Let $Y_t$ be a killed process defined by \begin{eqnarray} Y_t = X_t \quad \mbox{if } t<\xi,\\ Y_t = 0 \quad \mbox{if } t\geq\xi. \end{eqnarray} where $\xi$ is a random time such that $$ \xi=\inf \{ t\geq 0 : \int_0^t h(X_u)du>e\} $$ $e$ is a exponential distribution with mean 1.
$X_t$ is governed by the following SDE: $dX_t=\mu(X_t)dt+\sigma(X_t)dW_t.$
I am solving some exercise to compute $$ \int_0^\infty e^{-\lambda T}P(\xi>T)dT. $$ That is, the Laplace transform of survival probability.
In addition, function $h(x)$ is given by $$ h(x)=\beta+x^{\alpha}. $$
I couldn't proceed with this step.. $$ \int_0^\infty e^{-\lambda T}P(\xi>T)dT=\int_0^\infty e^{-\lambda T}e^{-\int_0^Th(X_u)du}dT. $$
I think this problem should be linked to ODE form ... Hlep please