Interacting Markov chain Monte Carlo methods for solving nonlinear measure-valued equations (Q968777)

Let \((S^{(l)},{\mathcal S}^{(l)})_{l\geq 0}\) be a sequence of measurable spaces, \({\mathcal P}(S^{(l)})\) the set of probability measures on \(S^{(l)}\), \(\pi^{(l)}\in{\mathcal P}(S^{(l)})\), \(l\geq 0\), where \(\pi^{(0)}\) is known. For \(l\geq 1\) consider the equations \(\pi^{(l)}= \Phi_l(\pi^{(l-1)})\) for some mappings \(\Phi_l:{\mathcal P}(S^{(l-1)})\to{\mathcal P}(S^{(l)})\). The authors develop a new class of interacting Markov chain Monte Carlo (i-MCMC) methods for a numerical approximation of the solution. At level \(l=0\) a Markov chain Monte Carlo (MCMC) algorithm is run to obtain a chain \(X^{(0)}= (X^{(0)}_n)_{n\geq 0}\) targeting \(\pi^{(0)}\). (The ``time'' index \(n\) corresponds to the number of iterations of the i-MCMC algorithm.) The occupation measure of the chain \(X^{(0)}\) at time \(n\) is used to design a second MCMC algorithm to generate \(X^{(1)}= (X^{(1)}_n)_{n\geq 0}\) at level \(1\) targeting \(\pi^{(1)}\): The elementary transition \(X^{(1)}_n\rightsquigarrow X^{(1)}_{n+1}\) of the chain \(X^{(1)}\) at ``time'' \(n\) depends on the occupation measure of \((X^{(0)}_0, X^{(0)}_1,\dots, X^{(0)}_n)\). Similarly, the empirical measure of \(X^{(l-1)}\) at level \(l-1\) is used to ``feed'' an MCMC algorithm generating \(X^{(l)}\) targeting \(\pi^{(l)}\) at level \(l\). Denote \(\overline X^{(m)}_n:= (X^{(0)}_n,\dots, X^{(m)}_n)\), \(\overline\pi^{[m]}:= \pi^{(1)}\otimes\cdots\otimes \pi^{(m)}\), let \(\delta_{X,n,m}\), \(\delta_{\overline X,n,m}\) be the Dirac measures at \(X^{(m)}_n\), \(\overline X^{(m)}_n\), respectively, and set \(\eta^{[m]}_n:= (n+1)^{-1}\cdot \sum^n_{p=1} \delta_{X,p,m}\), \(\overline\eta^{[m]}_n:= (n+1)^{-1}\cdot \sum^n_{p=1} \delta_{\overline X,p,m}\). The main results may then be written as follows: For every \(r\geq 1\), \(m\geq 1\), and any bounded measurable function \(f\) on \(S^{(0)}\times\cdots\times S^{(m)}\) we have \(\sup_{n\geq 1}\sqrt{n}{\mathbf E}(|\overline\eta^{[m]}_n(f)- \overline\pi^{[m]}(f)|^r)< \infty\) and, under additional regularity conditions, \[ \forall t\limsup_{n\to\infty}(1/n)\log{\mathbf P}(|\overline\eta^{[m]}_n(f)- \overline\pi^{[m]}(f)|> 1)< -t^2/2\overline\sigma^2_m \] with some positive constant \(\overline\sigma<\infty\). Finally, for some \(\alpha\in(0,1]\) and any sequence of bounded measurable functions \(f_n|S^{(n)}\) with \(|f_n|\leq 1\), \(\sup_{k\geq 0}\,\sup_{n\geq 0} n^{\alpha/2}{\mathbf E}(|\eta^{(k)}_n(f_n)- \pi^{(k)}(f_n)|^r)< \infty\).