Martingales associated to peacocks using the curtain coupling

Abstract

We consider right-continuous peacocks, that is, families of real probability measures $(\mu _t)_{t\in [0,1]}$ that are increasing in convex order. Given a sequence of time partitions we associate the sequence of martingales characterised by the fact that they are Markovian, constant on the partition intervals $[t_k,t_{k+1}[$, and such that the transition kernels at times $t_{k+1}$ are the curtain couplings of marginals $\mu _{t_k}$ and $\mu _{t_{k+1}}$. We study the limit curtain processes obtained when the mesh of the partition tends to zero and study existence, uniqueness and relevancy with respect to the original data. For any right-continuous peacock we show there exist sequences of partitions such that a limit process exists (for the finite-dimensional convergence).

Under certain additional regularity assumptions, we prove that there is a unique limit curtain process and that it is a Markovian martingale. We first study by elementary methods peacocks whose marginals correspond to uniform distributions in convex order. In this case, the results and techniques complete the results and techniques used in a parallel work by Henry-Labordère, Tan and Touzi [9]. We obtain the same type of results for all limit curtain processes associated to a class of analytic discrete peacocks, i.e., the measures $\mu _t$ are finitely supported and vary analytically in $t$.

Finally, we give examples of peacocks and sequences of partitions such that the limit curtain process is a non-Markovian martingale.