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The classical and the soft-killing Inverse First-Passage Time Problem: a stochastic order approach / von Alexander Klump ; angefertigt unter der Betreuung von Prof. Dr. Martin Kolb. Paderborn, 2022
Content
Abstract
Contents
Introduction
Problems and motivation
Main results and structure of the thesis
Related results for the inverse first-passage time problem
The inverse first-passage time problem
Semicontinuous functions and existence of solutions
Barriers and boundary functions
Existence of solutions by discrete approximation
Properties of Gaussian convolution and truncation
Usual stochastic order: Gaussian convolution, truncation
Likelihood ratio order: Gaussian convolution, truncation
Wasserstein distance: Gaussian convolution, truncation
Uniqueness, properties and examples
Auxiliary results: boundary functions, survival distribution and marginal distributions
The lower approximation and uniqueness of continuous solutions
The upper approximation and uniqueness
Comparison principle and properties of solutions
The shape of the exponential boundary and further examples
Simulation and interacting particle representation
Generalization of the N-Branching Brownian motion
A particle system without branching
Simulation of solutions
The inverse first-passage time problem with soft-killing
Properties of Markovian evolution and reweighting
Usual stochastic order: Markovian evolution, reweighting
Total variation distance: Markovian evolution, reweighting
Existence and uniqueness of continuous solutions
Auxiliary results: boundary functions, survival distribution and marginal distributions
The upper approximation: existence, uniqueness and comparison principle
Simulation of solutions for the soft-killing problem
Outlook
Appended proofs
Alternative proof of Lemma 2.3.25
The Fredholm integral equation connecting g, b and
Total positivity of the quasi-stationary distribution
Truncation and reweighting with respect to other probability distances
Total variation version of Lemma 2.3.30 for continuous survival distributions
Sufficient criteria on Markov processes for the conditions of Theorem 3.0.1
Background tools
Probability
Analysis
List of Symbols
Bibliography
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