The Robbins-Monro type SDE and recursive estimation (Q2769690)

Let NEWLINE\[NEWLINEdz_t=H_t(z_{t-}) dK_t+M(dt,z_{t-}),\;z_0,\tag{1}NEWLINE\]NEWLINE be a stochastic differential equation, where \(H(u)=(H_t(u))_{t\geq 0}\) and \(M(u)= (M(t,u))_{t\geq 0}\), \(u\in R\), are random fields and \((K_t)_{t\geq 0}\) is an increasing predictable process. The drift coefficient \(H_t(u)\) satisfies the following conditions: NEWLINE\[NEWLINE\text{for all }t\geq 0,\;H_t(0)=0, \quad H_t(u)u<0 \quad\text{if }u\neq 0\;(P-\text{a.s.}).NEWLINE\]NEWLINE In this case, equation (1) is called Robbins-Monro type stochastic differential equation (RM type SDE). The authors study the asymptotic properties of strong solutions \(z=(z_t)_{t\geq 0}\) of equation (1). They also consider a filtered statistical model and a recursive estimation procedure which is reduced to the RM type SDE (1) is described.NEWLINENEWLINEFor the entire collection see [Zbl 0968.00043].