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Title: The empirical estimator of the boundary in an inverse firstexit problem 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 $\tau$ has to be computed, in most cases numerically. Inverse boundary crossing probabilities assume that the distribution of $\tau$ is given and the boundary $b$ has to be found. The analysis is based on the fact that the boundary and the density of $\tau$ satisfy a Volterra integral equation. We propose and analyze estimators of $b$, when a sample $\tau_1,\ldots,\tau_n$ of first exit times is given. The first class of estimators are solutions of stochastic versions of the Volterra equation. The second class of estimators are approximate likelihood methods, using the idea of approximating the boundary $b(t)$ by a piecewise boundary $b_m(t)$. Define $W^m=(W_{t_1},\ldots,W_m)$. The density of $\tau$ for $b_m$ conditional on $W^m=w^m$ is available in closed form. The Bayesian estimator chooses a prior on $b$ and then uses Gibbs sampling to iterate the generation of $b\mid (W^m, \tau_1,\ldots,\tau_n)$ and $W^m\mid (b, \tau_1,\ldots,\tau_n)$. 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 $(V_t)$.