Inhomogeneous random evolutions and their applications. (2019)

• Record Type: Book
• Further Information: Note: By Anatoliy Swishchuk.
• Authors: Svishchuk, A. V (Anatoliĭ Vitalʹevich)
• Contents: 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
• Edition: 1st
• Publisher Details: Boca Raton : Chapman & Hall/CRC
• Publication Date: 2019
• Extent: 1 online resource, illustrations (black and white)
• Subjects: 515.732; Finance -- Mathematical models; Insurance -- Mathematical models; Stochastic processes; Banach spaces
• Languages: English
• ISBNs: 9780429855047; 9780429855054; 9780429855030; 9780429457548
• Related ISBNs: 9781138313477
• Notes: Note: Includes bibliographical references and index.; Note: Description based on CIP data; resource not viewed.
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.