1st Edition

Inhomogeneous Random Evolutions and Their Applications

By Anatoliy Swishchuk Copyright 2020
    252 Pages 10 B/W Illustrations
    by Chapman & Hall

    252 Pages 10 B/W Illustrations
    by Chapman & Hall

    252 Pages 10 B/W Illustrations
    by Chapman & Hall

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    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
        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
       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



    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.