# Addendum to `Asymptotic inference for a linear stochastic differential equation with time delay'

Abstract: In Gushchin and Kiichler (1999) we considered the stochastic differential equation dX(t) = aX(t)dt + bX(t - 1)dt + dW(t), t > 0, (1) with initial condition X(t) = Xo(t), t E [-1, 0], where (Xo(t), t E [-1, 0]) is a given continuous stochastic process independent of a standard Wiener process (W(t), t > 0). The parameter 9 = (a, b)* is unknown and supposed to be estimated based on (X(t), t < T). (The sign * denotes matrix or vector transposition.) The solutions (X(t), t E [-1, T]) of the differential equation (1) generate a family (Pt, 9 E Rl2) of distributions on C([-1, T]). We studied local asymptotic properties of this family and corresponding properties of the maximum likelihood estimators of 9 for T -- oc.

## Summary (1 min read)

### Summary

- (The sign denotes matrix or vector transposition.).
- The authors studied local asymptotic properties of this family and corresponding properties of the maximum likelihood estimators of W for T !1.

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##### Citations

12 citations

### Cites background from "Addendum to `Asymptotic inference f..."

...Gushchin and Küchler [12] - [14] derived local asymptotic properties of the likelihood process in (two-parameter) models connected with a special case of affine stochastic delay differential equations....

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6 citations

### Cites methods from "Addendum to `Asymptotic inference f..."

...Küchler and Kutoyants [27] investigated in 2000 the equation dX(t) = bX(t− θ) dt+ dW (t), where b is a known negative parameter and the delay parameter θ ∈ R is unknown....

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...Further, by Corollary 4.12 in Gushchin and Küchler [16], (Y T (1), [Y T , Y T ](1)) D−→ (Y(1), [Y ,Y ](1)) as T →∞....

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...[28] Küchler, U. and Sørensen, M. (2010)....

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...By the functional central limit theorem, W T D−→W as T →∞, hence∣∣∣∣ 1T 2 ∫ T 0 W (t)X(t) dt ∣∣∣∣ 6 √( 1 T 2 ∫ T 0 W (t)2 dt )( 1 T 2 ∫ T 0 X(t)2 dt ) = √(∫ 1 0 W T (t)2 dt )( 1 T 2 ∫ T 0 X(t)2 dt ) P−→ 0 as T →∞, and the claim follows from Corollary 4.12 in Gushchin and Küchler [16]....

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...[16] Gushchin, A. A. and Küchler, U. (1999)....

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1 citations

### Cites background from "Addendum to `Asymptotic inference f..."

...Gushchin and Küchler [12] - [14] derived local asymptotic properties of the likelihood process in (two-parameter) models connected with a special case of affine stochastic delay differential equations....

[...]

##### References

38 citations