Random Evolutions and their Applications

1. Front Matter

   Pages i-xvi

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2. Introduction

   • Anatoly Swishchuk

   Pages 1-20

3. Random Evolutions (RE)

   • Anatoly Swishchuk

   Pages 21-62

4. Stochastic Evolutionary Systems

   • Anatoly Swishchuk

   Pages 63-84

5. Random Evolution Equations Driven by Space-Time White Noise

   • Anatoly Swishchuk

   Pages 85-105

6. Analogue of Dynkin’s Formula (ADF) for Multiplicative Operator Functionals (MOF), RE and SES

   • Anatoly Swishchuk

   Pages 106-118

7. Boundary Value Problems (BVP) for RE and SES

   • Anatoly Swishchuk

   Pages 119-126

8. Stochastic Stability of RE and SES

   • Anatoly Swishchuk

   Pages 127-155

9. Stochastic Optimal Control of Random Evolutions and SES

   • Anatoly Swishchuk

   Pages 156-179

10. Statistics of SES

    • Anatoly Swishchuk

    Pages 180-187

11. Random Evolutions in Financial Mathematics Incomplete Market

    • Anatoly Swishchuk

    Pages 188-224

12. Random Evolutions in Insurance Mathematics. Incomplete Market

    • Anatoly Swishchuk

    Pages 225-249

13. Stochastic Stability of Financial and Insurance Stochastic Models

    • Anatoly Swishchuk

    Pages 250-262

14. Stochastic Optimal Control of Financial and Insurance Stochastic Models

    • Anatoly Swishchuk

    Pages 263-279

15. Statistics of Financial Stochastic Models

    • Anatoly Swishchuk

    Pages 280-284

16. Back Matter

    Pages 285-294

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The book is devoted to the new trends in random evolutions and their various applications to stochastic evolutionary sytems (SES). Such new developments as the analogue of Dynkin's formulae, boundary value problems, stochastic stability and optimal control of random evolutions, stochastic evolutionary equations driven by martingale measures are considered. The book also contains such new trends in applied probability as stochastic models of financial and insurance mathematics in an incomplete market. In the famous classical financial mathematics Black-Scholes model of a (B,S)­ market for securities prices, which is used for the description of the evolution of bonds and stocks prices and also for their derivatives, such as options, futures, forward contracts, etc., it is supposed that the dynamic of bonds and stocks prices are set by a linear differential and linear stochastic differential equations, respectively, with interest rate, appreciation rate and volatility such that they are predictable processes. Also, in the Arrow-Debreu economy, the securities prices which support a Radner dynamic equilibrium are a combination of an Ito process and a random point process, with the all coefficients and jumps being predictable processes.

• Martingale
• Stochastic model
• Stochastic models
• Stochastic processes
• operator
• statistics
• stochastic process

• Institute of Mathematics, National Academy of Sciences, Kiev, Ukraine

  Anatoly Swishchuk

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