On some stochastic differential equations with jumps subject to small positives coefficients

1.   Introduction

Specially, let us consider $\varepsilon$ (viscosity coefficient) and $\delta_{\varepsilon}$ (scaling parameter, written as a function of $\varepsilon$) both be non-negatives, such that

$\textbf{H.1}$ $\quad \lim_{\varepsilon\to 0}\dfrac{\delta_{\varepsilon}}{\varepsilon}: = \gamma$ $\ \left(\gamma\in\mathbb{R}_{+}^{*}\right)$.

Our concern is the following $d$-dimensional stochastic differential equations (SDE) :

where $\left\lbrace W_{t}:\ t\geqslant 0\right\rbrace$ is a $d$-dimensional standard Brownian motion and $L^{\varepsilon, \delta_{\varepsilon}}: = \left\lbrace L_{t}^{\varepsilon, \delta_{\varepsilon}}:\ t\geqslant 0\right\rbrace$ is a Poisson point process with continuous compensator, independent of $W$, both defined on a given filtered probability space $\left(\Omega, \mathcal{F}, \mathbb{P}, \mathbb{F}\right)$ with $\mathbb{F}: = \left\lbrace\mathcal{F}_{t}:\ t\geqslant 0 \right\rbrace$ being the $\mathbb{P}$-completion of the filtration $\mathcal{F}$. More precisely, we assume that $L^{\varepsilon, \delta_{\varepsilon}}$ takes the form:

where $k$ is a given predictable function (see ^[10], Chap. Ⅳ Sect. 9), and $N$ is a Poisson random counting measure on $\mathbb{R}^{d}$ with mean Lèvy measure (or intensity measure) $\nu$. The coefficients $b, c, \sigma$ are subject to suitable conditions.

The family of equation (1.1) subject to a combination effect of homogenization and large deviations, is a classical problem which goes back to Paolo Baldi ^[1] at the end of 20'th century. Such a problem, when $L_{t}^{\varepsilon, \delta_{\varepsilon}}\equiv 0$, has been extensively investigated by Freidlin and Sowers ^[9]. They have shown three classical regimes depending on the relative rate at which the small viscosity coefficient $\varepsilon$ and the scaling parameter $\delta_{\varepsilon}$ tend to zero. They have provided some effective rate functions associated to a large deviation principle (LDP) for SDE which have been used as direct applications to wavefront propagation. Recently, we have focused the problem (1.1) in the sense that homogenization dominates (see ^[4]), at the same time, we have provided lower and upper bounds of LDP for SDE which allows jumps processes. The purpose of the present article is to carry out the program outlined in ^[4] regarding the following case: The small coefficients ($\varepsilon$ and $\delta_{\varepsilon}$) go at the same rate, which is technically more difficult. Indeed, in this case, the logarithm moment generating function of $X^{x, \varepsilon, \delta_{\varepsilon}}_{T}$, noted $g_{T, x}^{\varepsilon}$ below, may not have an explicit value limit. Thanks to Baxendale and Stroock ^[2], we overcome this challenge by using the superior and inferior limits of the function $g_{T, x}^{\varepsilon}$.

Definition 1. Let $X^{x, \varepsilon, \delta}$ be a $\mathbb{R}^{d}$-valued random variable and let $\mathbb{P}_{\varepsilon, \delta}$ denote its distribution on the Borel subsets of $\mathbb{R}^{d}$, that is, $\mathbb{P}_{\varepsilon, \delta }\left(A\right) = \mathbb{P}\left(X^{x, \varepsilon, \delta}\in A \right)$. The family $\left\lbrace X^{x, \varepsilon, \delta}; \varepsilon, \delta\geqslant 0\right\rbrace$ satisfies an LDP if there exists a lower semi-continuous function $I: \mathbb{R}^{d}\longrightarrow\left[0, +\infty\right]$ such that

● for each open set $A\subseteq\mathbb{R}^{d}$ $\liminf\limits_{\varepsilon\to 0}\varepsilon\log\mathbb{P}\left(X^{x, \varepsilon, \delta_{\varepsilon}}\in A\right)\geqslant-\inf\limits_{x\in A}I(x),$

● for each closed set $B\subseteq\mathbb{R}^{d}$ $\limsup\limits_{\varepsilon\to 0}\varepsilon\log\mathbb{P}\left(X^{x, \varepsilon, \delta_{\varepsilon}}\in B\right)\leqslant-\inf\limits_{x\in B}I(x).$

$I$ is called the rate function for the LDP. A rate function is good if for each $a\in\mathbb{R}_{+}, \ \left\lbrace x: I(x)\leqslant a\right\rbrace$ is compact.

Our program will be as follows. In Section (2) we set up some notation, make precisely our hypothesis and state the main result. In Section (3) we give an overview of some existing characterizations of large deviations, comparing them with ours, and we give the outline of the proofs.

Finally, we wish to say that the subject related to this topic, besides those of the references mentioned earlier can be found in ^[12,15] and references therein.

2.   Preliminaries

2.1.   Notations and formulation

Denote expectation with respect to $\mathbb{P}$ by $\mathbb{E}$ and the gradient operator by $\nabla$. We have already defined $\left\langle., .\right\rangle$ as the standard Euclidean inner product on $\mathbb{R}^{d}$, and $\left\|.\right\|$ as the associated norm. Let $C_{p}\left(\mathbb{R}^{d}, \mathbb{R}^{d}\right)$ be the collection of continuous mapping from $\mathbb{R}^{d}$ into $\mathbb{R}^{d}$ which are periodic of period $1$ in each coordinate of the argument and let $\left\|.\right\|_{C_{p}\left(\mathbb{R}^{d}, \mathbb{R}^{d} \right) }$ be the associated sup norm. Let $\mathbb{T}^{d}$ be the $d$-dimensional torus of size $1$, and let $\left\|. \right\|_{C\left(\mathbb{T}^{d}, \mathbb{R}^{d}\right) }$ be the standard sup norm on $C\left(\mathbb{T}^{d}, \mathbb{R}^{d}\right)$, the space of continuous mapping from $\mathbb{T}^{d}$ into $\mathbb{R}^{d}$. Also we define $\mathcal{P}\left(\mathbb{T}^{d}\right)$ as the collection of all probability measure on $\mathbb{T}^{d}$.

The $\left\lbrace \sigma_{i}:\ 1\leqslant i\leqslant d \right\rbrace$ in (1.1) are assumed to be in $C_{p}\left(\mathbb{R}^{d}, \mathbb{R}^{d}\right)$, and we also assume that

We assume that $b, c$ in (1.1) are in $C_{p}\left(\mathbb{R}^{d}, \mathbb{R}^{d}\right)$.

We now turn our attention to the Poisson part. We first consider a Poisson random measure $N^{\varepsilon^{-1}}(., .)$ on $\left[0, T\right) \times\mathbb{R}^{d}$ defined on the space probability $\left(\Omega, \mathcal{F}, \mathbb{P}\right)$, with Lèvy measure $\varepsilon^{-1}\nu$ such that the standard integrability condition holds:

The compensator of $\varepsilon N^{\varepsilon^{-1}}$ is thus the deterministic measure $\varepsilon\hat{N}^{\varepsilon^{-1}}(dtdy): = dt\nu(dy)$ on $\left[0, T\right) \times\mathbb{R}^{d}$. In this paper we shall be interested in Poisson point process of class (QL), namely a point process whose counting measure has continuous compensator (see Ikeda and Watanabe ^[10]). More precisely, in light of the representation theorem of the Poisson point process (^[10], Chap. Ⅱ, Theorem $7.4$), we shall assume that $L^{\varepsilon, \delta_{\varepsilon}}$ is a pure jump process of the following form :

where $k$ is $C_{p}\left(\mathbb{R}^{d}\times\mathbb{R}^{d}, \mathbb{R}^{d}\right)$ with respect to first variable, integrable with respect to $dtdy$, so that the counting measure of $L^{\varepsilon, \delta_{\varepsilon}}$, denoted by $N_{L^{\varepsilon, \delta_{\varepsilon}}}(dtdy)$ takes the form :

and its compensator is therefore $\hat{N}_{{L}^{\varepsilon, \delta_{\varepsilon}}}(dtdy): = k\left(\frac{.}{\delta_{\varepsilon}}, y\right)(s)\mathbb{E}\left[N_{{L}^{\varepsilon, \delta_{\varepsilon}}}(dtdy)\right] = k\left(\frac{.}{\delta_{\varepsilon}}, y\right)(s)\nu(dy)dt$, and hence continuous, i.e., $L^{\varepsilon, \delta_{\varepsilon}}$ has continuous statistic.

The Markov processes $X^{\varepsilon, \delta_{\varepsilon}}$ that we consider include jump processes and diffusion. Next let's write down its generator on twice continuously differentiable functions with compact support by

where the matrix $a: = \left(a_{ij}\right)$ is factored as $a: = \sigma\sigma^{*}$, and $^{*}$ denotes the transpose. We set

and require the followings :

By requirement there exists a $\mathcal{L}_{\gamma}$-diffusion with jumps on $\mathbb{R}^{d}$ and by periodicity assumption on the coefficients such a process induces process $\overline{X}$ which has both a diffusion component and a jump component on the $d$-dimensional torus $\mathbb{T}^{d}$, moreover the $\mathcal{L}_{\gamma}$-diffusion-part process is ergodic. We denote by $m_{\gamma}$ its unique invariant measure. To move the SDE (1.1) to the torus $\mathbb{T}^{d}$, we define the pull-back $\overline{X}^{\varepsilon, \delta_{\varepsilon}}_{t}: = \frac{1}{\delta_{\varepsilon}}X^{x, \varepsilon, \delta_{\varepsilon}}_{\left(\delta_{\varepsilon}/\sqrt{\varepsilon}\right)^{2}t}$ which satisfies the SDE :

where

The infinitesimal generator $\overline{\mathcal{L}}_{\varepsilon, \delta_{\varepsilon}}$ of the diffusion component associated to the process $\overline{X}^{\varepsilon, \delta_{\varepsilon}}$ is

This generator tends to $\mathcal{L}_{\gamma}$ defined in (2.5), as $\varepsilon\to 0$.

2.2.   Usual formulas

The proof of the main Theorem 4 (below) relies on explicit calculation of the logarithm moments of $X^{\varepsilon, \delta_{\varepsilon}}$ and the followings Girsanov's formula and Itô's formula. Before proceeding, let us introduce some space.

For $E$ locally compact, let $\mathcal{H}^2(T, \lambda)$ be the linear space of all equivalence classes of mappings $F:[0, T] \times E \times \Omega \longrightarrow \mathbb{R}$ which coincide almost everywhere with respect to $dt \otimes d\lambda \otimes d\mathbb{P}$ and which satisfy the following conditions :

● $F$ is predictable;

● $\int_0^T \int_{E\backslash\{0\}}\mathbb{E}\Big(| F(t, z)|^2\Big) dt \lambda(dz) < + \infty.$

We endow $\mathcal{H}^2(T, \lambda)$ with the inner product $\left\langle F, G \right\rangle_{T, \lambda}: = \int_0^T \int_{E\backslash\{0\}}\mathbb{E}\Big(F(t, z)G(t, z)\Big) dt \lambda(dz).$ Then, it is well know that $\left(\mathcal{H}^2(T, \lambda); \left\langle., . \right\rangle_{T, \lambda}\right)$ is a real separable Hilbert space.

Let $N_{p}$ be a Poisson random measure on $\mathbb{R}_{+}\times\left(E\backslash\{0\}\right)$ with intensity measure $\lambda$, according to a given $\mathcal{F}_{T}$-adapted, $\sigma$-finite point process $p$ which is independent of the Brownian motion $W$. Let $\widetilde{N}_{p}$ be the associated compensated Poisson random measure. Now we have (see, D. Applebaum ^[3] Chapter $5$, Section $2$)

Lemma 2 (Girsanov's formula). Let $X$ be a Lèvy process such that $e^{X}$ is a martingale, i.e.,

with

We suppose that there exists $C > 0$ such that

For $L \in \mathcal{H}^2 (T, \lambda)$ we define

Set

and we suppose that

Finally, we define

Let $\boldsymbol{Q}$ be the probability measure on $(\Omega, \mathcal{F}_T)$ defined as:

Then under $\boldsymbol{Q}$, $B_t$ is a Brownian motion and $N_t$ is a $\boldsymbol{Q}$-martingale.

Next define a $d$-dimensional semi-martingale $Y_{t}: = \left(Y_1, \ldots, Y_d\right)$ by

where

● $M_{t}$ is locally continuous square integrable $\left(\mathcal{F}_{t}\right)$-martingale and $M_{0}: = 0$;

● $A_{t}$ is a continuous $\left(\mathcal{F}_{t}\right)$-adapted process whose almost all sample functions are of bounded variation on each finite interval and $A_{0}: = 0$;

● $g$ is $\left(\mathcal{F}_{t}\right)$-predictable and for $t > 0$, $\int_{0}^{t}\int_{E}\left\| g(s, z, .)\right\| \lambda(dz)ds < +\infty\ a.s$;

● $f$ is $\left(\mathcal{F}_{t}\right)$-predictable and for $t > 0$, $\int_{0}^{t}\int_{E}\left\| f(s, z, .)\right\| ^{2}\lambda(dz)ds < +\infty\ a.s$.

We have (see, for example Ikeda and Watanabe ^[10] Theorem $5.1$)

Lemma 3 (Itô's formula). Let $F$ be a function of class $C^{2}$ on $\mathbb{R}^{d}$ and $Y_t$ a $d$-dimensional semi-martingale given in (2.8). Then the stochastic process $F\left(Y_{t} \right)$ is also a $\left(\mathcal{F}_{t}\right)$-semi-martingale and the following formula holds :

2.3.   The main result

A version of the next theorem (without jumps) can be seen in Baxendale and Stroock ^[2] (Corollary 1.12). This is a very long one and we now have all the necessary arguments to be able to copy it, mutatis mutandis, by replacing it back in our frame. For this reason, we will content ourselves with the statement of the theorem, referring the careful reader to the aforementioned paper.

Before proceeding, we give some definitions. Denote by $C^{\infty}\left(\mathbb{T}^{d}\right)$ the space of functions of class $C^{\infty}$ on ${\mathbb{T}}^{d}$. For all function $\phi\in C^{\infty}\left(\mathbb{T}^{d}\right)$ and all probability measure $\mu\in\mathcal{P}\left(\mathbb{T}^{d}\right)$, let us set

Define

Now fix $\mu\in\mathcal{P}\left(\mathbb{T}^{d}\right)$, fix also $\phi\in C^{\infty}\left(\mathbb{T}^{d}\right)$, by assumption on $a$, the matrix $a_{\mu}(\phi)$ is strictly positive-definite. Then letting $a^{-1}_{\mu}(\phi)$ be its inverse matrix, we define the norm $\Big\|\theta \Big\|_{a_{\mu}^{-1}(\phi)}: = \sqrt{\left\langle\theta, a_{\mu}^{-1}(\phi)\theta\right\rangle }$ for all $\theta\in\mathbb{R}^{d}$. Next, we define

where $S_{\nu}$ denotes the support of $\nu$. Let ${\rm con}\left(Q_{a_{\mu}(\phi), \nu}\right)$ be the smallest convex cone that contains $Q_{a_{\mu}(\phi), \nu}$ and define

Let ${\rm ri}(A)$ denotes the relative interior of a set $A$. In light of the characterization lemma (see Ikeda and Watanabe ^[10], Lemma $10.2.3$) to the effective domain of $\mathcal{J}(\theta): = \inf\limits_{\theta'\in\mathbb{R}^{d}}\left\lbrace \left\langle\theta, \theta' \right\rangle-\overline{\mathcal{J}}\left(\theta'\right)\right\rbrace$, we have

By the requirements (2.1) and (2.2), it is well know that

where $\ \varrho(r): = r\log r-r+1, \quad r\in\left(0, +\infty\right)$.

Finally, let us define, for $T > 0$ and $x\in\mathbb{R}^{d}$,

Now we state our main result.

Theorem 4 (Main result). For all $T > 0$, we assume that the hypothesis $(\boldsymbol{H.1})$ and $(\boldsymbol{H.2})$ hold true. Suppose that

exists uniformly with respect to $x\in {\mathbb{R}}^{d}$. In addition

Then we have

where

and we have used $\Psi_{\theta}$ to denote the unique $u\in C^{\infty}\left({\mathbb{T}}^{d}\right)$ which satisfies

and with

Finally, set

Then for every set $\Gamma$ on $\mathcal{B}\left(\mathbb{R}^{d}\right)$,

3.   Outline of the LDP characterization

3.1.   Overview and discussion

Freidlin-Wentzell's theory is interested in the behaviour of a stochastic process seen as the Brownian perturbation of an ordinary differential equation (ODE). For the description and the background for ODE over-amortised by a Brownian martingale, we refer to ^[5,8,13] and the papers therein. At the same time, non-linear stochastic evolution equations have been studied in various literatures. However, there are still few results on the large deviation for stochastic evolution equations with jumps (see, for example ^[12,15]).

In all the references listed above the question is: What do SDE's trajectories look like when the viscosity parameter $\varepsilon$ is small? The answer is twofold depending on the time scales considered: Finite time interval or increasingly large time scale. Another question can also be considered: "When the process is ergodic and has an invariant distribution $\mu_{\varepsilon}$, what is the asymptotic behaviour of $\mu_{\varepsilon}$ when $\varepsilon\to 0$?"

This is the question in all the references available for us in this direction, with the one important exception given by the paper ^[9] (see also ^[1]). In ^[9] large deviations problems arising in stochastic homogenization are discussed. The scope of our approach based on the LDP with combination of homogenization, differs substantially from that of ^[9].

Our version of SDE contains a substantial novelty, which makes it possible to handle the Freidlin-Wentzell's large deviation arising in stochastic homogenization for non-linear evolution equations with Poisson jumps and Brownian motions. To explain this modified construction in the most transparent way, we took the "jump component" in a relatively simple form, that is, we consider Poisson point process of class (QL).

There are several toolbox for the large deviations (see, for example ^[6]). The way we choose allows us to characterize the LDP with help of the analysis of the logarithmic moment generating function.

We refer to Baxendale et al. ^[2] for the characterization and the general results on the LDP with help of the analysis of the logarithmic moment generating function. However, on this way we meet substantial difficulties already when we try to prove that the LDP holds. Initially the corresponding rate function (this one is denoted by $I$) is identified as the Legendre transform (or convex conjugate) of the logarithmic moment generating function defined, when the limit exists, as

Subsequently, the Cameron-Martin-Girsanov transformation is used to find an alternative expression for $I$. Although from the standpoint of large deviation theory it is the rate function $I$ which is of paramount interest, it is the logarithmic moment generating function $g^{T, x}_{\gamma}$ which is most important for the analysis of Lyapunov exponents. Indeed, one sees that $g^{T, x}_{\gamma}$ is here what is called the moment Lyapunov function.

3.2.   LDP results

The proof of the main result can be conducted in six steps, which we outline below.

Proof. (Theorem 4)

Step 1: (calculation of the moment Lyapunov function).

The calculation of this limit is inspired by Freidlin ^[8], Lemma 2.1 (without jumps) Section 7.2.

For any $\theta\in {\mathbb{R}}^{d}$

exists uniformly in $x\in {\mathbb{R}}^{d}$, with

being the largest eigenvalue of the differential operator

Step 2: (change of probability measure).

We set $\hat{X}_{t}^{\varepsilon, \delta_{\varepsilon}}: = \dfrac{1}{\delta_{\varepsilon}}X_{t}^{x, \varepsilon, \delta_{\varepsilon}}$, by the Itô's formula we have for all $\phi\in C^{\infty}\left({\mathbb{T}}^{d}\right)$

Let us define for $z\in {\mathbb{T}}^{d}$

So, we are going to consider the logarithm moment generating function of $X^{x, \varepsilon, \delta_{\varepsilon}}$, $g_{T, x}^{\varepsilon}$ (2.10). By Girsanov's formula, we have

where $\widetilde{ {\mathbb{E}}}$ is the expectation operator with respect to the probability $\widetilde{\mathbb{P}}$ defined as

Step 3: (ergodicity).

Let us set, for all $z\in {\mathbb{T}}^{d}$, for all $\theta\in {\mathbb{R}}^{d}$ :

and let us set $\Psi_{\theta}\in C^{\infty}\left({\mathbb{T}}^{d}\right)\ $ be the unique solution of

Such a solution $\Psi_{\theta}$ must exist again by the assumptions on the coefficients (see, for instance, Pardoux et al. ^[11]). So applying Itô formula to $\left(\frac{\delta_{\varepsilon}}{\sqrt{\varepsilon}}\right)^{2} \Psi_{\theta}\left(\overline{X}\right)$, we have

Then putting (3.3) into the formula (3.1), we have

Step 4: (upper bound).

First, we have

Secondly, we can see

Thirdly, we observe

In fact, we notice that

and

From (3.5) we have

Then, for all $\theta\in {\mathbb{R}}^{d}, \ T > 0$, we have

Step 5: (lower bound).

As it has already been seen previously, we then have

Step 6: (Fentchel-Legendre transform).

Let us set for $\ \theta\in {\mathbb{R}}^{d}$, $X_{1}$ be a Gaussian random vector with logarithm moment generating function

and $X_{2}$ be a stationary Poisson process on ${\mathbb{R}}^{d}$ independent of $X_{1}$ with logarithm moment generating function

We notice that

Let $\Lambda_{1}(\theta)$ and $\Lambda_{2}(\theta)$ denote respectively the Fenchel-Legendre transform of $\overline{\Lambda}_{1}$ and $\overline{\Lambda}_{2}$, we have

and

Since $\overline{\mathcal{J}}(\theta)$ is the logarithm moment generating of $X_{1}+X_{2}$, it follows that its Fenchel-Legendre transform is

For the final assertion of the Theorem 4, we can observe that $\mathcal{J}_{\gamma}$ is convex.

We also need to know that $X_{t}^{x, \varepsilon, \delta_{\varepsilon}}$ is exponentially tight in the path space. Indeed, let $\mathcal{D}\left([0, T], {\mathbb{R}}^{d}\right)$ be the space of functions that map $[0, T]$ into ${\mathbb{R}}^{d}$, which are right continuous and having left hand limits. $\mathcal{D}\left([0, T], {\mathbb{R}}^{d}\right)$ is metricated by the Skorohod metric, with respect to which it is complete and separable. Then, as in Proposition 3.2 ^[9], we have

Remark 5. We assume that the hypothesis $(\boldsymbol{H.1})$ and $(\boldsymbol{H.2})$ hold true. For any fixed $T > 0, \ x\in {\mathbb{R}}^{d}$ and $\alpha\in(0, 1/2)$,

To complete this part, we point out that the previous proof also provides the LDP in the path space $\mathcal{D}\left([0, T], {\mathbb{R}}^{d}\right)$. Let us consider some definitions :

Since the function $\mathcal{J}$ is convex we can show that

So we have

Remark 6. For all $T > 0$, we assume that the hypothesis $(\boldsymbol{H.1})$ and $(\boldsymbol{H.2})$ hold true. Then for every $x\in {\mathbb{R}}^{d}$, the family $\left\lbrace X^{x, \varepsilon, \delta_{\varepsilon}}_{T}: \varepsilon > 0\right\rbrace\ $ of $\ \mathcal{D}\left([0, T], {\mathbb{R}}^{d}\right)$-valued random variables has a large deviations principle with good rate function $\ S_{0, T}(\varphi)\ $ for all $\varphi\in\mathcal{D}\left([0, T], {\mathbb{R}}^{d}\right)$.

4.   Conclusion

We have derived the general expression of the action functional for stochastic processes with jumps. In particular, we generalize the Regime 2 considered in Freidlin-Sowers ^[9] for some classical SDE, as well as the version in Baldi ^[1] for a family of measures on a topological vector space. Furthermore, we have analysed a general LDP arising in stochastic homogenization and we have extended the result in ^[4].

Before finishing, we notice that the large deviations established in this paper generalizes Schilder's theorem, which is useful both to probabilist who are interested in the trajectories consequences of stochastic process and to statisticians who are interested in the weights of small balls given by the Wiener measure. Such results can have implications in both random optimization and Bayesian statistics when studying trajectories of posterior density estimates or regression problems. For example, one may consult the following works to see how highly theoretical probabilistic developments can rebound on unexpected statistical applications ^[7,14].

Acknowledgements

The authors would like to thank, in advance, the anonymous referee(s) for their valuable comments that helped clarify and refine this paper.

Conflict of interest

The authors declare no conflict of interest.