2011/05/28 by John M. Noble, Noble, John M.
Computer Science · Decision Sciences · Economics, Econometrics and Finance · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR
paper · pdf · doi:10.48550/arxiv.1105.5694
44 pages
openalex publication_date 2011/05/28 · arxiv created 2012/09/28 · arxiv updated 2012/10/01 · openalex created_date 2022/10/07 · openalex updated_date 2026/07/28
In this article, it is proved that for any cumulative distribution function with compact support and a specified t > 0, there exists a diffusion martingale which has this law at time t. The article proves existence; no claims are made about uniqueness. After a discussion on strings and associated semigroups, the article gives a re-working of a standard approach to the problem of constructing an explicit discrete time martingale diffusion on a finite state space which, for a random geometrically distributed time that is independent of the diffusion, the law of the diffusion stopped at this random time has the prescribed law. This argument is developed, using a fixed point theorem, to determine conditions under which there is a discrete time martingale diffusion that has a prescribed law at an independent random time with negative binomial distribution. The step length for the time discretisation is then reduced and in the limit it is shown that for a finite state space, there exists a continuous time martingale diffusion such that Xτ has law μ, where τ has a Gamma distribution. For fixed t = E[τ], the parameters of the Gamma distribution may be altered, reducing the coefficient of variation of τ to zero, to show that there is a martingale diffusion X such that Xt has law μ. The argument is then extended to obtain the result for any state space that is a bounded measurable subset of R.