On parameter estimation of stochastic delay differential equations with guaranteed accuracy by noisy observations

Abstract

Let (X(t), t ≥ −1) and (Y (t), t ≥ 0) be stochastic processes satisfying dX(t) = aX(t)dt + bX(t − 1)dt + dW (t) and dY (t) = X(t)dt + dV (t), respectively. Here (W (t), t ≥ 0) and (V (t), t ≥ 0) are independent standard Wiener processes and ϑ = (a, b) is assumed to be an unknown parameter from some subset Θ of R 2. The aim here is to estimate the parameter ϑ based on continuous observation of (Y (t), t ≥ 0). Sequential estimation plans for ϑ with preassigned mean square accuracy ε are constructed using the so-called correlation method. The limit behaviour of the duration of the estimation procedure is studied if ε tends to zero.

On Guaranteed Parameter Estimation of Stochastic Delay Differential Equations by Noisy Observations

“…Sequential estimators of ϑ with a prescribed least square accuracy we have already constructed in [10,12]. But in these articles the set of possible parameters ϑ were restricted to…”

“…To construct a sequential plan for estimating ϑ based on the observation of Y(·) we follow the line of [10,12]. We shall use a single equation for Y of the form:…”

“…The set Θ is defined to be the intersection of the set Θ with an arbitrary but fixed ball B 0,R ⊂R 2 . Sequential estimation problem has been solved for sets Θ of a different structure in [7]- [9], [11,13,14,16] by observations of the process (1) and in [10,12,15] -by noisy observations (2).…”

“…In the papers [10,12] the problem described above was solved for the two special sets of parameters Θ I (a straight line) and Θ II (where X(·) satisfies (1) is stable or periodic (unstable)) respectively. The general sequential estimation problem for all ϑ =( a, b) ′ from R 2 except of two lines was solved in [13,14,16] for the equation (1) based on the observations of X(·).…”

“…In this chapter the sequential estimation method developed in [10,12] for the system (1), (2) is extended to the case, considered by [13,14,16] for the equation (1) (as already mentioned, for all ϑ from R 2 except of two lines for the observations without noises).…”