Two-Scale Stochastic Systems

We consider here a two-scale system with the “slow” dynamics given by a one-dimensional conditionally Gaussian process X ^ε with the drift modulated by a “fast” finite-state Markov process θ ^ε . When θ ^ε is in the state i the process X ^ε behaves like the Wiener process with drift λ ⁱ . If θ ^ε is stationary, it is natural to expect that the process X ^ε approximates in distribution the Wiener process with drift obtained by averaging of λ ⁱ with weights proportional to the time spent by θ ^ε in corresponding states.

In this chapter we present results on the growth of solutions of SDEs driven by the multidimensional Wiener process W = (W _t ) under the assumption of exponential asymptotic stability at zero of solutions of the ordinary differential equation

Let us consider the following initial value problem for the system of ordinary differential equations where the “slow” variable x takes values in R ^k and the “fast” variable y takes values in R ⁿ , ε ∈]0, 1] is a small parameter. The reduced problem corresponding to the formal substitution of the zero value of ε has the form

In Sections 2.1 and 2.2 we established Theorems 2.1.1 and 2.2.1 on the accuracy of approximation on a finite time interval [T ₀, T] of the solution y ^ε of the singularly perturbed SDE (which is a “fast” process) by the deterministic function \({\tilde y_{./\varepsilon }}\) where \(\tilde y\) satisfies the ordinary differential equation

We continue here the study of asymptotic expansions started in Chapter 2. Our results are inspired by the Vasil’eva theorem providing, for the Tikhonov system an asymptotic expansion for both variables, uniform on the whole interval [0, T]. This expansion has the form where \(\tilde x_t^{k,\varepsilon } = \tilde x_{t/\varepsilon }^k,\tilde y_{t/\varepsilon }^k\). The essential property of the “boundary layer functions” \({\tilde x^k}\) and \({\tilde x^k}\) is that they are exponentially decreasing at infinity and this requirement allows us to define them uniquely, using a rather simple algorithm (but calculations are tedious).

In this chapter we study the limiting behavior of the optimal value of a cost functional for controlled two-scale stochastic systems with a small parameter tending to zero. In Section 5.1 we consider the Bolza problem where the cost functional contains an integral part (“running cost”) and a part depending only on terminal values of the phase variables of both types: slow and fast. The result is proved for the case where the model is “semilinear” in the sense that the coefficients depend linearly on the fast variable. This structure is essential: even in the deterministic setting the general nonlinear models are hardly tractable. It is assumed also that the diffusion coefficient of the fast variable is βε ^1/2 with β = o(|ln ε|^−1/2) which is our usual hypothesis. The admissible controls are open loop, i.e., adapted to the driving Wiener process. Such a setting seems to be the most developed: it allows to consider SDEs in the strong sense and use techniques similar to that of the classical theory of optimal control of ordinary differential equations. In the next sections we discuss more delicate subjects, namely, the behavior of the attainability sets for SDEs, aiming to prove a stochastic version of the Dontchev–Veliov theorem. In Section 5.2 we consider rather general models with open loop and closed loop (feedback) controls and compare the structure of attainability sets in these two settings. The main theorem of Section 5.2 clarifies the difference between these two concepts and shows why the model with closed loop controls (where solutions of SDEs are understood in the weak sense) suits more to the question we address later.

It is well-known that various results for stochastic differential equations can be translated, via probabilistic representations, to results for PDEs and vice versa, enriching both theories. In this section we apply our stochastic Tikhonov theorem to a study of asymptotics of boundary-value problems for the second-order PDEs with small parameters.