Inhomogeneous Random Evolutions and Their Applications  book cover
1st Edition

Inhomogeneous Random Evolutions and Their Applications

ISBN 9781032082295
Published August 2, 2021 by Chapman and Hall/CRC
252 Pages 10 B/W Illustrations

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Book Description

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.

Table of Contents

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
    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

2. Convergence of Random Bounded Linear Operators in the Skorokhod Space
    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

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

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)

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
6. Applications of IHRE in High-frequency Trading: Limit Order
    Books and their Semi-Markovian Modeling and Implementations
    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

7. Applications of IHREs in Insurance: Risk Model Based on General Compound Hawkes Process
   Hawkes, General Compound Hawkes Process
   Hawkes Process
   General Compound Hawkes Process (GCHP)
   Risk Model based on General Compound Hawkes Process
   LLN and 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


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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.