Calculate $\mathbb{E}(\exp (- \lambda T_x ))$ for $\lambda > 0$ where $X$ is a Brownian Motion with drift

Question

In a course I am studying on Stochastic Processes, I encountered the following exercise:

Let $X_t = B_t + ct$ for some $c \in \mathbb{R}$ and where $B$ is a standard Brownian Motion. Now define $T_x$ to be the first hitting time of $x$. More formally: $$ T_x = \inf \{ t > 0 : X_t = x \}$$ Calculate $\mathbb{E}(\exp (- \lambda T_x ))$ for $\lambda > 0$ (ie. find the Laplace transform of $T_x$).

I have found that $M_t = \exp (\theta X_t - \lambda t)$ the choices of $\theta$ that ensure that $M$ is a martingale are $$\theta_1 = -c + \sqrt{c^2 + 2 \lambda} \space \text{ and } \space \theta_2 = -c - \sqrt{c^2 + 2 \lambda}$$

I know that the provided solution to problem is $$\mathbb{E}(\exp (- \lambda T_x )) = \exp (x(c+\sqrt{c^2 + 2 \lambda})) = \exp (-x \theta_2)$$

However, I am unsure of how to prove the result.

It feels as though I ought to begin by substituting the hitting time into the definition of $M$ to yield: $$M_{T_x} = \exp (\theta X_{T_x} - \lambda T_x)$$ and work on some simplification from here.

However, I am conscious of the fact that if I use the Martingale property, then I can no longer take $\theta = 0$ which would simplify $M_{T_x}$ to $\exp ( - \lambda T_x)$ (the desired expectation).

Is there an alternative approach that I am missing? I would be grateful for any help with this problem.

Answer 1

We have

$$E[e^{\theta B_t+\theta ct-\lambda t}]=e^{\theta^{2}t/2+\theta c t-\lambda t}$$

and so we indeed need $\theta^{2}/2+\theta c-\lambda=0\Rightarrow \theta=-c\pm \sqrt{c^{2}+2\lambda}$ to get martingale. For simplicity we just set $\theta= \sqrt{c^{2}+2\lambda}-c>0$.

So by the optional-stopping-theorem we have

$$E[e^{\theta X_{T_{x}}-\lambda T_{x}}]=1\Rightarrow E[e^{-\lambda T_{x}}]=e^{-(\sqrt{c^{2}+2\lambda}-c) x}.$$

(To be clear one also needs to truncate $T_{x}\wedge n$ and use that

$$|E[e^{\theta X_{T_{x}\wedge n}-\lambda T_{x}\wedge n}]|\leq e^{\theta |x|}$$ in order to apply dominated convergence theorem.).

This question has been answered already eg. Expectation stopped Brownian motion with drift