Markov Processes, Feller Semigroups and Evolution Equations

The following sections are included:

• Preface

• Contents

The following sections are included:

• Weak and strong solutions to stochastic differential equations

• Stochastic differential equations in the infinite- dimensional setting

• Martingales

• Operator-valued Brownian motion and the Heston volatility model

• Stopping times and time-homogeneous Markov processes

The following sections are included:

• Strict topology
  • Theorem of Daniell-Stone
  • Measures on Polish spaces
  • Integral operators on the space of bounded continuous functions

• Strong Markov processes and Feller evolutions
  • The operators ∨t, ∧t and ϑt
  • Generators of Markov processes and maximum principles

• Strong Markov processes: Main results
  • Some historical remarks and references

• Dini's lemma, Scheffé's theorem, and the monotone class theorem
  • Dini's lemma and Scheffé's theorem
  • Monotone class theorem
  • Some additional information

The following sections are included:

• Proof of the main results: Theorems 2.9 through 2.13
  • Proof of Theorem 2.9
  • Proof of Theorem 2.10
  • Proof of Theorem 2.11
  • Proof of Theorem 2.12
  • Proof of Theorem 2.13
  • Some historical remarks
  • Kolmogorov extension theorem

The following sections are included:

• Space-time operators

• Dissipative operators and maximum principle

• Korovkin proerty

• Continuous sample paths

• Measurability properties of hitting times
  • Some related remarks

The following sections are included:

• Introduction

• A probabilistic approach: Weak solutions
  • Some more explanation

• Existence and uniqueness of solutions to BSDE's

• Backward stochastic differential equations and Markov processes
  • Remarks on the Runge-Kutta method and on monotone operators

The following sections are included:

• Comparison theorems

• Viscosity solutions

• Backward stochastic differential equations in finance

• Some related remarks

The following sections are included:

• Introduction
  • Hypotheses on the generator and the algebra A
  • Some additional comments

• The Hamilton-Jacobi-Bellman equation and its solution

• The Hamilton-Jacobi-Bellman equation and viscosity solutions

• A stochastic Noether theorem
  • Classical Noether theorem
  • Some problems
    • Conclusion

The following sections are included:

• Introduction

• Kolmogorov operators and weak*-continuous semigroups

• Kolmogorov operators and analytic semigroups
  • Ornstein-Uhlenbeck process
  • Some stochastic differential equations

• Ergodicity in the non-stationary case

• Conclusions

• Another characterization of generators of analytic semigroups

• A version of the Bismut-Elworthy formula

The following sections are included:

• Coupling methods

• Some ergodic theorems

• Spectral gap

• Some related stability results

• Notes

The following sections are included:

• Markov Chains: Invariant measure
  • Some definitions and results

• Markov processes and invariant measures
  • Some additional relevant results
  • An attempt to construct an invariant measure
  • Auxiliary results
  • Actual construction of an invariant measure

• A proof of Orey's theorem

• About invariant (or stationary) measures
  • Possible applications
  • Conclusion

The following sections are included:

• Bibliography

• Index