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Title: Estimators of the boundary in inverse first exit problems Authors:  Klaus Poetzelberger - WU Vienna (Austria) [presenting]
Abstract: First-passage problems for the Brownian motion $(W_t)$ or general diffusion processes, have important applications. Given a boundary $b(t)$, the distribution of the first-exit time $\tau^b$ has to be computed, in most cases numerically. Inverse boundary crossing probabilities assume that the distribution of $\tau^b$ is given and the boundary $b$ has to be found. The boundary and the density of $\tau^b$ satisfy a Volterra integral equation. We propose and analyze estimators of $b$. The first class of estimators are solutions of stochastic versions of the Volterra equation. The second class of estimators uses the idea of approximating the boundary $b(t)$ by a piecewise boundary $\tilde{b}_m(t)$. Define $W^m=(W_{t_1},\ldots,W_{t_m})$. The density of $\tau^{\tilde{b}_m}$ given $W^m=w^m$ is available in closed form. The EM estimator iterates the estimation and maximization steps. The Bayesian estimator additionally chooses a prior on $b$ and then uses Gibbs sampling to iterate the generation of $b |(W^m, \tau_1,\ldots,\tau_n)$ and $W^m |(b, \tau_1,\ldots,\tau_n)$. Typical inverse problems are sequential testing in statistics or the estimation of a ruin boundary, for instance in credit risk modelling. A company defaults, if a process $(V_t)$, called the value of the firm, crosses a boundary $b(t)$. $(V_t)$ cannot be observed. It is correlated with $(S_t)$, which includes published relevant information on $(V_t)$.