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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 counts 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, that 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 some 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.

**Preface****I 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