The class of PPCSII is rather rich covering point processes with independent increments (among which non homogeneous Poisson processes and negative binomial processes), doubly stochastic Poisson (i.e., Cox processes) including mixed Poisson processes (among which processes with the order statistics property) and Markov modulated point processes. The stochastic process is a compound process with events of interest arriving according to an arbitrary point process with conditional stationary independent increments (PPCSII), and event severities with any possibly dependent joint distribution. In Chapter 4, we give explicit formulas and a numerically efficient FFT-based method for computing the probability that a non-decreasing, pure jump stochastic process will first exit from above the strip between two deterministic, possibly discontinuous, time-dependent boundaries, within a finite-time interval with an overshoot (not) exceeding a positive value. We further demonstrate that our FFT-based method is computationally efficient and can be successfully applied in the context of inventory management (to determine an optimal replenishment policy), ruin theory (to evaluate ruin probabilities and related quantities) and double-barrier option pricing or simply computing non-exit probabilities for Brownian motion with general boundaries. In Chapter 3, we develop an efficient method based on FFT, for computing the probability that a non-decreasing, pure jump (compound) stochastic process stays between arbitrary upper and lower boundaries (i.e., deterministic functions, possibly discontinuous) within a finite time period. The numerical performance of the proposed FFT-based method, implemented both in C++ and in the R package KSgeneral, available from /package=KSgeneral, is illustrated when F(x) is mixed, purely discrete, and continuous. Secondly, we developed a C++ and an R implementation of the proposed method, which fills in the existing gap in statistical software. In Chapter 2, we provide a fast and accurate method based on fast Fourier transform (FFT), to compute the (complementary) cumulative distribution function (CDF) of the Kolmogorov-Smirnov (KS) statistic when the CDF under the null hypothesis, F(x), is purely discrete, mixed or continuous, and thus obtain exact p values of the KS test. The expected improvements in IPS reach probability estimation are also illustrated through simulation results for a simple GSHS example.In this thesis, we focus on the problem that a stochastic process crossing (or not crossing) upper and/or lower deterministic boundaries and its application in statistics, inventory management, finance, risk and ruin theory and queueing. To formally make this state component unobservable for IPS, this paper also develops an enriched GSHS transformation prior to transforming spontaneous jumps to forced jumps. This paper will show that the latter transformation yields an extra Markov state component that should be treated as being unobservable for the IPS process. An alternative is to first transform the spontaneous jumps of a GSHS to forced transitions, and then to simulate executions of this transformed version. One approach is direct simulation of a GSHS execution. From literature, two basic approaches in simulating GSHS execution are known. In applying IPS to a GSHS, simulation of the GSHS execution plays a central role. The continuous-time executions of a GSHS evolve in a hybrid state space under influence of combinations of diffusions, spontaneous jumps and forced jumps. This paper studies IPS based reach probability estimation for General Stochastic Hybrid Systems (GSHS). (TU Delft Air Transport & Operations)įor diffusions, a well-developed approach in rare event estimation is to introduce a suitable factorization of the reach probability and then to estimate these factors through simulation of an Interacting Particle System (IPS). (TU Delft Air Transport & Operations Northwestern Polytechnical University)īlom, H.A.P. Interacting Particle System based estimation of reach probability of General Stochastic Hybrid Systems
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