On Sequential Parameter Estimation for Some Linear Stochastic Differential Equations With Time Delay

Abstract

We consider the parameter estimation problem for the scalar di usion type process described by t h e s t o c hastic equation Here we construct a sequential MLE with preassigned least square accuracy for the so-called stationary and the periodic cases of the solution X( ): The limit behaviour of the duration of the procedure with given accuracy is obtained.Keywords: stochastic di erential equations time delay maximum likelihood estimator sequential analysis least square accuracy.

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

“…The problem of sequential estimation of ϑ by observation without noise under the condition (3) was considered in [5,6].…”

“…We have reduced the system (1)-(2) to a single differential equation 6for the observed process (∆Y (t), t ≥ 2) depending on the unknown parameters a and b. The term∆ξ(t) also contains a and b, but its variance is controllable in certain sense (see formula (5)).…”

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

“…Similar to the proof of Propositions 3.1, 3.2 and [7]- [16] we get for ϑ ∈ Θ 3 the needed asymptotic as t → ∞ relations for the processesΔY(t), Z(t) andZ(t). To this end we introduce the following notation:…”

“…The properties (59) can be established by using the asymptotic properties of the process (X(t), Y(t)) (see proofs of Propositions 3.1-3.4 and [3], [7]- [16]). …”

“…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).…”