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 getdeterministic 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

Dr. Anatoliy Swishchuk is a Professor in financial mathematics at the Department of Mathematics and Statistics, University of Calgary in Canada. Hereceived 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 receiveda gold medal for a series of research publications in random evolutions and their applications. Dr. Swishchuk isthe chair of finance at the Department of Mathematics and Statistics (15 years) where heleads the energy finance seminar Lunch at the Lab. Heworks, also, with theCalgary Site Director of Postdoctoral Training Center in Stochastics. He was a steering committee member of the Professional Risk Managers International Association, Canada (2006-2015), andsince 2015, has beena 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 boardsof four research journals andis the author of 13 books and more than 100 articles in peer-reviewed journals. Recently, he received a Peak Scholar award.