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
- 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
- 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
- List of Symbols
- Bibliography