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Let $\sigma \in C(\mathbb R)$, and $X$ be a solution of \begin{equation}\label{eq:1} X_{t} = x + t + \int_{0}^{t} \sigma(X_{s}) dB_{s} \end{equation} where $B$ is 1-d Brownian motion under filtered probability space $(\Omega, \mathcal F, \mathbb P, \{\mathcal F_{t}\}_{t\ge 0})$. We denote by $\mathbb P^{x}$ the probability on $C([0, \infty), \mathbb R)$ induced by the process $X$ starting from $x$. Consider a stopping time $\tau = \inf\{t>0: X(t) <0\}$.

[Q.] What is the sufficient condition on $\sigma$ to have the continuity of $u(x) = \mathbb E^{x}[e^{-\tau}]$?

[A.] $\sigma \in C(0,1)$ and $\sigma(0) \neq 0$.

[Sketch of Proof.] If $\sigma \in C^{0,1}$, then the following two conditions are standard:

(C1) The above SDE has unique strong solution;

(C2) $\mathbb P^{x_{n}} \Rightarrow \mathbb P^{x}$ whenever $x_{n} \to x$;

Moreover, according to the [Discussion 2] of the post (regularity of zero point), we also have

(C3) $\mathbb P^{0} (\tau = 0) = 1$.

In fact, (C3) implies that the mapping $\omega \mapsto \tau(\omega)$ is $\mathbb P^{0}$ almost surely continuous function on $C([0, \infty), \mathbb R)$ w.r.t. Skorohod topology. (In fact, it's equivalent to the a.s. continuity w.r.t. max norm on $C([0, T], \mathbb R)$ for each $T<\infty$ in this case.) Together with (C2), we have $\tau^{x_{n}}$ converge to $\tau^{x}$ in distribution by mapping theorem. This shows the continuity of $u$. END.

My question is then, can we have weaker sufficient condition on $\sigma$ to have continuity of $u$?

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Referring to $X(t)$ and $\tau(x) = \inf \{ t>0 : X(t)<0 \mid X(0) = x \}$ stated above, a Feynman-Kac formula implies that the function $u(x) = \mathbb{E}^x \exp(-\tau)$ satisfies a second-order, linear differential equation: $$ \begin{cases} \frac{1}{2} \sigma(x)^2 u''(x) + u'(x) - u(x) = 0 \\ u(0)=1\;, \quad u(\infty) = 0 \end{cases} $$ If $\sigma(x)^{-2}$ is integrable, then these equations can be put in the form of a classical Sturm-Liouville problem on a semi-infinite interval. For properties of their solutions see, e.g., Part 4 of Zettl, Anton (2005). Sturm–Liouville Theory. Providence: American Mathematical Society.

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  • $\begingroup$ Thanks for your reply. Since I am not familiar with this book, would you please indicate which theorem indicates solvability in C^2 with the above integrability condition? $\endgroup$ – kenneth Aug 21 '16 at 9:57
  • $\begingroup$ Part 4 of Zettl's book treats the semi-infinite interval SL problem (dubbed the singular SL problem). In particular, Zettl is able to relax the usual smoothness conditions imposed on the coefficients in the SL equation by much weaker integrability conditions, like the one given in my answer. $\endgroup$ – Nawaf Bou-Rabee Aug 21 '16 at 11:56
  • $\begingroup$ PS: your case is particularly simple since the boundary conditions are Dirichlet. $\endgroup$ – Nawaf Bou-Rabee Aug 21 '16 at 13:57

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