We extend the Longstaff-Schwartz algorithm for approximately solving optimal stopping problems on... more We extend the Longstaff-Schwartz algorithm for approximately solving optimal stopping problems on high-dimensional state spaces. We reformulate the optimal stopping problem for Markov processes in discrete time as a generalized statistical learning problem. Within this setup we apply deviation inequalities for suprema of empirical processes to derive consistency criteria, and to estimate the convergence rate and sample complexity. Our results strengthen and extend earlier results.
The geodesic flow of a compact Finsler manifold with negative flag curvature is an Anosov flow [2... more The geodesic flow of a compact Finsler manifold with negative flag curvature is an Anosov flow [23]. We use the structure of the stable and unstable foliation to equip the geodesic ray boundary of the universal covering with a Hölder structure. Gromov's geodesic rigidity and the Theorem of Dinaburg--Manning on the relation between the topological entropy and the volume entropy are generalized to the case of Finsler manifolds.
This paper concerns the pricing of American options with stochastic stopping time constraints exp... more This paper concerns the pricing of American options with stochastic stopping time constraints expressed in terms of the states of a Markov process. Following the ideas of Menaldi et al., we transform the constrained into an unconstrained optimal stopping problem. The transformation replaces the original payoff by the value of a generalized barrier option. We also provide a Monte Carlo method to numerically calculate the option value for multidimensional Markov processes. We adapt the Longstaff–Schwartz algorithm to solve the stochastic Cauchy–Dirichlet problem related to the valuation problem of the barrier option along a set of simulated trajectories of the underlying Markov process.
We introduce new quantile estimators with adaptive importance sampling. The adaptive estimators a... more We introduce new quantile estimators with adaptive importance sampling. The adaptive estimators are based on weighted samples that are neither independent nor identically distributed. Using a new law of iterated logarithm for martingales, we prove the convergence of the adaptive quantile estimators for general distributions with nonunique quantiles thereby extending the work of Feldman and Tucker [Ann. Math. Statist. 37 (1996) 451--457]. We illustrate the algorithm with an example from credit portfolio risk analysis.
We extend the Longstaff-Schwartz algorithm for approximately solving optimal stopping problems on... more We extend the Longstaff-Schwartz algorithm for approximately solving optimal stopping problems on high-dimensional state spaces. We reformulate the optimal stopping problem for Markov processes in discrete time as a generalized statistical learning problem. Within this setup we apply deviation inequalities for suprema of empirical processes to derive consistency criteria, and to estimate the convergence rate and sample complexity. Our results strengthen and extend earlier results.
The geodesic flow of a compact Finsler manifold with negative flag curvature is an Anosov flow [2... more The geodesic flow of a compact Finsler manifold with negative flag curvature is an Anosov flow [23]. We use the structure of the stable and unstable foliation to equip the geodesic ray boundary of the universal covering with a Hölder structure. Gromov's geodesic rigidity and the Theorem of Dinaburg--Manning on the relation between the topological entropy and the volume entropy are generalized to the case of Finsler manifolds.
This paper concerns the pricing of American options with stochastic stopping time constraints exp... more This paper concerns the pricing of American options with stochastic stopping time constraints expressed in terms of the states of a Markov process. Following the ideas of Menaldi et al., we transform the constrained into an unconstrained optimal stopping problem. The transformation replaces the original payoff by the value of a generalized barrier option. We also provide a Monte Carlo method to numerically calculate the option value for multidimensional Markov processes. We adapt the Longstaff–Schwartz algorithm to solve the stochastic Cauchy–Dirichlet problem related to the valuation problem of the barrier option along a set of simulated trajectories of the underlying Markov process.
We introduce new quantile estimators with adaptive importance sampling. The adaptive estimators a... more We introduce new quantile estimators with adaptive importance sampling. The adaptive estimators are based on weighted samples that are neither independent nor identically distributed. Using a new law of iterated logarithm for martingales, we prove the convergence of the adaptive quantile estimators for general distributions with nonunique quantiles thereby extending the work of Feldman and Tucker [Ann. Math. Statist. 37 (1996) 451--457]. We illustrate the algorithm with an example from credit portfolio risk analysis.
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Papers by Daniel Egloff