$ and so
$$ dZ_t = L_t(1-X_tu(\omega, t))dB_t.$$
So if you show that $b(\omega, t) := L_t(\omega)(1-X_t(\omega)u(\omega, t))$ is such that $b \in L^2(\Omega
\times [0, T])$ then you obtain that
$$Z_t = Z_0 + \int_0^t L_s(1-X_su(\omega, s))dB_s$$
is a martingale.
Last thing, I doubt that the Girsanov theorem is a hint to solve this exercise. I would rather say that, this exercise is a simplified Girsanov with a weaker conclusion. The assumptions in this exercise are similar to the ones in the Girsanov theorem. The theorem would imply that $(X_t)$ (if assumed to have $X_0=0$) is a standard Brownian motion with respect to the new measure $\mathbb{Q}$ defined on $\mathcal{F}_T$ via
$$ d\mathbb{Q}(\omega) = L_T(\omega)d\mathbb{P}(\omega).$$
To sum up,
- if $u$ is non-zero then $(X_t)$ cannot be a martingale w.r.t
$(\mathcal{F}_t)$ and $\mathbb{P}$. However, if we multiply it by a
`special' exponential martingale $(L_t)$ then the product $(X_tL_t)$
is martingale w.r.t $(\mathcal{F}_t)$ and $\mathbb{P}$.
- (Girsanov Theorem) $(X_t)$ could be viewed as a martingale w.r.t
$(\mathcal{F}_t)$ and $\mathbb{Q}$, where $\mathbb{Q}$ is defined on $\mathcal{F}_T$ via
$$ d\mathbb{Q}(\omega) = L_T(\omega)d\mathbb{P}(\omega).$$
This is essentially the definition of $\langle X,X\rangle_t$, but I guess it can also be verified using Ito's formula.
We apply Ito's formula to the function $f(x)=x^2$ and compute
\begin{align*}
d(X_t^2) &= 2X_t dX_t + dX_t dX_t = 2 X_tH_tdW_t + H_t^2dt
\end{align*}
so by integrating we have $X_t^2 - \int_0^t H_s^2ds = X_0^2 + 2\int_0^t X_s dX_s$. Since stochastic integrals are martingales, this implies $X_t^2 - \int_0^t H_s^2ds = X_t^2 - \langle X,X \rangle_t$ is a martingale.
Best Answer
Assume $(X_t)$ is a martingale, under your assumptions regarding $b$ the process $M_t=X_0+\int_0^t b dW_s$ is a martingale as well. The linear combination between martingales is again a martingale hence $$X_t-M_t=\int_0^t a(s) ds$$ is a martingale. This process has bounded variation and continuous sample path, it's a well known result that a martingale satisfying this two conditions is constant, and hence the $a=0$. The other direction is immediate.