Introduction

The main objective of consideration is a statistical experiment (SE), defined as the averaged sum SN (·) of independent and identically distributed random samples, which take a finite number of values, in particular, the binary values ±1:

The basic assumption 3 (Proposition 1.3.3) is derived by using the regression function of increments (RFI), given by the predictable component of SEs [1.3.10].

SE presentation as averaged sums of independent and identically distributed random samples (which take a finite number of values) means that an SE is defined by two components: evolutionary processes (EPs) (predictable components) and martingale differences (stochastic components). So, the SE can be considered as special semi-martingales (Jacod and Shiryaev 1987).

An essential feature of specifying SEs is their characterization by stochastic difference equations (SDEs) consisting of two parts: a predictable component defined by RFI and a stochastic component characterized by its first two moments.

Proposition 1.2.3 (Basic assumption 3). The SEs given by the averaged sums of the sample values are determined by the solutions of the SDEs:

The stochastic component of the SDEs [1.2.34] and [1.2.35] is characterized by their quadratic characteristics.

The existence of equilibria ?± of the predictable component provides the convergence with probability 1 (Theorem 1.2.2).

However, the stochastic component defined by the martingale differences generates, as N 8, a series (by k) of normally distributed random variables with certain quadratic variations which depend on the state of SEs (Theorem 1.2.3). Asymptotic representation of the normalized martingale differences provides a basic stochastic approximation by SDEs, with the stochastic part represented by the normally distributed martingale differences (Propositions 1.2.4 and 1.2.5).

The multivariate statistical experiments (MSEs) in section 1.3 are considered in the assumption of a finite number of the possible values E = {e0, e1, ., eM}, M = 1. The linear RFI is determined as follows.

Proposition 1.3.1 (Basic assumption 1)

That is, the linear RFI is given by the fluctuation of increments with respect to the equilibrium value.

The essential result is given in Proposition 1.3.2 where the multivariate frequencies with Wright-Fisher normalization are considered for models in population genetics (Ethier and Kurtz 1986). The nonlinear RFI are represented as in canonical representation [1.3.9] with the fluctuation of increments with respect to the ratio of the values of EPs and the corresponding equilibria.

Proposition 1.3.3. (Basic assumption 3.) The frequencies of multivariate EPs are determined by difference evolutionary equations (DEEs) with nonlinear RFIs having the fluctuation representation [1.3.10].

The presence of the equilibrium state ensures the convergence of EPs, as k 8 (Theorem 1.3.1).

The SDE for MSEs is given with martingale difference [1.3.16] as the stochastic component with quadratic characteristics (Lemma 1.3.1; also see Proposition 1.3.4). A representation of MSEs as normalized sums of independent and identically distributed random variables [1.3.28] provides the convergence with probability 1 (Theorem 1.3.2).

In sections 1.3.1 and 1.3.2, the martingale differences are approximated by normally distributed random variables (Theorem 1.3.3). The MSEs are determined by the solutions of the SDEs with normally distributed stochastic components and the "constant" quadratic characteristics defined by equilibrium points (Proposition 1.3.5).

The SEs with Wright-Fisher normalization are developed with RFIs given by the fluctuations with respect to equilibrium introduced in section 1.4. The binary EPs are transformed in the increment probabilities with RFIs in terms of the fluctuations [1.4.12]. In this connection, the RFIs are postulated as basic assumption 2 (Proposition 1.4.1).

Section 1.5 is dedicated to the exponential SE (ESE) defined as a product of random variables (Definition 1.5.1; also see [1.5.7]).

The ESE is investigated in two normalized schemes: Theorem 1.5.1-the convergence, in probability, of ESEs with normalization ?N = ?/N, N 8, is realized using Le Cam's approximation (Borovskikh and Korolyuk 1997) and Theorem 1.5.2-the convergence, in distribution, of ESEs with normalization , to geometric Brownian motion.

Theorems 1.5.1 and 1.5.2 give us an opportunity to represent ESEs in exponential approximation scheme [1.5.41] using only three parameters for the normal process of autoregression. Note that the ESE has important interpretation in financial mathematics (Shiryaev 1999).

Chapter 2 is dedicated to the diffusion approximation of SEs in discrete-continuous time.

Discrete Markov processes (DMPs), determined by the solutions of the SDEs in discrete-continuous time, are approximated, as N 8, by diffusion processes with evolution, given by differential stochastic equations.

The discrete-continuous time is determined by the connection of discrete instants of time k = 0, with continuous time t = 0, t ? R+ = [0, +8) by the formula k = [Nt], t = 0. The integer part [Nt] = k defines a discrete sequence . The adjacent moments of the continuous time and as N 8, k 8.

DMPs in discrete-continuous time are considered with the normalized fluctuation [2.1.12]. Basic assumption 2.1.2 supposes that the DMP, given by the solution of the SDE [2.1.15] with nonlinear RFIs, can be approximated by a linear SDE [2.1.20] (see basic assumption 2.1.3). The finite-dimensional distributions of DMPs converge, as N 8, to a diffusion process with evolution of Ornstein-Uhlenbeck type given by the solution of the SDE [2.1.24] (Theorem 2.2.1, Conclusion 2.2.1).

The diffusion approximation of DMPs is based on the operator characterization of Markov processes (Korolyuk and Limnios 2005).

Propositions 2.2.1 and 2.2.2 are determined the SDEs with predictable components, as well as the quadratic characteristics of stochastic component convergence, as N 8, to equilibrium values (Lemma 2.2.1) provide the linear character of the SDEs for limit processes, characterized only by the local fluctuations (Proposition 2.2.3).

The DMP in discrete and continuous Markov random environments (MREs) are considered in section 2.3. Using the method of singular perturbation (Korolyuk and Limnios 2005), Theorem 2.3.1 states the convergence of the finite-dimensional distributions to those of the limit diffusion processes with evolution, defined by averaged drift and diffusion parameters and [2.3.10], respectively. The averages are intend over the stationary distribution of the embedded Markov chain. Theorem 2.3.2 states the limit diffusion process with evolution, with parameters and averaged by the stationary distribution of a Markov ergodic environment.

Essential applications can give the technique of singular perturbation problem for a reducible invertible operator acting on a perturbed test function f1(c, x) that provides the solvability condition of a certain operator equation (see [2.3.38]-[2.3.39]).

One specific model in section 2.4 is DMP in a balanced Markov random environment. The approximation of DMP in discrete-continuous time with balance condition [2.4.9] is given in Theorem 2.4.1. The limit Ornstein-Uhlenbeck diffusion process [2.4.20] is determined by the parameters and calculated by the explicit formulas [2.4.15]-[2.4.16]. Here the solution of a singular perturbation problem for the truncated operator [2.4.35] is realized on the perturbed test function [2.4.36] that consists of three differently scaled components.

In section 2.5, the adapted SEs are combined with a random time change, which transforms a discrete stochastic basis into a continuous stochastic basis .

The adapted SE with continuous basis are studied in the series scheme with a small series parameter e 0 (e > 0). The limit diffusion process with evolution is determined in Theorem 2.5.1 by the predictable characteristics and .

In section 2.6, the DMPs are considered in an asymptotic diffusion environment, generated by the DEE [2.6.6] with the balance condition [2.6.7]. Theorem 2.6.1 states the convergence by e 0 to an Ornstein-Uhlenbeck diffusion process.

In section 2.7, the DMPs with asymptotically small diffusion are considered. Its exponential generator is determined in Theorem 2.7.2 by relation [2.7.2] on the test functions f(c) ? C3(R). The action functional for DMPs is introduced, following the method developed in monograph (Freidlin and Ventzell 2012).

Chapter 3 is devoted to statistically estimating the drift parameter V0 and verifying the hypothesis...