Dirichlet Brownian Motions

In this work we introduce a Brownian motion in random environment which is a Brownian constructions by an exchangeable sequence based on Dirichlet processes samples. We next compute a stochastic calculus and an estimation of the parameters is computed in order to classify a functional data.


Introduction
The Brownian motion is a very interesting tool for both theoretical and applied math.Brownian motion is among the simplest of the continuous-time stochastic processes, and it is a limit of both simpler and more complicated stochastic processes.In this paper we construct a new process called Dirichlet brownian motion by the usual i.i.d.Gaussian sequence used in Brownian motion constructions is replaced by an exchangeable sequence.
Despite its recent introduction to the literature, hierarchical models with a Dirichlet prior, shortly Dirichlet hierarchical models, were used in probabilistic classification applied to various fields such as biology [1], astronomy [2] or text mining [3] and finance [4]- [6].Actually, these models can be seen as complex mixtures of real Gaussian distributions fitted to non-temporal data.
The aim of this paper is to extend these models and estimate their parameters in order to deal with temporal data following a stochastic differential equation (SDE).
The paper is organized as follows.In Section 2 we briefly recall Ferguson-Dirichlet process.In Section 3 we consider a different construction of the Brownian motion based on an exchangeable sequence from Dirichlet processes samples which is shown to be a limit of a random walk in Dirichlet random environment.In Section 4, we prove the regularity of the new process and in the Section 5 we give some stochastic calculus and an estima-tion of the parameters of DBM.

Ferguson-Dirichlet Process
Let ( ) , , Ω   be a fixed probability space.Let  be a Polish space and let ( ) P  denote the set of all probability measures defined on  .The distribution of a random variable, say Z , will be denoted by ether . The following celebrated random distribution defined by Ferguson [7] plays a central role in our construction.
Let α be a finite positive measure on  .A random distribution ( )

( )
PDir c denote the Poisson-Dirichlet distribution with parameter c (Kingman [8]) which support is the set ( ) , , , , : , 0, 1 Ferguson has also shown that for a.a.ω , ( ) P ω is a discrete probability measure: there exist an i.i.d.se- quence of random variables on  , say , and a sequence of random weights such that ( ) ( ) ( ) , , Ω   be a probability space on which are defined all the random variables (r.v.) mentioned in this paper.The probability distribution of a r.v.X will be denoted X  .Equality in distribution is denoted by d = .For any integer 2 n ≥ , let n Σ denote the group of permutations of { } 1, 2, , n  .

Exchangeable Random Variables
, , , , for all Using transpositions, first notice that (2) implies that all the n X have the same distribution, say  : and also The variables n X are assumed to take their values on a separable space  and ( )   denote the separable set (for weak convergence topology) of all probability measures defined on  .
An i.i.d.sequence is of course exchangeable but an exchangeable sequence needs neither be independent nor Markov.
be an exchangeable sequence with values in  .Then there exists a probability measure µ on ( ) ( ) ( ) In other words, (5) shows that the distribution of an exchangeable sequence is a mixture with mixing measure µ , (6) shows that µ is the distribution of the weak limit empirical measure and finally (7) shows that if Q is considered as a parameter In the example of a sample from the Dirichlet process ( ) by definition of such a sample [7], while 0 .
For the rest of the paper it is assumed that =   the real line.
For any integer 1 n ≥ and 0 2 1 The functions , t s and , k n s consitute the so called Schauder system.Now, let ( ) ( ) , , for 1 and 0 2 1 for one (and any) n .
Notice that ( 3) and ( 4) then imply that , , are constants which do not depend on , n i and j . Let The series with general term ( ) defines a stochastic process.
Proof: Due to (10) we have and then (12) applied to the sequence 0 1 , , , k n N N N and (11) give Now, consider the following condition on the tails of  : There exists a convergent series with positive general term n c such that the series with general term Proposition 3 If condition (13) holds then a.a.paths of ( ) t X are continuous.Proof: Due to (10) and (11) we have ( ) the preceding inequality being due to the inequality ( ) for any 0 1 q ≤ ≤ which is a conse- quence of finite increments theorem.
Due to (13) we then get that the series with general term  ( ) Then by Borel-Cantelli lemma, we have for a.a.ω , ( ) ( ) for n large enough so that the series which is the general term of a convergent series.■ which are more explicitly described by the following hierarchical model

DBM Based on Random Walks
We will rather consider centered variables Now, consider the following random walk ( ) S ∈ in Dirichlet random environment, starting from 0: , Var Since the i X 's are independent with zero mean, we , For any integer 1 n ≥ and real number 0 where [ ] x denotes the integer part of x .Let B σ = denote a zero mean Brownian motion with variance 2 σ , B denoting the standard Brownian motion.Proposition 5 For any , we have in the space of distributions (

DBM
A Brownian motion in Dirichlet random environment (BMDE) is a process Z such that are orthogonal, are mixtures of Gaussians but need not be independent.Indeed, since

Regularity
, where ( ) , then it is a gaussian random variable with 0 mean and variance

Sethuraman Stick-Breaking Construction
Sethuraman (1994) [13] has shown that the sequence of random distributions ( ) ( ) ( ) converges to the Dirichlet process when the random weights ( ) i Q are defined by the following stick-breaking construction: ( ) , for any 2, , .

Simulation Algorithm
A path of the BMDE ( ) , c σ process ( ) can be simulated as follows: Let , , , K q q q q =  from (19) Draw ( )

Estimation
Using proposition 6 we can show that where since for every { } 0, , j k ∈  , j η and ( ) fixed.First, let ( ) i Y be a sequence of random variables such that be small enough and let K be the stick-breaking precision Draw ( ) This enables us to define with standard techniques, the stochastic integral converges uniformly and defines a continuous function of t .Thus for a.a.ω , ( ) ( ) n t S ω t X are continuous.
The set of random step processes will be denoted by Observe that the assumption that the j η are to be step  t The assumption that the j η are square integrable ensures that ( )f t is square integrable for each .tThe stochastic integral of step f ∈  is defined as Let t and s two reals numbers such that s t ≤ , let k ∈  such that