- Research Article
- Open Access
A Criterium for the Strict Positivity of the Density of the Law of a Poisson Process
© Rémi Léandre. 2011
- Received: 26 August 2010
- Accepted: 9 January 2011
- Published: 24 January 2011
We translate in semigroup theory our result (Léandre, 1990) giving a necessary condition so that the law of a Markov process with jumps could have a strictly positive density. This result express, that we have to jump in a finite number of jumps in a "submersive" way from the starting point to the end point if the density of the jump process is strictly positive in . We use the Malliavin Calculus of Bismut type of (Léandre, (2008;2010)) translated in semi-group theory as a tool, and the interpretation in semi-group theory of some classical results of the stochastic analysis for Poisson process as, for instance, the formula giving the law of a compound Poisson process.
- Stochastic Differential Equation
- Implicit Function Theorem
- Stochastic Analysis
- Jump Process
- Semigroup Theory
We are interested in this paper in the following problem.
Let be a random variable given by the solution of a stochastic differential equation, with law . For what is bounded below by , where is strictly positive continuous near ?
This problem was solved by using the Malliavin Calculus. See the survey paper of Léandre  on that. For various applications of the Malliavin Calculus on heat kernels, we refer to the review of Kusuoka , Léandre , and Watanabe .
If is a submersion in some sense in , then we can apply in some sense the implicit function theorem in order to get a lower bound of the law of by a measure having a strictly positive density in the values of in with respect of the Lebesgue measure on . The problem is that the solution of the stochastic differential equation (1.3) is only almost surely defined. So the use of the implicit theorem leds to some difficulties which were overcome by Bismut in . The use of Bismut's procedure allows to  to solve Problem*. See  for a translation of the proof of  in semigroup theory.
Plenty of the standard tools of stochastic analysis were translated recently by Léandre in semigroup theory. See the review [9, 10] on that. Problem* was solved for a diffusion by using the Malliavin Calculus of Bismut type in semigroup theory in .
Tortrat  studied the support of the law of : if and , then belong to . If has a finite mass , then the process has the law of a compound Poisson process: is sum of his jumps. There is only a finite number of jumps. The jumps are all independents with law and the times where the jumps occur follow the law of a standard Poisson process with parameter . We will give a proof, uniquely based upon algebraic computations on semi groups, of this fact in the paper.
Problem* was solved in  by using the Malliavin Calculus for jump processes (see [15–17] for related works). is called the Lévy measure. For that we need some regularity on the Lévy measure . Under regularity assumption on ,  used another time the implicit function theorem, when we can jumps in a finite number of jumps in a "submersive" way between the starting point and the end point. Recently we have translated in semigroup theory plenty of tools of the stochastic analysis for Poisson processes [18–23]. Our goal is to translate in semigroup theory the result of .
For material on stochastic differential equations driven by jump processes, we refer to the books [13, 24, 25]. For the analytic side of the theory of Markov processes with jumps, we refer to the books [11–13].
This paper enters in a general program which would like that stochastic analysis tools become available for partial differential equation different of the parabolic equations whose generators satisfy the maximum principle [26, 27].
Let us consider functions positive with compact support on continuous except in 0 equal to near 0 with .
Let us introduce functions with bounded derivatives at each order, equal to 0 in 0 with values in .
We do the following hypothesis.
There exists an such that the family of vectors generates .
If , then there exists , such that such that,
is a submersion in .
has only a finite number of jumps because its Lévy measure is of finite mass and its law gives a good approximation of the law of if is small enough!
if is bounded differentiable. If , the is classically related to fractional powers of the Laplacian .
We do the following Hypothesis.
The condition implies that there exists , , , and , such that:
(ii) is a submersion in .
The law of is a good approximation of the law of if is small enough. This express the fact that by a finite number of jumps, has to pass from to in a submersive way if .
It is a bounded operator on the space of continuous bounded functions endowed with the uniform norm. It generates therefore a semigroup .
Theorem 3.1 (compound Poisson process).
In order to simplify the exposition, we suppose .
We suppose that the total mass of is finite and is equal to the constant quantity and that depends continuously of for the strong topology. generates a semigroup on the space of continuous functions endowed with the uniform norm.
It is a bounded operator on the set of uniformly bounded functions endowed with the uniform topology. Therefore it generates a semigroup on the set of bounded continuous functions. We get the following translation of (2.20) in semigroup theory.
The measure has a smooth density , and, when , uniformly.
when when . Therefore the result is obtained.
has a density , and, when , tends uniformly to .
where is a polynomial in , and of valuation 1 in and its derivatives. The result will come from the next lemma.
and the result goes. It remains to remark that under the previous condition has a polynomial behaviour whose component tends to zero and to apply Theorem 3 of . This comes from the fact that is a polynomial in and is differentiable bounded in because we keep only bounded values of due to the apparition of .
Proof of Theorem 2.2.
If , there exists a , such that .
To simplify the exposition, we suppose that .
One of the measure in the above sum has a stricly positive density in , and, therefore, nearby . So there exists for close from , such that
The matrix has an inverse bounded by .
It remains to remark that the Gram matrix associated to is equal to is larger to and to apply the implicit function theorem.
belong to , the space of invertible matrices, and belong to . is called the Malliavin matrix.
and generate Markov semigroup and .
It has a density . When the density of tends uniformly to in . When , the density tends uniformly in to . Therefore, if , we can find and such that .
We suppose to simplify the exposition that .
is strictly positive in !
From (5.11), we see that there exists such that, for some , we have for close from
is bounded by .
But the Gram matrix associated to is equal to . It has therefore an inverse bounded by . The result arises by the implicit function theorem.
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