1. 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 estimation of the parameters of DBM.
2. Ferguson-Dirichlet Process
Let
be a fixed probability space. Let
be a Polish space and let
denote the set of all probability measures defined on
. The distribution of a random variable, say
, will be denoted by ether
or
.
The following celebrated random distribution defined by Ferguson [7] plays a central role in our construction. Let
be a finite positive measure on
. A random distribution
is a Dirichlet process
if for every
and every measurable partition
of
, the joint distribution of the random vector
has a Dirichlet distribution with parameters
Ferguson proved that this definition satisfies the Kolmogorov criteria which yields the existence of such random distributions.
For
, let
denote the Poisson-Dirichlet distribution with parameter
(Kingman [8] ) which support is the set
![]()
Ferguson has also shown that for a.a.
,
is a discrete probability measure: there exist an i.i.d. sequence of random variables on
, say
, and a sequence of random weights
verifying:
(1)
such that
![]()
Let
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.
will be denoted
. Equality in distribution is denoted by
.
For any integer
, let
denote the group of permutations of
.
Exchangeable Random Variables
Definition 1 A sequence
of r.v.s is said to be exchangeable if for all ![]()
(2)
Using transpositions, first notice that (2) implies that all the
have the same distribution, say
:
(3)
and also
(4)
The variables
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.
For example a sequence
of centered Gaussian variables with
and
is exchangeable but not i.i.d.
Another interesting example of exchangeable sequence is a sample
from a Dirichlet process
with precision parameter
and mean parameter
[7] :
![]()
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) [9] in the case of Bernoulli variables and by Hewitt-Savage (1955) [10] in the general case. Very elegant proofs can be found in Meyer (1966) [11] p. 191- 192 and Kingmann (1978) [8] .
Theorem 1 (de Finetti-Hewitt-Savage) Let
be an exchangeable sequence with values in
. Then there exists a probability measure
on
such that
(5)
(6)
(7)
In other words, (5) shows that the distribution of an exchangeable sequence is a mixture with mixing measure
, (6) shows that
is the distribution of the weak limit empirical measure and finally (7) shows that if
is considered as a parameter
, then
is a sufficient statistic for estimating
.
Applying (5) with
it is seen that the mean
of
, defined as
, is equal to the common distribution of the
:
(8)
In the example of a sample from the Dirichlet process
,
is nothing but the Dirichlet process itself, by definition of such a sample [7] , while
(9)
For the rest of the paper it is assumed that
the real line.
3. DBM Constructions
3.1. DBM Based on Ciesielski Construction
We follow L. Gallardo [12] pp. 79-80 and 206-208.
Let
![]()
For any integer
and
let
![]()
that is
,
.
The functions
and
for
and
constitute what is called the Haar Hilbertian basis of
.
Let
![]()
Note that
is a nonnegative triangle function with support in
so that
(10)
and
(11)
The functions
and
consitute the so called Schauder system.
Now, let
be a an exchangeable sequence such that
for one (and any)
.
Notice that (3) and (4) then imply that
(12)
are constants which do not depend on
and
.
Let
![]()
Then
Proposition 2 The series with general term
converges in
and
![]()
defines a stochastic process.
Proof: Due to (10) we have
![]()
and then (12) applied to the sequence
and (11) give
![]()
Then
and
so that
converges in
. ■
Now, consider the following condition on the tails of
:
There exists a convergent series with positive general term
such that the series with general term
(13)
Proposition 3 If condition (13) holds then a.a. paths of
are continuous.
Proof: Due to (10) and (11) we have
![]()
and (5) implies
![]()
the preceding inequality being due to the inequality
for any
which is a conse- quence of finite increments theorem.
Due to (13) we then get that the series with general term
converges.
Then by Borel-Cantelli lemma, we have for a.a.
,
for
large enough so that the series
converges uniformly and defines a continuous function of
. Thus for a.a.
,
is continuous. ■
As a corollary observe that
Proposition 4 For a sample of
, a.a. paths of
are continuous.
Proof: Condition (13) holds for
with
. Indeed, since
![]()
that is
![]()
holds for any positive number
, we have for any ![]()
![]()
which is the general term of a convergent series. ■
3.2. DBM Based on Random Walks
Let
and
be fixed.
First, let
be a sequence of random variables such that
![]()
which are more explicitly described by the following hierarchical model
(14)
We will rather consider centered variables
![]()
Now, consider the following random walk
in Dirichlet random environment, starting from 0:
![]()
so that we have
![]()
It is straightforward that
![]()
Since the
’s are independent with zero mean, we have
![]()
Therefore
is finite a.e. or equivalently, for
a.a. ![]()
(15)
For any integer
and real number
let
(16)
where
denotes the integer part of
.
Let
denote a zero mean Brownian motion with variance
,
denoting the standard Brownian motion.
Proposition 5 For any
, we have in the space of distributions
![]()
where
is defined in (15).
3.3. DBM
A Brownian motion in Dirichlet random environment (BMDE) is a process
such that
![]()
Proposition 6 If
is BMDE then its conditional increments are independent Gaussians
![]()
The increments
are orthogonal, are mixtures of Gaussians but need not be independent. Indeed, since
![]()
we see that
![]()
4. Regularity
Theorem 7 Let
be as in (ref) then
![]()
so that there exist a continuous version of (Zt)
Proof:
![]()
Since
then
![]()
where
Conditional to the
,
is a linear combina- tion of
, then it is a gaussian random variable with 0 mean and variance
![]()
conditional to
.
5. Simulation and Estimation
5.1. Sethuraman Stick-Breaking Construction
Sethuraman (1994) [13] has shown that the sequence of random distributions
(17)
converges to the Dirichlet process when the random weights
are defined by the following stick-breaking construction:
(18)
(19)
5.2. Simulation Algorithm
A path of the BMDE
process
can be simulated as follows:
Let
be small enough and let
be the stick-breaking precision
Draw
from (19)
Draw
with ![]()
Compute
by truncating (15)
Put
and draw
points
such that ![]()
5.3. Estimation
Using proposition 6 we can show that
(20)
6. Stochastic Calculus
Consider
the natural filtration defined by
, that is
the sigma algebra generated by
A random process
is a step process if there exist a finite sequence of numbers
and square integrable random variables
such that
(21)
where
is
-measurable for
The set of random step processes will be denoted by
Observe that the assumption that the
are to be
-measurable ensures that
is adapted to the filtration
The assumption that the
are square integrable ensures that
is square integrable for each
The stochastic integral of
is defined as
(22)
Proposition 8 For
, we have
and
![]()
where ![]()
This enables us to define with standard techniques, the stochastic integral
![]()
for any continuous function
.
Proposition 9 The stochastic process
is a
-martingale
Proof:
Let
and
two reals numbers such that
, let
such that
, let ![]()
![]()
where
since for every
,
and
is
-measurable then,
![]()
On the other hand for every
using the zero means of increment
conditional to
.
![]()
consequently,
![]()
Itô Formulae
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
and for any ![]()
![]()
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
![]()
7. Conclusion
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.