N. Lazrieva and T. Toronjadze

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

THE ROBBINS–MONRO TYPE SDE. ASYMPTOTIC PROPERTIES OF SOLUTION AND AVERAGING PROCEDURES

We consider the Robbins–Monro family (RM) type SDE introduced in [1], that is,

z_{t} = z_{0} + ∫_{0}^{t} H_{s}(z_{s−}) dK_{s} + ∫_{0}^{t} M(ds, z_{s−}),

where the random field {H_{t}(ω, u), t ≥ 0, u ∈ R^{1}} is such that (H_{t}(u))_{t≥0} is a predictable process for each u ∈ R^{1} and

H_{t}(0) = 0,
uH_{t}(u) ≤ 0 if u ≠ 0,

for each t ≥ 0, P -a.s., K = (K_{t})_{t≥0} is a predictable increasing process,

M(u) = (M(t, u))_{t≥0} ∈ M_{loc}^{2} for each u ∈ R^{1}, ∫_{0}^{t} M(ds, z_{s−}) is the stochastic

line integral w.r.t. the family {M(t, u), t ≥ 0, u ∈ R^{1}} along the predictable curve u = (u_{t})_{t≥0}.

Assuming that there exists a unique strong solution z = (z_{t})_{t≥0} of this equation, the general results concerning the asymptotic behavior of z = (z_{t})_{t≥0} as t → ∞ are presented. In particular, the rate of convergence z_{t} → 0 P -a.s. is established. Moreover, it is shown that z admits an asymptotic expansion which enables one to obtain the asymptotic distribution of the randomly normed solution from a martingale limit theorems.

Further, we consider the Polyak weighted averaging procedure

\overline{z}_{t} = E(−g ∘ K) ∫_{0}^{t} z_{s} dE_{s}^{−1}(−g ∘ K),

where E(−g∘K) is the Dolean exponential of the process g∘K_{t} = − ∫_{0}^{t} g_{s} dK_{s},

g_{t} ≥ 0.

The asymptotic properties of \overline{z} = (\overline{z}_{t})_{t≥0} as t → ∞ are investigated.

References

1. N. Lazrieva, T. Sharia, and T. Totonjadze, The Robbins–Monro type SDE, I, Convergance of solutions. Stochastics and Stochastics Reports 61(1997), 67–87.