currently I am trying to solve the following exercise.
Exercise: Let $b : \mathbb R → \mathbb R$ be Lipschitz, and let $t \mapsto X(t)$ be the unique strong solution of the 1-dimensional SDE given by $$ dX(t) = b(X(t)) dt + dB(t), \quad X(0) = x \in \mathbb R $$
- Use Girsanov's theorem to show that for any $M < \infty$, $x \in \mathbb R$ and $T > 0$, we have that $P(X(t) > M) > 0$.
- Choose $b(x) = -r$, where $r>0$ is a constant. Prove that, for all x, we have that $\lim_{t \to \infty} X(t) = -\infty$ almost surely. Compare this with the result in part (a).
My Solution: My idea is that under the probability space $(\Omega, \mathcal{F}, \mathbb P)$ the solution to the SDE is given by a Brownian motion + (non-deterministic) drift. Thus using Girsanov we want to find another probability space $(\Omega, \mathcal{F},Q)$ together with the Radon Nikodym derivative $dQ^{T}/dP^{T}$ such that we can trivially bound the event.
Calculation: Take arbitary $M>0$ and $T>0$, we want to find the Radon Nikodym derivative of $(X_t + H_t : t \leq T)$ with respect to $(X_t : t \leq T)$. Check the following.
- Take $N_T = \int^{T}_0 b(X_s) dB_s$, then since $b \colon \mathbb R \to \mathbb R$ is Lipschitz, one can easily show that $\mathcal{E}(N)$ (exponential martingale) is u.i.
- Using the Girsanov theorem we know that the Radon Nikodym derivative $Z_T$ is given by
$$ Z_T := \frac{dQ^{T}}{dP^{T}} = \exp\left(-\int^{1}_0 b(X_s) dB_s + \frac{1}{2} \int^{1}_0 b^2(X_s) ds \right) $$ - Applying this via Girsanov, while denoting $A = \{X(T) > M\}$ gives that $$ P(A) = \mathbf{E}_{P}\left(\mathbf{1}_{A} Z^T \right) = Q\left(\{B(T) > M\}\right) > 0, $$ where $B$ is a $Q$-Brownian motion and the last inequality follows easily using the definition of Brownian motion. For the second item, we can easily verify the limit via for example Borel Cantelli. We don't get a contradiction with respect to (a), since Girsanov theorem only gives a Radon Nikodym derivative up to some fixed time $t \leq T$.
Is this essentially correct? I am not 100% sure, since the drift of my underlying process also depends on the process itself and is non-deterministic. How does this compare to the usual case when one has deterministic drift?
