Markov Processes, Feller Semigroups And Evolution Equations

by Jan A Van Casteren

Overview

The book provides a systemic treatment of time-dependent strong Markov processes with values in a Polish space. It describes its generators and the link with stochastic differential equations in infinite dimensions. In a unifying way, where the square gradient operator is employed, new results for backward stochastic differential equations and long-time behavior are discussed in depth. The book also establishes a link between propagators or evolution families with the Feller property and time-inhomogeneous Markov processes. This mathematical material finds its applications in several branches of the scientific world, among which are mathematical physics, hedging models in financial mathematics, and population models.

Product Details

ISBN-13:	9789814322188
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	11/26/2010
Series:	Series On Concrete And Applicable Mathematics , #12
Pages:	824
Product dimensions:	6.30(w) x 9.00(h) x 1.90(d)

Table of Contents

Preface vii

Introduction 1

1 Introduction: Stochastic differential equations 3

1.1 Weak and strong solutions to stochastic differential equations 3

1.2 Stochastic differential equations in the infinite-dimensional setting 45

1.3 Martingales 90

1.4 Operator-valued Brownian motion and the Heston volatility model 95

1.5 Stopping times and time-homogeneous Markov processes 104

Strong Markov Processes 107

2 Strong Markov processes on Polish spaces 109

2.1 Strict topology 109

2.1.1 Theorem of Daniell-Stone 110

2.1.2 Measures on Polish spaces 116

2.1.3 Integral operators on the space of bounded continuous functions 128

2.2 Strong Markov processes and Feller evolutions 138

2.2.1 The operators ν_t, $$$_t and $$$_t 142

2.2.2 Generators of Markov processes and maximum principles 143

2.3 Strong Markov processes: Main results 147

2.3.1 Some historical remarks and references 158

2.4 Dini's lemma, Scheffé's theorem, and the monotone class theorem 159

2.4.1 Dini's lemma and Scheffé's theorem 159

2.4.2 Monotone class theorem 162

2.4.3 Some additional information 164

3 Strong Markov processes: Proof of main results 167

3.1 Proof of the main results: Theorems 2.9 through 2.13 167

3.1.1 Proof of Theorem 2.9 167

3.1.2 Proof of Theorem 2.10 192

3.1.3 Proof of Theorem 2.11 195

3.1.4 Proof of Theorem 2.12 199

3.1.5 Proof of Theorem 2.13 219

3.1.6 Some historical remarks 222

3.1.7 Kolmogorov extension theorem 224

4 Space-time operators and miscellaneous topics 227

4.1 Space-time operators 227

4.2 Dissipative operators and maximum principle 240

4.3 Korovkin property 260

4.4 Continuous sample paths 280

4.5 Measurability properties of hitting times 282

4.5.1 Some related remarks 299

Backward Stochastic Differential Equations 301

5 Feynman-Kac formulas, backward stochastic differential equations and Markov processes 303

5.1 Introduction 304

5.2 A probabilistic approach: Weak solutions 327

5.2.1 Some more explanation 330

5.3 Existence and uniqueness of solutions to BSDE's 335

5.4 Backward stochastic differential equations and Markov processes 371

5.4.1 Remarks on the Runge-Kutta method and on monotone operators 379

6 Viscosity solutions, backward stochastic differential equations and Markov processes 385

6.1 Comparison theorems 386

6.2 Viscosity solutions 392

6.3 Backward stochastic differential equations in finance 399

6.4 Some related remarks 405

7 The Hamilton-Jacobi-Bellman equation and the stochastic Noether theorem 407

7.1 Introduction 407

7.2 The Hamilton-Jacobi-Bellman equation and its solution 411

7.3 The Hamilton-Jacobi-Bellman equation and viscosity solutions 420

7.4 A stochastic Noether theorem 436

7.4.1 Classical Noether theorem 446

7.4.2 Some problems 448

Long Time Behavior 451

8 On non-stationary Markov processes and Dunford projections 453

8.1 Introduction 453

8.2 Kolmogorov operators and weak*-continuous semigroups 455

8.3 Kolmogorov operators and analytic semigroups 460

8.3.1 Ornstein-Uhlenbeck process 477

8.3.2 Some stochastic differential equations 503

8.4 Ergodicity in the non-stationary case 518

8.5 Conclusions 537

8.6 Another characterization of generators of analytic semigroups 543

8.7 A version of the Bismut-Elworthy formula 550

9 Coupling methods and Sobolev type inequalities 555

9.1 Coupling methods 555

9.2 Some ergodic theorems 597

9.3 Spectral gap 602

9.4 Some related stability results 611

9.5 Notes 644

10 Invariant measure 647

10.1 Markov Chains: Invariant measure 647

10.1.1 Some definitions and results 648

10.2 Markov processes and invariant measures 660

10.2.1 Some additional relevant results 665

10.2.2 An attempt to construct an invariant measure 671

10.2.3 Auxiliary results 684

10.2.4 Actual construction of an invariant measure 702

10.3 A proof of Orey's theorem 731

10.4 About invariant (or stationary) measures 752

10.4.1 Possible applications 754

10.4.2 Conclusion 754

Bibliography 759

Index 789