**Introduction **

**Deterministic Dynamical Systems and Stochastic Perturbations**

Deterministic dynamical systems

Stochastic perturbations of deterministic dynamical systems

**Random Dynamical Systems and Random Maps**

Random dynamical systems

Skew products

Random maps: Special structures of random dynamical systems

Necessary and sufficient conditions for the existence of invariant measures for a general class of random maps with constant probabilities

Support of invariant densities for random maps

Smoothness of density functions for random maps

Applications in finance

**Position-Dependent Random Maps**

Random maps with position dependent probabilities

Markov switching position dependent random maps

Higher dimensional Markov switching position dependent random maps

Approximation of invariant measures for position dependent random maps

Applications in finance

**Random Evolutions as Random Dynamical Systems**

Multiplicative operator functionals (MOF)

Random evolutions

Limit theorems for random evolutions

**Averaging of the Geometric Markov Renewal Processes (GMRP)**

Introduction

Markov renewal processes and semi-Markov processes

The GMRP

Averaged geometric Markov renewal processes

Rates of convergence in ergodic averaging scheme

Merged geometric Markov renewal processes

Security markets and option prices using generalized binomial models induced by random maps

Applications

**Diffusion Approximations of the GMRP and Option Price Formulas**

Introduction

Diffusion approximation of the GMRP

Proofs

Merged diffusion geometric Markov renewal process in the case of two ergodic classes

European call option pricing formulas for diffusion GMRP

Applications

**Normal Deviation of a Security Market by the GMRP**

Normal deviations of the GMRP

Applications

European call option pricing formula for normal deviated GMRP

Martingale property of GMRP

Option pricing formulas for stock price modelled by GMRP

Examples of option pricing formulas modelled by GMRP

**Poisson Approximation of a Security Market by the GMRP**

Averaging in Poisson scheme

Option pricing formula under Poisson scheme

Application of Poisson approximation with a finite number of jump values

**Stochastic Stability of Fractional RDS in Finance**

Fractional Brownian motion as an integrator

Stochastic stability of a fractional (*B, S*)-security market in Stratonovich scheme

Stochastic stability of fractional (*B, S*)-security market in Hu and Oksendal scheme

Stochastic stability of fractional (*B, S*)-security market in Elliott and van der Hoek scheme

Appendix

**Stability of RDS with Jumps in Interest Rate Theory**

Introduction

Definition of the stochastic stability

The stability of the Black-Scholes model

A model of (*B, S*)- securities market with jumps

Vasicek model for the interest rate

The Vasicek model of the interest rate with jumps

Cox-Ingersoll-Ross interest rate model

Cox-Ingersoll-Ross model with random jumps

A generalized interest rate model

A generalized model with random jumps

**Stability of Delayed RDS with Jumps and Regime-Switching in Finance**

Stochastic differential delay equations with Poisson bifurcations

Stability theorems

Application in finance

Examples

**Optimal Control of Delayed RDS with Applications in Economics**

Introduction

Controlled stochastic differential delay equations

Hamilton-Jacobi-Bellman equation for SDDEs

Economics model and its optimization

**Optimal Control of Vector-Delayed RDS with Applications in Finance and Economics**

Introduction

Preliminaries and formulation of the problem

Controlled stochastic differential delay equations

Examples: optimal selection portfolio and Ramsey model

**RDS in Option Pricing Theory with Delayed/Path-Dependent Information**

Introduction

Stochastic delay differential equations

General formulation

A simplified problem

Appendix

**Epilogue **

**Index**