Large Deviations for a Slow–Fast System with Jump-Diffusion Processes

A slow–fast system with jump-diffusion processes is considered. The large deviations are established via the weak convergence approach, which is based on the variational representations for functional of Poisson random measure and Brownian motion. We present an example to verify that the level of the asset price satisfies large deviations with small volatility.


Introduction
In this paper, we consider the following slow-fast system where is associated Poisson measure, and (dz) × 1 dt is the intensity measure with the Lévy measure satisfying ∫ (1 ∧ z 2 ) (dz) < ∞ with a locally compact Polish space [1]. W t is a d-Brownian motion independent of N 1 (⋅, ⋅) . Two small positive parameters and , satisfying = o( ) , are used to describe the separation of time scale between the slow variables x ( , ) t and the fast variables y ( , ) t . The initial data (x ( , ) 0 , y ( , ) 0 ) of this system are given by (x 0 , y 0 ) ∈ ℝ d × ℝ d . Equation (1.1) can be widely-used in diverse areas, for instance, modeling the level of the asset price in financial economics [2]. The purpose of this paper is to prove the slow variables of the system (1.1) satisfy large deviations with some rate function.
The study of large deviations results for slow-fast systems with diffusions has attracted many researchers' attention (cf. e.g., [3] Chap. 7, [4] and references therein). The authors in [5] use methods from weak convergence and stochastic control to obtain large deviation properties of stochastic dynamical systems with rapidly fluctuating coefficients. If the systems are coupled in the sense that the coefficients may depend on both the slow and the fast variables, a large deviation principle is obtained for the case of systems in [6]. As for the case of jump-diffusions, there are very few large deviation results. In [7], the authors applied PDE method to study the large deviations for a special case of dynamical systems in which the slow variable is a diffusion, while the fast variable is a mean-reverting process driven by a Lévy process. For a general slow-fast system where the fast process is ergodic and the slow one is perturbed by small noise, large deviations results were established in [8,9] . Their method based on nonlinear semigroups and viscosity solution theory proposed in [10,11]. However, it is a non-trivial task to verify the corresponding conditions when applying the viscosity solution theory to jump-diffusion systems.
In our work, a weak convergence argument in [12] is applied. This weak convergence method is constructed based on a bounded Laplace transformation variation on Polish space [13], and it has been applied to a wide variety of stochastic models [14,15]. The advantage of this approach is that the required conditions are easy to be verified. With the variational representation for functional of Brownian motion and Poisson random measure, one can get that the exponential functional of the original slow component is a variational infimum over a family of controlled slow component. Accordingly, the study on large deviation principle is turned into basic qualitative properties for the controlled system, which is so-called weak convergence approach. However, the presence of controls in the system makes the asymptotic analysis challenging, an interplay between large deviation for the slow component and averaging with respect to the fast component. It is to be observed in [16,17] that a Khasminskii type averaging principle plays an key role in the weak convergence for the controlled system as = o( ) → 0 with → 0 . In our work, the results depend on the limit of ∕ . Thanks to the particular regime that = o( ) , in the limit there is no control in the fast component. To reach our aim, we divide into two parts. Besides, some additional technical treatments are needed. Firstly, with the Itô's formula, Burkholder-Davis-Gundy inequality, and so on, the controlled slow component weakly converges to the auxiliary slow component. Then with aid of the exponential ergodicity of auxiliary uncontrolled fast component, the weak convergence of auxiliary slow component is obtained. To the best of our knowledge, this is the first large deviations result for a slow-fast system with jump-diffusions based on the weak convergence argument.
The plan of this paper is as follows. In the next section, we establish notation, give some precise conditions for the slow-fast system (1.1) and review some preliminary results. In Sect. 3, we state and prove our main results. And we also give an application to the so-called Black-Scholes or Samuelson model with leverage effect in Sect. 4. Throughout this paper, c, C, c 1 , C 1 , … denote certain positive constants that may vary from line to line.

Preliminaries
Fix T ∈ (0, ∞) and let T = [0, T] × . We denote by B( ) the Borel -field on . And the space of all measure on ( , B( )) will be denoted as and denote by ℙ the unique probability measure on ( , B( )) under which the canonical map, N ∶ → , N(m) = m , is a Poisson random measure with intensity measure T . We also consider, for > 0 , the probability measures ℙ on ( , We can define the analogous canonical map Ñ and the analogous coordinate maps denoted again as is underlying probability space, we will write (t, w, m, z) , (w, m) ∈̃ , as (t, z) for simplicity. For any ∈Ã , define a counting process N on T by and a random variable L (1) . which is the local rate function for a standard Poisson process.
and U n be the space of controls To ensure existence and uniqueness of solutions to the system (1.1), we assume can deduce from [1,Chap. 6] that the slow-fast system (1.1) admits a unique solution ) . Moreover, to study the large deviation principle for the system (1.1) we need the following assumptions.

Main Result
Now, we give the statement of our main theorem.
Proof The subsequent proof consists of four steps.
Step 1. Let (v (j) , u (j) ) and (v, u) belong to S n t } is a family of solutions to the Eq. (2.1), that is, to the skeleton Eq. (2.1), which is uniformly bounded and equicontinuous. So x (j) t converges to x as j → ∞ , that is, Step 2. Considering the following controlled system associated to (1.1), where (v ( , ) , u ( , ) ) ∈ U n is called a pair of control, it is not hard to check that there exists a unique solution (x ( , ) t ,ŷ ( , ) t ) of the system (3.4) in × . In this part, we will establish the estimates Using the Itô's formula, one can get Since the final part in the right hand side of (3.9) is a martingale term, the Gronwall's inequality yields Similarly, by using the Itô's formula, we obtain where (3.7) (3.12) where From (3.11) and (3.12), Then by the Gronwall's inequality, we obtain By (A1) and the Hölder inequality, we have (3.14) sup Due to (A5) and the Burkholder-Davis-Gundy inequality, By using the Gronwall's inequality, it implies from (3.13), (3.14) and (3.15) Thus, we have proved the estimates (3.5). Moreover, considering the auxiliary processes similarly, we can also obtain Step 3. Assume (v ( , ) , u ( , ) ) ∈ U n such that (v ( , ) , u ( , ) ) weakly converges to (v, u) as → 0 . Reformulating the slow variables of system (3.4) as follows, we will show that as → 0 , x ( , ) t weakly converges to x t , that is, where By Assumption (A2) and using Hölder inequality, (3.20)