  We consider the problem of rare event importance sampling, where the random
variable of interest is a path functional of an It\^o diffusion computed up to
the first exit from a $d$-dimensional bounded domain. Dupuis and Wang
(\textit{Ann. Appl. Probab.}, 15 (2005), pp. 1-38) studied the importance
sampling problem by formulating it as a stochastic optimal control problem,
where the value function is related to the conditional cumulant generating
function of the random variable. In this paper, we show that the sufficient
conditions for the value function to be twice-differentiable and
$\alpha$-uniformly H\"older continuous on the closure of the domain are also
sufficient conditions for positive definiteness of the second variation of the
control functional on the space of differentiable, $\alpha$-uniformly H\"older
continuous $\mathbb{R}^d$-valued feedback controls. We derive an expression for
the second variation using Kazamaki's sufficient condition for
$L^q$-boundedness of exponential martingales, and using Fredholm theory to
prove the finiteness of the moment generating function of the first exit time
over any bounded interval containing the origin. The strict convexity result
suggests that one may be able to solve the corresponding
Hamilton-Jacobi-Bellman boundary value problem in a dimension-robust way, by
combining convex optimisation and Monte Carlo methods. We apply the result to
analyse a gradient descent algorithm proposed by Hartmann and Sch\"utte
(\textit{J. Stat. Mech. Theor. Exp.} (2012), P11004) for efficient rare event
simulation.
