| Autore |
Limnios Nikolaos
|
| Edizione | [1st ed. 2023.] |
| Pubbl/distr/stampa |
Cham : , : Springer Nature Switzerland : , : Imprint : Birkhäuser, , 2023
|
| Descrizione fisica |
1 online resource (206 pages)
|
| Disciplina |
519.233
|
| Altri autori (Persone) |
SwishchukAnatoliy
|
| Collana |
Probability and Its Applications
|
| Soggetto topico |
Stochastic processes
Probabilities
Mathematical statistics
Dynamical systems
Stochastic Processes
Probability Theory
Mathematical Statistics
Applied Probability
Dynamical Systems
Stochastic Systems and Control
|
| ISBN |
3-031-33429-9
|
| Formato |
Materiale a stampa  |
| Livello bibliografico |
Monografia |
| Lingua di pubblicazione |
eng
|
| Nota di contenuto |
Intro -- Preface -- Contents -- Acronyms -- Notation -- 1 Discrete-Time Stochastic Calculus in Banach Space -- 1.1 Introduction -- 1.2 Random Elements and Discrete-Time Martingales in a Banach Space -- 1.3 Martingale Characterization of Markov and Semi-Markov Chains -- 1.3.1 Martingale Characterization of Markov Chains -- 1.3.2 Martingale Characterization of Markov Processes -- 1.3.3 Martingale Characterization of Semi-Markov Processes -- 1.4 Operator Semigroups and Their Generators -- 1.5 Martingale Problem in a Banach Space -- 1.6 Weak Convergence in a Banach Space -- 1.7 Reducible-Invertible Operators and Their Perturbations -- 1.7.1 Reducible-Invertible Operators -- 1.7.2 Perturbation of Reducible-Invertible Operators -- 2 Discrete-Time Semi-Markov Chains -- 2.1 Introduction -- 2.2 Semi-Markov Chains -- 2.2.1 Definitions -- 2.2.2 Classification of States -- 2.2.3 Markov Renewal Equation and Theorem -- 2.3 Discrete- and Continuous-Time Connection -- 2.4 Compensating Operator and Martingales -- 2.5 Stationary Phase Merging -- 2.6 Semi-Markov Chains in Merging State Space -- 2.6.1 The Ergodic Case -- 2.6.2 The Non-ergodic Case -- 2.7 Concluding Remarks -- 3 Discrete-Time Semi-Markov Random Evolutions -- 3.1 Introduction -- 3.2 Discrete-time Random Evolution with Underlying Markov Chain -- 3.3 Definition and Properties of DTSMRE -- 3.4 Discrete-Time Stochastic Systems -- 3.4.1 Additive Functionals -- 3.4.2 Geometric Markov Renewal Chains -- 3.4.3 Dynamical Systems -- 3.5 Discrete-Time Stochastic Systems in Series Scheme -- 3.6 Concluding Remarks -- 4 Weak Convergence of DTSMRE in Series Scheme -- 4.1 Introduction -- 4.2 Weak Convergence Results -- 4.2.1 Averaging -- 4.2.2 Diffusion Approximation -- 4.2.3 Normal Deviations -- 4.2.4 Rates of Convergence in the Limit Theorems -- 4.3 Proof of Theorems -- 4.3.1 Proof of Theorem 4.1.
4.3.2 Proof of Theorem 4.2 -- 4.3.3 Proof of Theorem 4.3 -- 4.3.4 Proof of Proposition 4.1 -- 4.4 Applications of the Limit Theorems to Stochastic Systems -- 4.4.1 Additive Functionals -- 4.4.2 Geometric Markov Renewal Processes -- 4.4.3 Dynamical Systems -- 4.4.4 Estimation of the Stationary Distribution -- 4.4.5 U-Statistics -- 4.4.6 Rates of Convergence for Stochastic Systems -- 4.5 Concluding Remarks -- 5 DTSMRE in Reduced Random Media -- 5.1 Introduction -- 5.2 Definition and Properties -- 5.3 Average and Diffusion Approximation -- 5.3.1 Averaging -- 5.3.2 Diffusion Approximation -- 5.3.3 Normal Deviations -- 5.4 Proof of Theorems -- 5.4.1 Proof of Theorem 5.1 -- 5.4.2 Proof of Theorem 5.2 -- 5.5 Application to Stochastic Systems -- 5.5.1 Additive Functionals -- 5.5.2 Dynamical Systems -- 5.5.3 Geometric Markov Renewal Chains -- 5.5.4 U-Statistics -- 5.6 Concluding Remarks -- 6 Controlled Discrete-Time Semi-Markov Random Evolutions -- 6.1 Introduction -- 6.2 Controlled Discrete-Time Semi-Markov Random Evolutions -- 6.2.1 Definition of CDTSMREs -- 6.2.2 Examples -- 6.2.3 Dynamic Programming for Controlled Models -- 6.3 Limit Theorems for Controlled Semi-Markov Random Evolutions -- 6.3.1 Averaging of CDTSMREs -- 6.3.2 Diffusion Approximation of DTSMREs -- 6.3.3 Normal Approximation -- 6.4 Applications to Stochastic Systems -- 6.4.1 Controlled Additive Functionals -- 6.4.2 Controlled Geometric Markov Renewal Processes -- 6.4.3 Controlled Dynamical Systems -- 6.4.4 The Dynamic Programming Equations for Limiting Models in Diffusion Approximation -- 6.4.4.1 DPE/HJB Equation for the Limiting CAF in DA (see Sect.6.4.1) -- 6.4.4.2 DPE/HJB Equation for the Limiting CGMRP in DA (see Sect.6.4.2) -- 6.4.4.3 DPE/HJB Equation for the Limiting CDS in DA (see Sect.6.4.3) -- 6.5 Solution of Merton Problem for the Limiting CGMRP in DA -- 6.5.1 Introduction.
6.5.2 Utility Function -- 6.5.3 Value Function or Performance Criterion -- 6.5.4 Solution of Merton Problem: Examples -- 6.5.5 Solution of Merton Problem -- 6.6 Rates of Convergence in Averaging and Diffusion Approximations -- 6.7 Proofs -- 6.7.1 Proof of Theorem 6.1 -- 6.7.2 Proof of Theorem 6.2 -- 6.7.3 Proof of Theorem 6.3 -- 6.7.4 Proof of Proposition 6.1 -- 6.8 Concluding Remarks -- 7 Epidemic Models in Random Media -- 7.1 Introduction -- 7.2 From the Deterministic to Stochastic SARS Model -- 7.3 Averaging of Stochastic SARS Models -- 7.4 SARS Model in Merging Semi-Markov Random Media -- 7.5 Diffusion Approximation of Stochastic SARS Models in Semi-Markov Random Media -- 7.6 Concluding remarks -- 8 Optimal Stopping of Geometric Markov Renewal Chains and Pricing -- 8.1 Introduction -- 8.2 GMRC and Embedded Markov-Modulated (B,S)-Security Markets -- 8.2.1 Definition of the GMRC -- 8.2.2 Statement of the Problem: Optimal Stopping Rule -- 8.3 GMRP as Jump Discrete-Time Semi-Markov Random Evolution -- 8.4 Martingale Properties of GMRC -- 8.5 Optimal Stopping Rules for GMRC -- 8.6 Martingale Properties of Discount Price and Discount Capital -- 8.7 American Option Pricing Formulae for embedded Markov-modulated (B,S)-Security markets -- 8.8 European Option Pricing Formula for Embedded Markov-Modulated (B,S)-Security Markets -- 8.9 Proof of Theorems -- 8.10 Concluding Remarks -- A Markov Chains -- A.1 Transition Function -- A.2 Irreducible Markov Chains -- A.3 Recurrent Markov Chains -- A.4 Invariant Measures -- A.5 Uniformly Ergodic Markov Chains -- Bibliography -- Index.
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| Record Nr. | UNINA-9910735778203321 |
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Discrete-Time Semi-Markov Random Evolutions and Their Applications [[electronic resource] /] / by Nikolaos Limnios, Anatoliy Swishchuk
