Transportation distance between the Lévy measures and stochastic equations for Lévy-type processes

The notion of the transportation distance on the set of the Lévy measures on R is introduced. A Lévy-type process with a given symbol (state dependent analogue of the characteristic triplet) is proved to be well defined as a strong solution to a stochastic differential equation (SDE) under the assumption of Lipschitz continuity of the Lévy kernel in the symbol w.r.t. the state space variable in the transportation distance. As examples, we construct Gammatype process and u�-stable like process as strong solutions to SDEs.


Introduction
Recently a wide variety of models were proposed in physics, chemistry, biology and econometrics, where the stochastic fluctuations are distributed according to the Lévy law, instead of the more traditional Gaussian one (see for example [1] and the list of references therein).If the parameters in such models are state dependent, then one should deal with a Lévy-type process.A typical example here is model with so-called Lévy flights, where the correspondent Lévy measure is alpha-stable The particular case corresponds to the model with porous environment [2], and in this case the parameter  = () is state dependent.By definition, the Lévy-type process is such a Markov process that its generator on the functions of the class  2 ∞ = { ∶  ∈  2 ,  () → 0,  ′ () → 0, as || → ∞} takes the form  () = () ′ () + Here functions  ∶ ℝ → ℝ,  2 ∶ ℝ → [0; ∞) and a Lévy kernel (, ) (a measurable function w.r.t. and a Lévy measure for any  ∈ ℝ) are the state dependent analogue of the characteristic triplet of a Lévy process.Lévy-type processes also may appear as limiting distributions in theorems of Skorokhod's invariance principletype with the convergence of step-wise processes constructed via Markov chains (see [8,7]).One of the most natural ways to construct and characterise such processes is to use an SDE approach, e.g. to define the required process as a (strong) solution to a proper SDE.Naturally, one could expect that the uniqueness and existence of solution to such SDE remain valid under Lipschitz continuity type assumptions on the members of the (state-dependent) characteristic triplet, which leads us to the question: what should be the proper form for the Lipschitz continuity condition for the Lévy kernel?To answer this question we consider a transportation distance on the set of Lévy measures on ℝ.
The main idea of the transportation distance construction is to present any Lévy measure  uniquely as a transformation of a fixed infinite Lévy measure  0 by means of a function  from some prescribed family of functions, and then compare the correspondent functions for two measures.Below we show that for a given  the respective function  is unique in the properly chosen class and the transportation distance is a metrics on the set of Lévy measures on ℝ.The main gain of this method is that we can obtain any Poisson point measure, with a Lévy kernel as a Lévy measure by means of such function , from the measure  0 with the Lévy measure  0 .This technique allows one to write a Lévy-type process with a given state dependent characteristic triplet as a solution to an SDE with  0 as a Poisson random noise.The solvability of such SDEs and the solution properties are the questions under consideration in this paper.
The idea to present any Lévy kernel as an image of some fixed measure goes back to the works of Skorokhod [9] and Strook [10].The authors therein treat the equation with indirectly defined function .The conditions for uniqueness and existence of the solution to such SDE are quite implicit.It appears that in our approach the explicit transportation distance construction allows one to give the conditions in more transparent form.This paper is organised as follows: first we introduce the notion of the transportation distance and formulate the theorem about the uniqueness and existence of the strong solution to an SDE for Lévy type process.As examples we construct Gammatype and -stable like processes as strong solutions to SDEs.

Transportation distance
Recall that a Lévy measure on ℝ is a measure  on ℝ such that It is convenient for our further purposes to introduce two slightly unusual conventions.First, we admit that a Lévy measure assigns a non-trivial mass to the point 0; that is, it contains a term  0 ().This term can be even infinite; that is  ∈ [0; ∞] in general.Second, for two such measures  1 ,  2 we write if they coincide up to the term  0 ().In other words, Proposition 2. The function  defined above is a metric on the set of the Lévy measures on ℝ.
Taking into account the proposition above, we call the function  as a transportation distance.Proofs of Propositions 1 and 2 are of a technical nature, and we postpone them to Appendix A in order not to overload the text.
Theorem 1. Assume that, for some decomposition (5) and some  ⩽ , the following conditions hold true.
1.The functions ã and  satisfy the Lipschitz condition w.r.t., i.e. there exists a constant  1 > 0 such that 2. There exists a constant  2 > 0 such that Then there exists a unique strong solution to the equation ( 4).This solution is a Markov process, and its generator on the class  2 ∞ takes the form (1). Proof of Theorem 1: uniqueness.In order not to overload the notation we prove only the case where  =  = 1; that is, in fact we shall deal with equation (3).This will not restrict generality, because (a) dealing with  Lipschitz terms instead of one can be made literally in the same way; (b) it is a standard observation that adding the terms with bounded "total intensity" of jumps does not spoil an existence and uniqueness result because we can separate the time instants where the jumps with the bounded total intensity occur, and resolve consequently the SDE on the intervals between these time instants.
Consider two solutions of the equation ( 3) with the same starting point: The standard argument here, say, for SDE's with square integrable noise, would be to use the Itô formula and the Gronwall lemma to prove that ( 1 () −  2 ()) 2 = 0. Now because of possible lack of square integrability we shall modify this argument.Namely, we shall prove that for every  ∈ [0, 1] To do this, we apply the smoothing cut-off technique developed in [3].Namely we shall apply the Itô formula to After rearrangements, we get finally the formula where is a martingale, and Observe that  ∧ 1 ⩽ 4  arctan ,  ⩾ 0. (13) In addition, for the function  and its derivatives, we have the following explicit expressions and bounds: Using that, we bound every term in the r.h.s. of (12) for any triple (,  1 ,  2 ) which satisfy  =  1 −  2 .Observe that  () =  1 () −  2 (), hence to estimate the right hand side of ( 10) we can restrict our consideration to this class of triples (,  1 ,  2 ).Estimate ( 16) and condition 1 yield Estimates (15), (13) and condition 1 yield To estimate  3 (,  1 ,  2 ), we re-write it in the following way where and note that in any case the absolute value of the  (,  1 ,  2 ) does not exceed  + 4.
In the case when |( 1 , ) − ( 2 , )| ⩽ 1, we have the inequality which comes from (17).Hence, after simple re-arrangements, we get Next, the function  is Lipschitz with the constant 1, hence the absolute value of the inner function in the last integral is bounded by Recall that  =  1 −  2 , hence by the Lipschitz condition on (, ⋅) we have Next, combining the first inequality in ( 14) and the second one in (15), we get Hence, using (13) once again, we can write Consequently by (13) we get (7).This completes the proof of the uniqueness part.

Proof of
Using the same arguments as for the estimate (22) we can get for the processes  +1 () =  +1 () −  () the following Using the estimates (13) and where Therefore there exists a process  such that it satisfies equation (3) and Passing to the limit in (24) will yield that  is a solution to (3); this argument is standard and we omit the details.The proof of the Markov property for  is standard as well, and it is omitted.The formula (1) for its generator follows from the Itô formula in a standard way, and again, we omit the details.

The Gamma-type process
Let us consider Gamma process with the Lévy measure () = − ,  > 0, ,  > 0.Here parameter  can be interpreted as a rate of jump arrivals, and  as an effective size of a jump.The proposition below gives the estimates for the transportation distance between two Gamma measures, with one of the above parameters being fixed and other one varying.
For the proof we refer the reader to Appendix B. We call a Gamma-type process a Lévy-type process with a characteristic triplet (0, 0, (, )) where a Lévy kernel is (, ) = ( ) − ( ) ,  > 0. If we assume that only the size of jumps varies, i.e. the parameter  is fixed, then the corresponding SDE can be easily written in the form: where  1 denotes the Gamma process with the same parameter  and () = 1.
Meanwhile the case where the intensity of jumps  = () varies is not so easy to treat because, heuristically, one should introduce the change of time into the Lévy-Itô formula.
Nevertheless using Theorem 1 in this case is also manageable.If () is Lipschitz continuous, then all the conditions of Theorem 1 are verified.The condition 2 follows from (30), the condition 3 is easy to verify straightforwardly.Thus Gamma-type process in this case can be written as a solution to equation (3) with (, ) = ( 1 ( ) )/.