Consider a stochastic differential equation:
$$\frac{dX}{dt} = b + \sigma \frac{dW}{dt}, X(0) = x$$
where $b,\sigma$ are constant, $x \in [0,1]$, and $W$ is a Wiener process. Let $\tau = \inf \{ t \geq 0 : X(t) \in \{ 0,1 \} \}$ and $u_c(x) = \mathbb{E}_x(e^{c \tau})$. $u_c$ is clearly finite for any $c \leq 0$, but may be infinite for sufficiently large $c>0$. The title arises because $u_c$, as a function of $c$, is the Laplace transform of the density of $\tau$ (if $\tau$ has one).
Let $(Lf)(x) = \frac{\sigma^2}{2} f''(x) + b f'(x)$ be the generator of this process. The Feynman-Kac formula implies that we have the following equation for $u_c$:
$$(L u_c)(x) + c u_c(x) = 0 \text{ if } x \in (0,1) \\ u_c(x) = 1 \text{ if } x \in \{ 0,1 \}.$$
In this simple case, the solution to this problem can be calculated explicitly. We calculate the roots of
$$\frac{\sigma^2}{2} \lambda^2 + b \lambda + c = 0.$$
Then we have our solution $u_c(x) = c_1 e^{\lambda_1 x} + c_2 e^{\lambda_2 x}$ with
$$\begin{bmatrix} 1 & 1 \\ e^{\lambda_1} & e^{\lambda_2} \end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}.$$
This equation has a solution if and only if $e^{\lambda_1} \neq e^{\lambda_2}$ or $\lambda_1$ and $\lambda_2$ are both integer multiples of $2 \pi i$. (We can ignore this latter case by assuming $b \neq 0$.)
My problem is that this only fails at discrete values of $c$. Yet it should fail for all $c$ larger than some critical $c^*$, by a straightforward probabilistic argument. So I think I've made an error, but I can't seem to find it. Any thoughts on where I made a mistake?
The set of values of $c$ such that $e^{\lambda_1}=e^{\lambda_2}$ is $\{\gamma_n\mid n\geqslant0\}$ where $\gamma_n=(b^2/2\sigma^2)+n^2\pi^2(\sigma^2/2)$.
When $c=\gamma_0$, $\lambda_1=\lambda_2$ hence $u_c(x)=(ax+1)\mathrm e^{\lambda x}$ for some constant $a$, and the limit conditions yield $u_c(x)=x\mathrm e^{-\lambda(1-x)}+(1-x)\mathrm e^{\lambda x}$ for every $x$ in $[0,1]$.
For every $c$ between $\gamma_0$ and $\gamma_1$, the roots $\lambda_1$ and $\lambda_2$ have a purely imaginary part and $u_c$ is well defined.
When $c\to\gamma_1^-$, $u_c(x)\to+\infty$ for every $x$ in $(0,1)$ and actually, $E_x(e^{c\tau})=+\infty$ for every $x$ in $(0,1)$ and every $c\geqslant\gamma_1$.
The solutions $u_c$ when $c\geqslant\gamma_1$ are artefacts with no "physical" reality, actually (if I remember correctly) none of them takes positive values on the whole interval.