Semimartingale stochastic approximation procedure and recursive estimation

Abstract

The semimartingale stochastic approximation procedure, namely, the Robbins-Monro type SDE is introduced which naturally includes both generalized stochastic approximation algorithms with martingale noises and recursive parameter estimation procedures for statistical models associated with semimartingales. General results concerning the asymptotic behaviour of the solution are presented. In particular, the conditions ensuring the convergence, rate of convergence and asymptotic expansion are established. The results concerning the Polyak weighted averaging procedure are also presented.1991 Mathematics Subject Classification. 62L20, 60H10, 60H30. t 0 E W (s, x, u s )(µ − ν)(ds, dx) (in case 3 • ) provided the latters are well-defined.Consider the following semimartingale stochastic differential equationWe call SDE (0.1) the Robbins-Monro (PM) type SDE if the drift coefficient H t (u), t ≥ 0, u ∈ R 1 satisfies the following conditions: for all t ∈ [0, ∞) P -a.s. (A)H t (0) = 0,The question of strong solvability of SDE (0.1) is well-investigated (see, e.g., [8], [9], [13]).

Two-Timescale Stochastic Gradient Descent in Continuous Time with Applications to Joint Online Parameter Estimation and Optimal Sensor Placement

“…The other two assumptions are unique to the continuoustime, two-timescale stochastic gradient descent algorithm with Markovian dynamics introduced in this paper. We should note, however, that similar assumptions have previously appeared in the analysis of the single-timescale stochastic approximation schemes in [87,88,89,90,91,92,103,140].…”

“…where {a 1 (t)} t≥0 , i = 1, 2, are predictable, increasing processes, and {ξ i (t)} t≥0 , i = 1, 2, are semi-martingales. This algorithm can be regarded, in some sense, as as a two-timescale extension of the Robbins-Monro type semimartingale SDEs studied in [87,88,89,90,91,92,103,140]. Obtaining asymptotic results under this somewhat more general framework is of considerable interest, as such results would apply to twotimescale stochastic gradient descent schemes in both discrete time and continuous time.…”

“…This is a standard assumption used in the convergence analysis of two-timescale stochastic approximation algorithms in discrete time [20,66,71], as well as singletimescale stochastic approximation algorithms in continuous time [37,103,121,149], although slightly weaker assumptions may also be possible (see, for example, [88]). This assumption implies, in particular, that the functions ∇ α f (•) and ∇ β g(•) satisfy linear growth conditions.…”

“…, with all constants defined as previously (e.g., [72].) 2 The case when the noise process is a local martingale is also considered by [87,88,89,91,103,140]. In these works, however, there is no requirement that this local martingale is continuous.…”

Update estimation of diffusion parameter observed at high frequency

“…We refer to Sharia [10], [11], [12] and [13] for asymptotics of time-series model recursive estimator. See Kutoyants [6, Section 2.6.6] and Lazrieva et al [7] as well as the references therein for the case of continuous-time data from a diffusion model.…”