1st Edition
Inhomogeneous Random Evolutions and Their Applications
Inhomogeneous Random Evolutions and Their Applications explains how to model various dynamical systems in finance and insurance with non-homogeneous in time characteristics. It includes modeling for:
- financial underlying and derivatives via Levy processes with time-dependent characteristics;
- limit order books in the algorithmic and HFT with counting price changes processes having time-dependent intensities;
- risk processes which count number of claims with time-dependent conditional intensities;
- multi-asset price impact from distressed selling;
- regime-switching Levy-driven diffusion-based price dynamics.
Initial models for those systems are very complicated, which is why the author’s approach helps to simplified their study. The book uses a very general approach for modeling of those systems via abstract inhomogeneous random evolutions in Banach spaces. To simplify their investigation, it applies the first averaging principle (long-run stability property or law of large numbers [LLN]) to get deterministic function on the long run. To eliminate the rate of convergence in the LLN, it uses secondly the functional central limit theorem (FCLT) such that the associated cumulative process, centered around that deterministic function and suitably scaled in time, may be approximated by an orthogonal martingale measure, in general; and by standard Brownian motion, in particular, if the scale parameter increases. Thus, this approach allows the author to easily link, for example, microscopic activities with macroscopic ones in HFT, connecting the parameters driving the HFT with the daily volatilities. This method also helps to easily calculate ruin and ultimate ruin probabilities for the risk process. All results in the book are new and original, and can be easily implemented in practice.
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
Biography
Dr. Anatoliy Swishchuk is a Professor in financial mathematics at the Department of Mathematics and Statistics, University of Calgary in Canada. He received his B.Sc. and M.Sc. degrees from Kyiv State University, Kyiv, Ukraine. He is a holder of two doctorate degrees - Mathematics and Physics (Ph. D. and D. Sc.) - from the prestigious National Academy of Sciences of Ukraine, Kiev, Ukraine, and is a recipient of the NASU award for young scientists. He received a gold medal for a series of research publications in random evolutions and their applications.
Dr. Swishchuk is the chair of finance at the Department of Mathematics and Statistics (15 years) where he leads the energy finance seminar Lunch at the Lab. He works, also, with the Calgary Site Director of Postdoctoral Training Center in Stochastics. He was a steering committee member of the Professional Risk Managers International Association, Canada (2006-2015), and since 2015, has been a steering committee member of Global Association of Risk Professionals, Canada. His research includes financial mathematics, random evolutions and applications, biomathematics, stochastic calculus. He serves on the editorial boards of four research journals and is the author of 13 books and more than 100 articles in peer-reviewed journals. Recently, he received a Peak Scholar award.
