Inhomogeneous Random Evolutions and Their Applications

PrefaceI Stochastic Calculus in Banach Spaces
1. Basics in Banach Spaces
    Random Elements, Processes and Integrals in Banach Spaces
    Weak Convergence in Banach Spaces
    Semigroups of Operators and Their Generators
    Bibliography
    Stochastic Calculus in Separable Banach Spaces
    Stochastic Calculus for Integrals over Martingale measures
    The Existence of Wiener Measure and Related Stochastic Equations
    Stochastic Integrals over Martingale Measures
    Orthogonal martingale measures
    Ito`s Integrals over Martingale Measure
    Symmetric (Stratonovich) Integral over Martingale Measure
    Anticipating (Skorokhod) Integral over Martingale Measure
    Multiple Ito`s Integral over Martingale Measure
    Stochastic Integral Equations over Martingale Measures
    Martingale Problems Associated with Stochastic Equations over Martingale Measures
    Evolutionary Operator Equations Driven by Wiener Martingale Measure
    Stochastic Calculus for Multiplicative Operator Functionals (MOF)
    Definition of MOF
    Properties of the characteristic operator of MOF 
    Resolvent and Potential for MOF
    Equations for Resolvent and Potential for MOF
    Analogue of Dynkin`s Formulas (ADF) for MOF
    ADF for traffic processes in random media
    ADF for storage processes in random media
    Bibliography
2. Convergence of Random Bounded Linear Operators in the Skorokhod Space
    Introduction
    D-valued random variables & various properties on elements of D
    Almost sure convergence of D-valued random variables
    Weak convergence of D-valued random variables
    Bibliography
II Homogeneous and Inhomogeneous Random Evolutions
3. Homogeneous Random Evolutions (HREs) and their Applications
    Random Evolutions
    Definition and Classification of Random Evolutions
    Some Examples of RE
    Martingale Characterization of Random Evolutions
    Analogue of Dynkin`s formula for RE (see Chapter 2)
    Boundary value problems for RE (see Chapter 2)
    Limit Theorems for Random Evolutions
    Weak Convergence of Random Evolutions (see Chapter 2 and 3)
    Averaging of Random Evolutions
    Diffusion Approximation of Random Evolutions
    Averaging of Random Evolutions in Reducible Phase Space. Merged Random Evolutions
    Diffusion Approximation of Random evolutions in Reducible Phase Space
    Normal Deviations of Random Evolutions
    Rates of Convergence in the Limit Theorems for RE
    Bibliography
    Index
4. Inhomogeneous Random Evolutions (IHREs)
   Propagators (Inhomogeneous Semi-group of Operators)
   Inhomogeneous Random Evolutions (IHREs): Definitions and Properties
   Weak Law of Large Numbers (WLLN)
   Preliminary Definitions and Assumptions
   The Compact Containment Criterion (CCC)
   Relative Compactness of {Ve}
   Martingale Characterization of the Inhomogeneous Random Evolution
   Weak Law of Large Numbers (WLLN)
   Central Limit Theorem (CLT)
   Bibliography
III Applications of Inhomogeneous Random Evolutions
5. Applications of IHREs: Inhomogeneous Levy-based Models
    Regime-switching Inhomogeneous Levy-based Stock Price Dynamics and Application to Illiquidity Modelling
    Proofs for Section 6.1:
    Regime-switching Levy Driven Diffusion-based Price Dynamics
    Multi-asset Model of Price Impact from Distressed Selling: Diffusion Limit
    Bibliography
6. Applications of IHRE in High-frequency Trading: Limit Order
    Books and their Semi-Markovian Modeling and Implementations
    Introduction
    A Semi-Markovian modeling of limit order markets
    Main Probabilistic Results
    Duration until the next price change
    Probability of Price Increase
    The stock price seen as a functional of a Markov renewal process
    The Mid-Price Process as IHRE
    Diffusion Limit of the Price Process
    Balanced Order Flow case: Pa (1; 1) = Pa (-1;-1) and Pb (1; 1) = Pb (-1;-1)
    Other cases: either Pa (1; 1) < Pa (-1;-1) or Pb (1; 1) < Pb (-1;-1)
    Numerical Results
    Bibliography
7. Applications of IHREs in Insurance: Risk Model Based on General Compound Hawkes Process
   Introduction
   Hawkes, General Compound Hawkes Process
   Hawkes Process
   General Compound Hawkes Process (GCHP)
   Risk Model based on General Compound Hawkes Process
   RMGCHP as IHRE
   LLN and FCLT for RMGCHP
   LLN for RMGCHP
   FCLT for RMGCHP
   Applications of LLN and FCLT for RMGCHP
   Application of LLN: Net Profit Condition
   Application of LLN: Premium Principle
   Application of FCLT for RMGCHP: Ruin and Ultimate Ruin Probabilities
   Application of FCLT for RMGCHP: Approximation of RMGCHP by a Diffusion Process
   Application of FCLT for RMGCHP: Ruin Probabilities
   Application of FCLT for RMGCHP: Ultimate Ruin Probabilities
   Application of FCLT for RMGCHP: The Distribution of the Time to Ruin
   Applications of LLN and FCLT for RMCHP
   Net Profit Condition for RMCHP
   Premium Principle for RMCHP
   Ruin Probability for RMCHP
   Ultimate Ruin Probability for RMCHP
   The Probability Density Function of the Time to Ruin
   Applications of LLN and FCLT for RMCPP
   Net Profit Condition for RMCPP
   Premium Principle for RMCPP
   Ruin Probability for RMCPP
   Ultimate Ruin Probability for RMCPP
   The Probability Density Function of the Time to Ruin for RMCPP
   Bibliography