Let X t = b t + W t + Z t X t = b t

Let $X_t=bt+W_t+Z_t$ where $W$ is a Brownian motion and $Z_t={\sum }_{k=1}^{N_t}{Y}_k$ a $(\lambda ,F)$-compound Poisson process independent of $W$. The first passage time above the level $x$ is

$$T_x=inf\{t:X_t\ge x\}$$

and the overshoot is $O_x=X_{T_x}-x$. Let ${\Phi }_x$ be the Laplace transform of the pair $(T_x,O_x)$, i.e.,

$${\Phi }_x(\theta ,\mu ,x)=E\left(e^{-\theta T_x-\mu O_x}{1}_{\{T_x<\infty \}}\right)$$

Let ${\tau }_1$ be the first jump time of $N$. We wish to establish an integral equation for ${\Phi }_x$ using the following computation:

(1) Prove that

$$E\left(e^{-\theta T_x-\mu O_x}{1}_{\{T_x<{\tau }_1\}}\right)=e^{(b-\alpha )x}$$

where $\alpha =\sqrt{b^2+2(\theta +\lambda )}$.

(2) Prove that

$$\begin{array}{cc}E\left(e^{-\theta T_x-\mu O_x}{1}_{\{T_x={\tau }_1\}}\right)& =\frac{e^{(b-\alpha )x}}{\alpha (\mu -b+\alpha )}{\int }_{[0,x[}(e^{(\alpha -b)y}\end{array}$$

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