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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.

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 [

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 estimation of the parameters of DBM.

Let

The following celebrated random distribution defined by Ferguson [

For

Ferguson has also shown that for a.a.

such that

Let

paper. The probability distribution of a r.v.

For any integer

Definition 1 A sequence

Using transpositions, first notice that (2) implies that all the

and also

The variables

An i.i.d. sequence is of course exchangeable but an exchangeable sequence needs neither be independent nor Markov.

For example a sequence

Another interesting example of exchangeable sequence is a sample

The following celebrated theorem states that an exchangeable sequence is somewhat conditionally i.i.d. as in the preceding example. It was first established by de Finetti (1931) [

Theorem 1 (de Finetti-Hewitt-Savage) Let

In other words, (5) shows that the distribution of an exchangeable sequence is a mixture with mixing measure

Applying (5) with

In the example of a sample from the Dirichlet process

For the rest of the paper it is assumed that

We follow L. Gallardo [

Let

For any integer

that is

The functions

Let

Note that

and

The functions

Now, let

Notice that (3) and (4) then imply that

are constants which do not depend on

Let

Then

Proposition 2 The series with general term

defines a stochastic process.

Proof: Due to (10) we have

and then (12) applied to the sequence

Then

Now, consider the following condition on the tails of

There exists a convergent series with positive general term

Proposition 3 If condition (13) holds then a.a. paths of

Proof: Due to (10) and (11) we have

and (5) implies

the preceding inequality being due to the inequality

Due to (13) we then get that the series with general term

Then by Borel-Cantelli lemma, we have for a.a.

As a corollary observe that

Proposition 4 For a sample of

Proof: Condition (13) holds for

that is

holds for any positive number

which is the general term of a convergent series. ■

Let

First, let

which are more explicitly described by the following hierarchical model

We will rather consider centered variables

Now, consider the following random walk

so that we have

It is straightforward that

Since the

Therefore

For any integer

where

Let

Proposition 5 For any

where

A Brownian motion in Dirichlet random environment (BMDE) is a process

Proposition 6 If

The increments

we see that

Theorem 7 Let

so that there exist a continuous version of (Z_{t})

Proof:

Since

where

conditional to

Sethuraman (1994) [

converges to the Dirichlet process when the random weights

A path of the BMDE

Let

Draw

Draw

Compute

Put

Using proposition 6 we can show that

Consider

where

Proposition 8 For

where

This enables us to define with standard techniques, the stochastic integral

for any continuous function

Proposition 9 The stochastic process

Proof:

Let

where

since for every

On the other hand for every

consequently,

In this paragraph we shall give an expression of Itô formulae of the process

Proposition 10

Proof:

Since

Suppose that

For almost surely

On the other hand for almost surely

Therefore according to the dominus convergence theorem,

this means that

as required. ■

Proposition 11 Let f be a bounded and 2 times derivable function, then

We have extended Brownian motion in dirichlet random environment for the application on the Dirichlet hierarchical models in order to deal with temporal data such as solutions of SDE with stochastic drift and volatility. It can be thought that the process on which are based these parameters belongs to a certain well-known class of processes, such as continuous time Markov chains. Then, we think that a Dirichlet prior can be put on the path space, that is a functional space. It seems to us that the estimation procedure in such a context is an interesting topic for future works.