Random Evolutions and Their Applications

by Svishchuk, A. V.

• ISBN13: 9780792362647
• ISBN10: 0792362640
• Format: Hardcover
• Copyright: 2000-07-01
• Publisher: Kluwer Academic Pub

Summary

This book is devoted to new trends in random evolution and their applications to the stochastic evolutionary system. It contains new developments such as an analogue of Dynkin's formula, boundary value problems, stability and control of random evolutions, stochastic evolutionary equations, and driven martingale measures. In addition, it treats statistics of random evolutions processes, statistics of financial stochastic models, and stochastic stability and control of financial markets. Audience: This volume will be of interest to research and applied mathematicians working in the fields of applied probability, stochastic processes, and random evolutions, as well as experts in statistics, finance and insurance.

Table of Contents

Preface	xiii
List of Notations	xv
Introduction	1	(1)
Markov and Semi-Markov Processes	2	(2)
Markov and Semi-Markov Random Evolutions	4	(2)
Random Evolution Processes (REP) or Stochastic Evolutionary Systems (SES)	6	(2)
Random Evolutions in Financial and Insurance Mathematics in a Incomplete Market	8	(3)
Stochastic Stability of Markov and semi-Markov Processes	11	(1)
Stochastic Stability of SES	11	(1)
Stochastic Stability of Random Evolutions	12	(1)
Stochastic Stability of Random Evolutions in Averaging and Diffusion Approximation Schemes	13	(1)
Stochastic Stability of SES in Averaging and Diffusion Approximation Schemes	14	(1)
Stochastic Optimal Control of Markov and Semi-Markov Processes	14	(3)
Stochastic Optimal Control of Markov and Semi-Markov Stochastic Evolutionary Systems (SES)	17	(1)
Stochastic optimal control of Markov SES	17	(1)
Stochastic optimal control of semi-Markov SES	18	(1)
Stochastic Optimal Control of Random Evolutions	18	(3)
Stochastic optimal control of Markov RE	19	(1)
Stochastic optimal control of semi-Markov RE	19	(1)
Organization of the Book	19	(1)
Structure of the Book	20	(1)
Random Evolutions (RE)	21	(42)
Definitions and classification of random evolutions	21	(3)
Definitions of random evolutions	21	(1)
Classification of Random Evolutions. Examples	22	(2)
Martingale methods in random evolutions	24	(14)
Martingale characterization of random evolutions	24	(5)
Martingale approach to random evolutions	29	(1)
Orthogonal local martingale measure	30	(1)
Stochastic integrals over martingale measure	30	(1)
The stochastic integral equation in Banach space	31	(1)
Martingale problem in Banach space	32	(1)
Connection between martingale problem and stochastic integral equation in Banach space	33	(4)
Martingale characterization of Markov processes and chains	37	(1)
Limit theorems for Random Evolution	38	(15)
Weak convergence of random evolutions	39	(2)
Averaging of random evolutions	41	(2)
Diffusion approximation of random evolutions	43	(2)
Averaging of random evolutions in reducible phase space. Merged random evolutions	45	(4)
Diffusion approximation of random evolutions in reducible phase space	49	(2)
Normal deviations of random evolutions	51	(2)
Rates of convergence in the limit theorems for SMRE	53	(3)
Evolutionary Equations	56	(7)
Stochastic Evolutionary Systems	63	(22)
Definition and examples of SES	63	(3)
Traffic processes in semi-Markov random media	63	(1)
Storage processes in semi-Markov random media	64	(1)
Diffusion processes in semi-Markov random media	65	(1)
Averaging and merging of SES	66	(3)
Traffic processes	66	(1)
Storage processes	67	(1)
Diffusion processes	68	(1)
Diffusion Approximation of Stochastic Evolutionary Systems	69	(8)
Traffic processes	69	(1)
Storage processes	70	(2)
Rates of convergence in the limit theorems for SES	72	(5)
Normal Deviations of SES	77	(2)
Martingale characterization of stochastic evolutionary systems	79	(6)
Martingale characterization of Markov processes and chains	80	(1)
Martingale characterization of semi-Markov process	81	(1)
Martingale characterization of SES	81	(4)
Random Evolution Equations Driven by Space-Time White Noise	85	(21)
The Existence of Wiener Measure and Related Stochastic Equations	85	(4)
Stochastic Integrals over Martingale Measures	89	(8)
Orthogonal martingale measures	89	(1)
Ito's integrals over martingale measure	90	(3)
Symmetric integral over martingale measure	93	(2)
Anticipating integral over martingale measure	95	(3)
Multiple Ito's integral over martingale measure	98
Stochastic Integral Equations over Martingale Measures	97	(2)
Martingale Problems Connected with Stochastic Equations over Martingale Measures	99	(2)
Stochastic Integral Equation for the Limiting Random Evolutions	101	(1)
Evolutionary Operator Equations Driven by Wiener Martingale Measure	102	(4)
Analogue of Dynkin's Formula (ADF) for Multiplicative Operator Functionals (MOF), RE and SES	106	(13)
Definitions and basic notations	106	(2)
Properties of the characteristic operator of MOF	108	(2)
Resolvent and potential for MOF	110	(1)
Equations for resolvent and potential for MOF	111	(1)
ADF for MOF	112	(1)
ADF for Markov RE	113	(1)
ADF for semi-Markov RE	114	(1)
Analogue of Dynkin's formulae for SES	115	(4)
ADF for traffic processes in random media	115	(1)
ADF for storage processes in random media	116	(1)
ADF for diffusion process in random media	117	(2)
Boundary Value Problems (BVP) for RE and SES	119	(8)
Boundary value problems for Markov RE	119	(3)
Boundary value problems for discontinuous Markov and semi-Markov RE	122	(2)
Boundary value problems for Stochastic Evolutionary Systems	124	(3)
Traffic, storage and diffusion processes in random media	124	(1)
BVP for traffic processes in random media	125	(1)
BVP for storage processes in random media	126	(1)
BVP for diffusion processes in random media	126	(1)
Stochastic Stability of RE and SES	127	(29)
Definitions of stochastic stability	127	(2)
Stochastic stability of Markov and semi-Markov processes	129	(4)
Stochastic stability of random evolutions	133	(7)
Stability of random evolutions w.p.1	134	(2)
Stability of random evolutions in averaging scheme	136	(2)
Stability of random evolutions in diffusion approximation scheme	138	(2)
Stability of stochastic evolutionary systems	140	(6)
Stability of traffic processes	140	(3)
Stability of storage processes	143	(2)
Stability of diffusion processes	145	(1)
Stability of SES in averaging and diffusion approximation schemes	146	(10)
Stability of impulse traffic process in averaging scheme	146	(3)
Stability of impulse traffic process in diffusion approximation scheme	149	(2)
Stability of diffusion processes in averaging scheme	151	(5)
Stochastic Optimal Control of Random Evolutions and SES	156	(24)
Definitions, Conditions and Preliminary Results	156	(2)
SOC of Markov Random Evolutions	158	(3)
SOC of semi-Markov Random Evolutions	161	(3)
SOC of Controlled Averaged Random Evolutions	164	(1)
SOC of Controlled Diffusion Random Evolutions	165	(1)
SOC of Controlled Merged Random Evolutions	166	(1)
Control of Stochastic Evolutionary Systems in Random Media	167	(13)
Functionals of Uncontrolled Processes	167	(1)
Cost functionals	168	(3)
Optimal stochastic control	171	(9)
Statistics of SES	180	(8)
Filtering problems for stochastic evolutionary systems	180	(3)
Definition of diffusion processes in random media and example	180	(1)
State of the problem and conditions	181	(1)
Formulation of the result	182	(1)
Proof of the result	183	(1)
Interpolation problems for stochastic evolutionary systems	183	(2)
State of the problem and conditions	183	(1)
Formulation of the result and proof	184	(1)
Extrapolation problems for stochastic evolutionary systems	185	(3)
State of the problem and conditions	186	(1)
Formulation of the result and proof	186	(2)
Random Evolutions in Financial Mathematics. Incomplete Market	188	(37)
Examples of Random Evolutions in financial mathematics	188	(2)
Discrete approximations of the random evolution processes	190	(6)
Approximation of continuous REP	190	(2)
Approximation of discontinuous REP	192	(3)
Discrete approximations of the dynamics of stocks prices	195	(1)
Dynamics of stocks prices in an incomplete market	196	(10)
Continuous dynamic of stocks prices	196	(1)
Discontinuous dynamic of stocks prices	196	(1)
Ito formula for random evolutions in financial mathematics	197	(2)
Analogue of Girsanov's result for random evolution in financial mathematics	199	(1)
Analogue of Feynman-Kac formula for random evolutions in financial mathematics	200	(2)
Mean value and probability of the time of reaching some bounds	202	(3)
Forecast of real increasing of stocks prices	205	(1)
Hedging of Options under mean-square criterion and with semi-Markov volatility	206	(8)
Description of the model and preliminary notions	207	(2)
The result	209	(2)
Random Evolution approach	211	(3)
Contingent claims valuations of the dynamic of stocks prices with jumps	214	(10)
Discontinuous trading model as discontinuous random evolution process	214	(4)
Contingent claims valuation of discontinuous trading model	218	(1)
Analogue of Black-Scholes formula for random evolutions in financial mathematics	219	(1)
Black-Scholes formula for market when the model is combined (B,S,X)-incomplete market and compound geometric Poisson process	220	(4)
Averaging and Merging of Securities Prices	224	(1)
Random Evolutions in Insurance Mathematics. Incomplete Market.	225	(25)
Examples of random evolutions in insurance mathematics	225	(1)
Stochastic models of the insurance mathematics under incomplete market	226	(18)
Semi-Markov risk processes	228	(3)
Ruin probabilities for semi-Markov risk processes	231	(2)
Risk process on the infinite insurance level	233	(2)
Averaged risk process	235	(1)
Merged risk process	236	(1)
Diffusion risk process	237	(2)
Normal deviated risk process	239	(1)
Diffusion risk process on the infinite insurance level	240	(4)
Ruin probabilities for semi-Markov risk processes	244	(6)
Ruin probabilities for averaged risk processes	244	(1)
Ruin probabilities for merged risk rpocesses	244	(1)
Ruin probabilities for diffusion risk processes	245	(2)
Ruin probability for normal deviated risk process	247	(1)
Ruin probabilities for risk processes on the infinite insurance level	248	(2)
Stochastic Stability of Financial and Insurance Stochastic Models	250	(13)
Definitions of stochastic stability	250	(1)
Stochastic stability of financial stochastic models	251	(2)
Stochastic stability of (B,S,X)-securities market	251	(1)
Stochastic stability of (B,S,X)-securities market with jumps	252	(1)
Stochastic stability of insurance stochastic models	253	(1)
Stochastic stability of Markov and semi-Markov risk processes	253	(1)
Stability of semi-Markov risk processes in averaging, diffusion approximation and normal deviations schemes	254	(9)
Stability of semi-Markov risk process in averaging scheme	254	(4)
Stability of semi-Markov risk process in diffusion approximation scheme	258	(2)
Stability of semi-Markov risk processes in normal deviations scheme	260	(3)
Stochastic Optimal Control of Financial and Insurance Stochastic Models	263	(17)
Stochastic optimal control of financial stochastic models	263	(12)
Functionals of uncontrolled dynamics of stocks prices	263	(1)
Cost functionals for stochastic financial models	264	(3)
Optimal control of stochastic financial models. Bellman principle	267	(8)
Stochastic optimal control of insurance stochastic models	275	(5)
Semi-Markov risk processes	275	(1)
Semi-Markov risk process as discontinuous semi-Markov random evolution	276	(1)
Stochastic optimal control of semi-Markov risk processes	277	(1)
Construction of stochastic optimal control for semi-Markov risk processes	278	(2)
Statistics of Financial Stochastic Models	280	(5)
State of the problem and conditions	280	(1)
Filtering problem for (B,S,X)-incomplete securities market	281	(1)
Interpolation problem for (B,S,X)-incomplete securities market	282	(1)
Extrapolation problem for (B,S,X)-incomplete securities market	283	(2)
Bibliography	285	(6)
Index	291