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Global well-posedness of a class of stochastic equations with jumps
Advances in Difference Equations volume 2013, Article number: 175 (2013)
The existence and uniqueness of a mild solution to stochastic equations with jumps are established, a stochastic Fubini theorem and a type of Burkholder-Davis-Gundy inequality are proved, and the two formulas are used to study the regularity property of the mild solution of a general stochastic evolution equation perturbed by jump processes. As applications, the regularity of a stochastic heat equation with jump is given.
In recent years, there have been many monographs concerning stochastic partial differential equations with Lévy jump and their applications in physics, economics, statistical mechanics, fluid dynamics and finance etc. For this theory and its applications, one can see [1–3] and references therein. In this article, the existence, uniqueness, regularity for the mild solution of stochastic partial differential equations with Lévy jump are studied. There are a lot of works dealing with existence and uniqueness for stochastic partial differential equations with jump processes. In , the existence and uniqueness for solutions of stochastic reaction diffusion equations driven by Poisson random measures are obtained. In , Malliavin calculus is applied to study the absolute continuity of the law of the solutions of stochastic reaction diffusion equations driven by Poisson random measures. In , a minimal solution is obtained for the stochastic heat equation driven by non-negative Lévy noise with coefficients of polynomial growth. In , a weak solution is established for the stochastic heat equation driven by stable noise with coefficients of polynomial growth. In , the existence and uniqueness for solutions of stochastic generalized porous media equations with Lévy jump are obtained. In , the strong solutions to a large class of stochastic equations with Lévy noise are obtained in variational framework. And it is shown in  that the results can be applied to stochastic reaction-diffusion equations, Burgers-type equations, 2D Navier-Stokes equations, p-Laplace equations and porous media equations with locally monotone perturbations.
The main aim of this paper is to study the existence, uniqueness and regularity of the stochastic equation
In , the intensity measure λ of is finite, while the intensity measure in this article is σ-finite, and also the classical Lipschitz condition (2.5) in  is relaxed to condition (A.5) in this article. The author of this article proves the existence and uniqueness of a mild solution of (1.1), the continuity of the solution with respect to initial data. And then a stochastic Fubini theorem is established for the compensated Poisson random measure whose intensity measure is σ-finite compared to a finite case in . Furthermore, a new type of the Burkholder-Davis-Gundy inequality, which is more precise than Lemma 2.2 in , is gotten. The two formulas are basic in stochastic analysis. Using the formulas, the author gets the regularity property of a mild solution of (1.1) without conditions (2.15) and (2.16) in  which are critical there.
The article is organized as follows. In Section 2, we present the framework. Existence, uniqueness and regularity are proved in Section 3. In Section 4, two examples including stochastic heat equations with Lévy jump are given.
2 Some preliminaries
Let , be separable Hilbert spaces and Z be a Banach space. Let denote the space of all Hilbert-Schmidt operators from U to H, and set . Let , be a cylindrical Wiener process in U on a probability space with a normal filtration , and let be a Poisson random measure associated with the compensated -martingale measure defined as , where defined on a measurable space Z is the intensity measure of . In this article we study the following equation:
with initial condition , and the coefficients in equation (2.1) satisfy the following conditions:
(A.1) is the infinitesimal generator of a -semigroup , .
(A.2) is -measurable, and there exists a positive constant C satisfying
(A.3) is strongly continuous, i.e. the mapping
is continuous from H to H for each .
(A.4) For all and , we have
(A.5) There is a square integrable mapping such that
for all and .
(A.6) There is a positive constant C like (A.2) such that
where is measurable.
For fixed , we define:
Obviously, is a Banach space. For technical reasons we define , on as
Then, for and , we have
For the reader’s convenience, before giving our main results, we cite some theorems here which will be needed later.
Contraction theorem (i) Let and be two Banach spaces. Let the mapping satisfy
for all and . Then there exists exactly one mapping such that
for all .
If we assume in addition that the mapping is continuous from Λ to E for all , we get that is continuous.
If the mapping is not only continuous from Λ to E for all , but there even exists an such that
for all , then the mapping is Lipschitz continuous.
If a process Ψ is adapted to , , and stochastically continuous with values in a Banach space E, then there exists a predictable version of Ψ.
Definition 2.1 An H-valued predictable process , is called a mild solution of (2.1) if
for each .
3 Existence, uniqueness and regularity
Theorem 3.1 Assume that conditions from (A.1) to (A.6) hold, then there exists a unique mild solution of (2.1) with the initial condition
In addition we get that the mapping
is Lipschitz continuous.
Proof Fix , and , and define
In the following two steps, it comes to prove that
The first step: It is proved that the mapping ℱ is well defined.
Let , as
By the strongly continuity of , we know is predictable. And together with (A.2), we know the Bochner integral , , is well defined.
Let and be an orthonormal basis respectively of U and H. Because
is predictable, and by (A.5), it is easy to see . So, the stochastic integral is well defined.
Similarly to 1, it is easy to see that is predictable, and by (A.6) we have
Therefore the stochastic integral is well defined.
The second step: We prove that for all and .
Obviously , , is an element of .
There is a version of the second summand , , which is an element of .
Similarly to the argument of the first step, we know is adapted for . Then we will show is continuous a.s. Let , . By (A.2), we have
In the above inequalities, we set and , where C is as in (A.2). As we can derive that
Letting , and then , by (A.2), strong continuity of and dominated convergence theorem, we know, for all most , we have
Similarly we can get the same conclusion for when , so we have proved is continuous a.s. for .
As we have
Therefore we have proved .
There is a version of , , which is in .
First we fix , then
Let us define
Then it is clear that . By (A.2), we have
Similarly we have and is -measurable. Next we are going to show
is continuous in the mean square and therefore stochastically continuous. Indeed, let , we get
The last step follows by a dominated convergence theorem since and , .
For , we can get the same conclusion. Therefore by Theorem  we know
has a predictable version for all . Furthermore, by (A.5), we have
So far we have proved has a predictable version which is an element in .
There is a version of , , which is in . It is easy to show is predictable. By (A.6), we have
Letting , we have
letting and , by a dominated convergence theorem, we have the following result:
If , similarly we can get
As is adapted, thus by Theorem  we know
has a predictable version which is an element in .
The third step: We are going to show
is a contraction mapping for all .
Let , and . Then we have
where . For the first term of the right-hand side of (3.1), we have
Dividing by both sides of (3.2), we have
Obviously as .
For the second term of (3.1), by the Burkholder-Davis-Gundy inequality and (A.5), we have
Dividing by both sides of (3.3), we have
By the Burkholder-Davis-Gundy inequality and (A.6), we have the following estimate for the third term of the right-hand side of (3.1):
Thus we have
Therefore we have finally proved that there exists an with
So, there exists a unique
which is the unique solution of (2.1).
The fourth step : We will show the Lipschitz continuity of . By contraction theorem (iii), we only have to prove that the mapping
is Lipschitz continuous for all , where the Lipschitz constant does not depend on Y.
But this is obvious for all and , as
Before giving the regularity of the mild solution of (2.1), we first need a stochastic Fubini theorem with respect to compensated Poisson measure. Set , let be σ-algebra generated by open subsets in Z, be the predictable σ-algebra of , be the product of the Lebesgue measure, P and on . .
Proposition 3.1 Let be a finite measure space and let be a -measurable mapping such that
the process indexed by
is progressively measurable and belongs to ,
the process indexed by
has an -measurable version such that
Proof (1) follows from the following inequality: Let f be a nonnegative -measurable function. Then
and we get (3.4) by the Schwarz inequality. Now suppose that m in (2) exists. Then, by taking Ω instead of in (3.4), we get
by the Burkholder inequality. Hence
is defined P-almost everywhere. Now take satisfying the assumption of the proposition such that the sequence of the integrals
converges to zero. Then there exists a subsequence such that:
in for μ-almost all .
in for μ-almost all .
Let us introduce the set of all -measurable processes ψ with
such that there exists an m satisfying (2) and (3). It is easy to see that is a linear space and if we can find such that
then we would finish the proof. Indeed, take the corresponding functions . Then the sequence is Cauchy in due to (3.5) and ψ belongs to due to (a) and (b). Now we will show how to construct the approximating sequence . We assume that is finite. By Lemma A.1.4 in  we can find mappings on H. The simple functions take values in the finite dimensional subspace of H. Moreover,
and if , , then . Now to show that , we will take advantage of the fact that each is bounded in H and
if and only if
for uniformly bounded in H. So as is of the form
where is a -decomposition of and , are elements in the finite dimensional subspace of H. We conclude that provided due to linearity of . Another reduction shows that this is true if
for every , , as can be approximated by in where is a disjoint union of sets of the type . Finally, as is a predictable process, it can be approximated by simple bounded real processes in . So, we will finish the proof by showing that
for , , but this is obvious. When is σ-finite, there exists a sequence satisfying and . Obviously,
therefore . □
In the following we give a type of the Burkholder-Davis-Gundy inequality which will play an important role in proving the regularity property of the mild solution of (2.1).
Lemma 3.1 If A is the infinitesimal generator of pseudo-contraction -semigroup , , and is an H-valued progressively measurable mapping with respect to such that
then for any ,
for some number .
Proof Let us define
Assume that takes values in and
Since A is the infinitesimal generator of a -semigroup, A is closed. And by (3.6) we can check that
for all and
By (3.6) and Proposition 3.1 with Y replaced with , we have
By Itô’s formula, we get
So, we have
Since A is pseudo-contraction, there exists such that
Define . Then by (3.7), (3.8) and B-D-G inequality, we can get the following estimate:
So, by Gronwall inequality, we have
By the Fatou lemma, we have
Without assuming takes values in , we define where , , is the resolvent of A, then we know takes values in and satisfies (3.6) under the following condition:
Then instead , satisfies (3.9). By the Burkholder-Davis-Gundy inequality and dominated convergence theorem, we have
Then it is easy to check
So, under condition (3.10), (3.9) also follows without assumption that takes values in . In order to relax (3.10), we define stopping times
Therefore by Fatou lemma, the assertion follows. □
In order to study the regularity property of the mild solution of (2.1), we introduce an approximation system of (2.1) in the following:
where which is the resolvent set of A, and , is the resolvent of A. We say a stochastic process is if each of it’s sample path is right continuous with left limit. So a stochastic process is called a strong solution of (2.1), if it is , -adapted, and satisfies (2.1).
Theorem 3.2 Let and -measurable. In addition to assumptions in Theorem 3.1 we assume A is dissipative with . Then the mild solution of (2.1) is RCLL.
Proof Let , obviously are bounded operators. So (3.11) has unique strong solution. In fact by Theorem 3.1 we know (3.11) has a unique mild solution denoted by and the following hold:
Thus by Fubini theorem, we have
On the other hand, by the stochastic Fubini theorem for Q-Wiener processes in , we have
And by Proposition 3.1
Hence, is integrable almost surely and
So far, we prove , , is the unique strong solution of (3.11). Let be the mild solution of (2.1). Then we consider
for any . We have that for any ,
When l is big enough, there exists constant M such that , so we have
Then we consider
where as . By (A.6) and Lemma 3.1 we have
Finally we consider
It is easy to check
By dominated convergence theorem, we have
By Burkholder-Davis-Gundy inequality, we have
for . So, by dominated convergence theorem and the fact that
for , we get
By Lemma 3.1, we get
After discussing about , , we can get the following estimate
where as . By Gronwall theorem, we have
Therefore we know is on a.s. □
Example 4.1 Consider the following stochastic heat equation
where , , is a real standard Brownian motion, is a Poisson compensated martingale measure with character measure on ℝ, and f is a real Lipschitz continuous function on H satisfying for some and . Let with , then we have
For the sake of simplicity, we assume . It is easy to check that (A.1)-(A.5) are satisfied. So Theorem 3.1 and Theorem 3.2 hold true for (4.1).
Example 4.2 Denote a process satisfying . Then we consider the semilinear stochastic partial differential equation:
for some negative constant , and some , is a bounded function with for all , is nonlinear and Lipschitz continuous with , , , , is an H-valued Q-wiener process with covariance operator Q, .
Let and with domain
So, it is easy to deduce
It is easy to check that (A.1)-(A.5) are satisfied. So, Theorem 3.1 and Theorem 3.2 hold true for (4.3).
The author read and approved the final manuscript.
Applebaum D: Lévy Processes and Stochastic Calculus. Cambridge University Press, Cambridge; 2004.
Protter PE: Stochastic Integration and Differential Equations. 2nd edition. Springer, New York; 2004.
Peszat S, Zabczyk J: Stochastic Partial Differential Equation with Lévy Noise: An Evolution Equation Approach. Cambridge University Press, Cambridge; 2007.
Albeverio S, Wu JL, Zhang TS: Parabolic SPDES driven by Poisson white noise. Stoch. Process. Appl. 1998, 74: 21-36. 10.1016/S0304-4149(97)00112-9
Fournier N: Malliavin calculus for parabolic SPDES with jumps. Stoch. Process. Appl. 2000, 87: 115-147. 10.1016/S0304-4149(99)00107-6
Mueller C: The heat equation with Lévy noise. Stoch. Process. Appl. 1998, 74: 67-82. 10.1016/S0304-4149(97)00120-8
Mytnik L: Stochastic partial differential equations driven by stable noise. Probab. Theory Relat. Fields 2002, 123: 157-201. 10.1007/s004400100180
Zhou GL, Hou ZT: Stochastic generalized porous media equations with Lévy jump. Acta Math. Sin. 2011, 27: 1671-1696. 10.1007/s10114-011-9194-8
Brzeźniak, Z, Liu, W, Zhu, J: Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise. arXiv:1108.0343. e-printatarXiv.org
Luo JW, Liu K: Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps. Stoch. Process. Appl. 2008, 118: 864-895. 10.1016/j.spa.2007.06.009
Röckner M: Introduction to Stochastic Partial Differential Equation. Purdue University Press, West Lafayette; 2007.
The author is thankful to the referee for careful reading and insightful comments which led to many improvements of the earlier version. The work was supported by Tianyuan Foundation of NSF (Grant No. 11126079), CPSF (Grant No. 2013M530559) and Fundamental Research Funds for the Central Universities (Grant No. CDJRC10100011).
The author declares that he has no competing interests.
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Cite this article
Zhou, G. Global well-posedness of a class of stochastic equations with jumps. Adv Differ Equ 2013, 175 (2013). https://doi.org/10.1186/1687-1847-2013-175
- stochastic evolution equation
- Lévy processes
- mild solution