1 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 $\alpha =\alpha (x)$ is state dependent. By definition, the Lévy-type process is such a Markov process that its generator on the functions of the class ${\mathcal{C}_{\infty }^{2}}=\{f:f\in {C}^{2},\hspace{2.5pt}f(x)\to 0,\hspace{2.5pt}{f^{\prime }}(x)\to 0,\hspace{2.5pt}\text{as}\hspace{2.5pt}|x|\to \infty \}$ takes the form
Here functions $a:\mathbb{R}\to \mathbb{R}$, ${\sigma }^{2}:\mathbb{R}\to [0;\infty )$ and a Lévy kernel $\varPi (x,dy)$ (a measurable function w.r.t. x and a Lévy measure for any $x\in \mathbb{R}$) 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 principle-type 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 $\mathbb{R}$.
(1)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle Af(x)& \displaystyle =a(x){f^{\prime }}(x)+\frac{1}{2}{\sigma }^{2}(x){f^{\prime\prime }}(x)\\{} & \displaystyle \hspace{1em}+\int _{\mathbb{R}}\big(f(x+y)-f(x)-{f^{\prime }}(x)y\mathbf{1}_{|y|\leqslant 1}\big)\varPi (x,dy)\\{} & \displaystyle \hspace{1em}+\int _{\mathbb{R}}\big(f(x+y)-f(x)\big)\mathbf{1}_{|y|>1}\varPi (x,dy),\hspace{1em}x\in \mathbb{R}.\end{array}\]The main idea of the transportation distance construction is to present any Lévy measure Π uniquely as a transformation of a fixed infinite Lévy measure $\varPi _{0}$ by means of a function c from some prescribed family of functions, and then compare the correspondent functions for two measures. Below we show that for a given Π the respective function c is unique in the properly chosen class and the transportation distance is a metrics on the set of Lévy measures on $\mathbb{R}$. 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 c, from the measure $\nu _{0}$ with the Lévy measure $\varPi _{0}$. This technique allows one to write a Lévy-type process with a given state dependent characteristic triplet as a solution to an SDE with $\nu _{0}$ 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 c. 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 Gamma-type and α-stable like processes as strong solutions to SDEs.
2 Main results
2.1 Transportation distance
Recall that a Lévy measure on $\mathbb{R}$ is a measure Π on $\mathbb{R}$ 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 $m\delta _{0}(du)$. This term can be even infinite; that is $m\in [0;\infty ]$ in general. Second, for two such measures $\varPi _{1}$, $\varPi _{2}$ we write
if they coincide up to the term $m\delta _{0}(du)$. In other words, $\varPi _{1}\doteq \varPi _{2}$ iff
for any $g:\mathbb{R}\to \mathbb{R}$ such that $|g(u)|\leqslant {u}^{2}\wedge 1$, $u\in \mathbb{R}$.
Proposition 1.
Let us fix a Lévy measure
and let Π be an arbitrary Lévy measure. Then there exists a unique function $c:\mathbb{R}\to \mathbb{R}$, such that $\varPi =\varPi _{0}\circ {c}^{-1}$ and c satisfies the additional assumptions
Example 1.
Let $\varPi (du)=\frac{du}{{u}^{2}}\mathbf{1}_{u>0}$, then the function c equals $c(v)=v\mathbf{1}_{v>0}$. Note that this function maps all the mass from $[-\infty ;0]$ to the point 0, consequently we get a measure $\tilde{\varPi }(du)=\frac{du}{{u}^{2}}\mathbf{1}_{u>0}+\infty \cdot \delta _{0}$ as an image of $\varPi _{0}$ under this c. This is exactly the reason for us to introduce the conventions we’ve discussed above.
2.2 An SDE for a Lévy-type process
Let us consider a state dependent characteristic triplet $(a(x),{\sigma }^{2}(x),\varPi (x,\cdot ))$ and the function $c(x,v)$ such that $\varPi (x,du)=\varPi _{0}(\{v:c(x,v)\in du\})$ and for every fixed x the function $c(x,\cdot )$ has the properties listed in Proposition 1. In the same spirit with §1, Chap. 4 [4], one can write the following SDE, whose solution, if exists, is a natural candidate to be a Levy-type process with the given state dependent characteristic triplet:
where $X(0)=x_{0}$, $x_{0}\in \mathbb{R}$, $\nu _{0}$ is a Poisson point measure with the intensity measure $\varPi _{0}$, W is independent of $\nu _{0}$ Wiener process and $\tilde{\nu }_{0}(dt,dv)=\nu _{0}(dt,dv)-\varPi _{0}(dv)dt$. However, a thorough look at some particular classes of processes (see Section 3.2 below) shows that it will be more flexible to consider more general SDE
where
is some decomposition of the given Lévy kernel, $c_{\varPi _{k}}(x,\cdot )$ are corresponding functions transporting $\varPi _{0}$ to $\varPi _{k}(x,\cdot )$, $\nu _{1},\dots ,\nu _{n}$ are independent copies of the Poisson point measure $\nu _{0}$, and $\tilde{\nu }_{1},\dots ,\tilde{\nu }_{n}$ are corresponding compensated Poisson point measures.
(3)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle dX(t)& \displaystyle =a\big(X(t)\big)dt+\sigma \big(X(t)\big)dW(t)\\{} & \displaystyle \hspace{1em}+\int _{\mathbb{R}}c\big(X(t-),v\big)\mathbf{1}_{|c(X(t-),v)|\leqslant 1}\tilde{\nu }_{0}(dt,dv)\\{} & \displaystyle \hspace{1em}+\int _{\mathbb{R}}c\big(X(t-),v\big)\mathbf{1}_{|c(X(t-),v)|>1}\nu _{0}(dt,dv),\end{array}\](4)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle dX(t)& \displaystyle =a\big(X(t)\big)dt+\sigma \big(X(t)\big)dW(t)\\{} & \displaystyle \hspace{1em}+{\sum \limits_{k=1}^{n}}\bigg[\int _{\mathbb{R}}c_{\varPi _{k}}\big(X(t-),v\big)\mathbf{1}_{|c_{\varPi _{k}}(X(t-),v)|\leqslant 1}\tilde{\nu }_{k}(dt,dv)\\{} & \displaystyle \hspace{1em}+\int _{\mathbb{R}}c_{\varPi _{k}}\big(X(t-),v\big)\mathbf{1}_{|c_{\varPi _{k}}(X(t-),v)|>1}\nu _{k}(dt,dv)\bigg],\end{array}\]Denote
Theorem 1.
Proof of Theorem 1: uniqueness.
In order not to overload the notation we prove only the case where $n=l=1$; that is, in fact we shall deal with equation (3). This will not restrict generality, because (a) dealing with l 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 $\mathbf{E}{(X_{1}(t)-X_{2}(t))}^{2}=0$. Now because of possible lack of square integrability we shall modify this argument. Namely, we shall prove that for every $t\in [0,1]$
To do this, we apply the smoothing cut-off technique developed in [3]. Namely we shall apply the Itô formula to
see Chap. 2, [5]. To use the Itô formula we need an auxiliary construction. For some $\varepsilon >0$ we can rewrite the process $Y(t)=X_{1}(t)-X_{2}(t)$ as follows
(6)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle dX_{i}(t)& \displaystyle =a\big(X_{i}(t)\big)dt+\sigma \big(X_{i}(t)\big)dW(t)\\{} & \displaystyle \hspace{1em}+\int _{\mathbb{R}}c\big(X_{i}(t-),v\big)\mathbf{1}_{|c(X_{i}(t-),v)|\leqslant 1}\tilde{\nu }_{0}(dt,dv)\\{} & \displaystyle \hspace{1em}+\int _{\mathbb{R}}c\big(X_{i}(t-),v\big)\mathbf{1}_{|c(X_{i}(t-),v)|>1}\nu _{0}(dt,dv),\hspace{1em}i=1,2.\end{array}\](8)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle Y(t)& \displaystyle ={\int _{0}^{t}}\big(\tilde{a}\big(X_{1}(s)\big)-\tilde{a}\big(X_{2}(s)\big)\big)ds+{\int _{0}^{t}}\big(\sigma \big(X_{1}(s)\big)-\sigma \big(X_{2}(s)\big)\big)dW(s)\\{} & \displaystyle \hspace{1em}+{\int _{0}^{t}}\int _{|v|\leqslant \varepsilon }\big(c\big(X_{1}(s-),v\big)-c\big(X_{2}(s-),v\big)\big)\tilde{\nu }_{0}(dv,ds)\\{} & \displaystyle \hspace{1em}+{\int _{0}^{t}}\int _{|v|>\varepsilon }\big(c\big(X_{1}(s-),v\big)-c\big(X_{2}(s-),v\big)\big)\nu _{0}(dv,ds)\\{} & \displaystyle \hspace{1em}-{\int _{0}^{t}}\int _{|v|>\varepsilon }\big[\tau \big(c\big(X_{1}(s),v\big)\big)-\tau \big(c\big(X_{2}(s),v\big)\big)\big]\varPi _{0}(dv)ds\\{} & \displaystyle \hspace{1em}+{\int _{0}^{t}}\int _{|v|\leqslant \varepsilon }\big[c\big(X_{1}(s),v\big)-\tau \big(c\big(X_{1}(s),v\big)\big)\big]\varPi _{0}(dv)ds\\{} & \displaystyle \hspace{1em}-{\int _{0}^{t}}\int _{|v|\leqslant \varepsilon }\big[c\big(X_{2}(s),v\big)-\tau \big(c\big(X_{2}(s),v\big)\big)\big]\varPi _{0}(dv)ds,\end{array}\]where
Using the Itô formula we get
After rearrangements, we get finally the formula
where
is a martingale, and
Observe that
In addition, for the function F 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 $(y,x_{1},x_{2})$ which satisfy $y=x_{1}-x_{2}$. Observe that $Y(s)=X_{1}(s)-X_{2}(s)$, hence to estimate the right hand side of (10) we can restrict our consideration to this class of triples $(y,x_{1},x_{2})$.
(9)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle F\big(Y(t)\big)& \displaystyle ={\int _{0}^{t}}\big(\tilde{a}\big(X_{1}(s)\big)-\tilde{a}\big(X_{2}(s)\big)\big){F^{\prime }}\big(Y(s)\big)ds\\{} & \displaystyle \hspace{1em}+{\int _{0}^{t}}\big(\sigma \big(X_{1}(s)\big)-\sigma \big(X_{2}(s)\big)\big){F^{\prime }}\big(Y(s)\big)dW(s)\\{} & \displaystyle \hspace{1em}+{\int _{0}^{t}}\frac{1}{2}{\big(\sigma \big(X_{1}(s)\big)-\sigma \big(X_{2}(s)\big)\big)}^{2}{F^{\prime\prime }}\big(Y(s)\big)ds\\{} & \displaystyle \hspace{1em}+{\int _{0}^{t}}\int _{|v|\leqslant \varepsilon }\big[F\big(Y(s-)+\big(c\big(X_{1}(s-),v\big)-c\big(X_{2}(s-),v\big)\big)\big)\\{} & \displaystyle \hspace{1em}-F\big(Y(s-)\big)\big]\tilde{\nu }_{0}(dv,ds)\\{} & \displaystyle \hspace{1em}+{\int _{0}^{t}}\int _{|v|\leqslant \varepsilon }\big[F\big(Y(s)+\big(c\big(X_{1}(s),v\big)-c\big(X_{2}(s),v\big)\big)\big)\\{} & \displaystyle \hspace{1em}-\big(c\big(X_{1}(s),v\big)-c\big(X_{2}(s),v\big)\big){F^{\prime }}\big(Y(s)\big)-F\big(Y(s)\big)\big]\varPi _{0}(dv)ds\\{} & \displaystyle \hspace{1em}+{\int _{0}^{t}}\int _{|v|>\varepsilon }\big[F\big(Y(s-)+\big(c\big(X_{1}(s-),v\big)-c\big(X_{2}(s-),v\big)\big)\big)\\{} & \displaystyle \hspace{1em}-F\big(Y(s-)\big)\big]\nu _{0}(dv,ds)\\{} & \displaystyle \hspace{1em}-{\int _{0}^{t}}\int _{|v|>\varepsilon }\big[\tau \big(c\big(X_{1}(s),v\big)\big)-\tau \big(c\big(X_{2}(s),v\big)\big)\big]{F^{\prime }}\big(Y(s)\big)\varPi _{0}(dv)ds\\{} & \displaystyle \hspace{1em}+{\int _{0}^{t}}\int _{|v|\leqslant \varepsilon }\big[c\big(X_{1}(s),v\big)-\tau \big(c\big(X_{1}(s),v\big)\big)\big]{F^{\prime }}\big(Y(s)\big)\varPi _{0}(dv)ds\\{} & \displaystyle \hspace{1em}-{\int _{0}^{t}}\int _{|v|\leqslant \varepsilon }\big[c\big(X_{2}(s),v\big)-\tau \big(c\big(X_{2}(s),v\big)\big)\big]{F^{\prime }}\big(Y(s)\big)\varPi _{0}(dv)ds.\end{array}\](10)
\[ \displaystyle F\big(Y(t)\big)=M_{t}+{\int _{0}^{t}}g\big(Y(s),X_{1}(s),X_{2}(s)\big)\varPi _{0}(dv)ds,\](11)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle M_{t}& \displaystyle ={\int _{0}^{t}}\big(\sigma \big(X_{1}(s)\big)-\sigma \big(X_{2}(s)\big)\big){F^{\prime }}\big(Y(s)\big)dW(s)\\{} & \displaystyle \hspace{1em}+{\int _{0}^{t}}\int _{\mathbb{R}}\big[F\big(Y(s-)+\big(c\big(X_{1}(s-),v\big)-c\big(X_{2}(s-),v\big)\big)\big)-F\big(Y(s-)\big)\big]\tilde{\nu }_{0}(dv,ds)\end{array}\](12)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle g(y,x_{1},x_{2})& \displaystyle =\big(\tilde{a}(x_{1})-\tilde{a}(x_{2})\big){F^{\prime }}(y)\\{} & \displaystyle \hspace{1em}+\frac{1}{2}{\big(\sigma (x_{1})-\sigma (x_{2})\big)}^{2}{F^{\prime\prime }}(y)\\{} & \displaystyle \hspace{1em}+{\int _{0}^{t}}\int _{\mathbb{R}}\big[F\big(y+\big(c(x_{1},v)-c(x_{2},v)\big)\big)\\{} & \displaystyle \hspace{1em}-\big[\tau \big(c(x_{1},v)\big)-\tau \big(c(x_{2},v)\big)\big]{F^{\prime }}(y)-F(y)\big]\varPi _{0}(dv)ds\\{} & \displaystyle =g_{1}(y,x_{1},x_{2})+g_{2}(y,x_{1},x_{2})+g_{3}(y,x_{1},x_{2}).\end{array}\](14)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle {F^{\prime }}(y)=\frac{2y}{1+{y}^{4}},\hspace{2em}{F^{\prime\prime }}(y)=2\frac{1-{y}^{4}}{{(1+{y}^{4})}^{2}},\end{array}\](15)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \big|{F^{\prime\prime }}(y)\big|\leqslant 2,\hspace{2em}\big|{F^{\prime }}(y)\big|\leqslant \frac{1+{y}^{2}}{1+{y}^{4}}\leqslant 2,\end{array}\]To estimate $g_{3}(y,x_{1},x_{2})$, we re-write it in the following way
Recall that $y=x_{1}-x_{2}$, hence by the Lipschitz condition on $\varPi (x,\cdot )$ we have
This gives finally
Summarising all the above we get
Putting
we’ll get that
since $M_{t}$ is a martingale and hence $\mathbf{E}M_{t}=0$. Thus by the Gronwall lemma and non-negativity of the function F we get
Consequently by (13) we get (7). This completes the proof of the uniqueness part. □
\[ g_{3}(y,x_{1},x_{2})=\int _{\{v:|c(x_{1},v)-c(x_{2},v)\}|\leqslant 1}+\int _{\{v:|c(x_{1},v)-c(x_{2},v)|>1\}}\big[f(y,x_{1},x_{2})\big]\varPi _{0}(dv),\]
where
\[ f(y,x_{1},x_{2}):=F\big(y+\big(c(x_{1},v)-c(x_{2},v)\big)\big)-F(y)-{F^{\prime }}(y)\big(\tau \big(c(x_{1},v)\big)-\tau \big(c(x_{2},v)\big)\big)\]
and note that in any case the absolute value of the $f(y,x_{1},x_{2})$ does not exceed $\pi +4$. In the case when $|c(x_{1},v)-c(x_{2},v)|\leqslant 1$, we have the inequality
\[ F\big(y+\big(c(x_{1},v)-c(x_{2},v)\big)\big)-F(y)\leqslant {F^{\prime }}(y)\big(c(x_{1},v)-c(x_{2},v)\big)+{\big(c(x_{1},v)-c(x_{2},v)\big)}^{2},\]
which comes from (17). Hence, after simple re-arrangements, we get
\[\begin{array}{r@{\hskip0pt}l}\displaystyle g_{3}(y,x_{1},x_{2})& \displaystyle \leqslant (\pi +4)\int _{\mathbb{R}}\big({\big(c(x_{1},v)-c(x_{2},v)\big)}^{2}\wedge 1\big)\varPi _{0}(dv)\\{} & \displaystyle \hspace{1em}+{F^{\prime }}(y)\int _{|c(x_{1},v)-c(x_{2},v)|\leqslant 1}\big[\big(c(x_{1},v)-c(x_{2},v)\big)\\{} & \displaystyle \hspace{1em}-\big(\tau \big(c(x_{1},v)\big)-\tau \big(c(x_{2},v)\big)\big)\big]\varPi _{0}(dv)\\{} & \displaystyle =(\pi +4){T}^{2}\big(\varPi (x_{1},\cdot ),\varPi (x_{2},\cdot )\big)\\{} & \displaystyle \hspace{1em}+{F^{\prime }}(y)\int _{|c(x_{1},v)-c(x_{2},v)|\leqslant 1}\big[\big(c(x_{1},v)-c(x_{2},v)\big)\\{} & \displaystyle \hspace{1em}-\big(\tau \big(c(x_{1},v)\big)-\tau \big(c(x_{2},v)\big)\big)\big]\varPi _{0}(dv).\end{array}\]
Next, the function τ is Lipschitz with the constant 1, hence the absolute value of the inner function in the last integral is bounded by
\[ \big|c(x_{1},v)-c(x_{2},v)\big|+\big|\tau \big(c(x_{1},v)\big)-\tau \big(c(x_{2},v)\big)\big|\leqslant 2\big(\big|c(x_{1},v)-c(x_{2},v)\big|\wedge 1\big),\]
since the domain of integration is $\{v:|c(x_{1},v)-c(x_{2},v)|\leqslant 1\}$. If, within this domain, we have $|c(x_{1},v)|\leqslant 1$, $|c(x_{2},v)|\leqslant 1$, then $\tau (c(x_{i},v))=c(x_{i},v)$, $i=1,2$ and the inner function in the last integral equals zero. Hence by the Cauchy inequality
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \int _{|c(x_{1},v)-c(x_{2},v)|\leqslant 1}\big[\tau \big(c(x_{1},v)-c(x_{2},v)\big)-\big(\tau \big(c(x_{1},v)\big)-\tau \big(c(x_{2},v)\big)\big)\big]\varPi _{0}(dv)\\{} & \displaystyle \hspace{1em}\leqslant 2\int _{|c(x_{1},v)|>1or|c(x_{2},v)|>1}\big(\big|c(x_{1},v)-c(x_{2},v)\big|\wedge 1\big)\varPi _{0}(dv)\\{} & \displaystyle \hspace{1em}\leqslant 2{\bigg(\int _{{\mathbb{R}}^{2}}\big({\big(c(x_{1},v)-c(x_{2},v)\big)}^{2}\wedge 1\big)\varPi _{0}(dv)\bigg)}^{1/2}\\{} & \displaystyle \hspace{2em}\times {\big(\varPi _{0}\big(\big\{v:\big|c(x_{1},v)\big|>1or\big|c(x_{2},v)\big|>1\big\}\big)\big)}^{1/2}.\end{array}\]
Denote
see condition 3. Then, using the elementary inequality $2ab\leqslant {a}^{2}+{b}^{2}$ we obtain
(18)
\[ \displaystyle g_{3}(y,x_{1},x_{2})\leqslant {\big({F^{\prime }}(y)\big)}^{2}+(\pi +4+2B){T}^{2}\big(\varPi (x_{1},\cdot ),\varPi (x_{2},\cdot )\big).\]
\[ {T}^{2}\big(\varPi (x_{1},\cdot ),\varPi (x_{2},\cdot )\big)\leqslant {L_{2}^{2}}\big(|x_{1}-x_{2}{|}^{2}\wedge 1\big)\leqslant \frac{4{L_{2}^{2}}}{\pi }F(y).\]
Next, combining the first inequality in (14) and the second one in (15), we get
Hence, using (13) once again, we can write
(19)
\[ \displaystyle {\big({F^{\prime }}(y)\big)}^{2}\leqslant 4\big({y}^{2}\wedge 1\big)\leqslant \frac{16}{\pi }F(y),\hspace{1em}y\in \mathbb{R}.\](21)
\[ \begin{array}{r@{\hskip0pt}l}\displaystyle g(y,x_{1},x_{2})& \displaystyle \leqslant \frac{4L_{1}+2{L_{1}^{2}}+16+4{L_{2}^{2}}(\pi +4+2B)}{\pi }F(y).\end{array}\](22)
\[ \displaystyle \mathbf{E}F\big(Y(t)\big)\leqslant Q{\int _{0}^{t}}\mathbf{E}F\big(Y(s)\big)ds\]Proof of Theorem 1: existence.
Consider the sequence of the successive approximations for the solution of the equation (3):
Using the same arguments as for the estimate (22) we can get for the processes $Y_{n+1}(t)=X_{n+1}(t)-X_{n}(t)$ the following
Using the estimates (13) and
we obtain
where $\tilde{Q}=2Q$. Consequently
Therefore there exists a process X such that it satisfies equation (3) and
Passing to the limit in (24) will yield that X is a solution to (3); this argument is standard and we omit the details.
(24)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle X_{0}(t)& \displaystyle =x_{0},\hspace{1em}t\in [0;1],\\{} \displaystyle X_{n+1}(t)& \displaystyle =x_{0}+{\int _{0}^{t}}a\big(X_{n}(s)\big)ds+{\int _{0}^{t}}\sigma \big(X_{n}(s)\big)dW(s)\\{} & \displaystyle \hspace{1em}+{\int _{0}^{t}}\int _{\mathbb{R}}c\big(X_{n}(s-),v\big)\mathbf{1}_{|c(X_{n}(s-),v)|\leqslant 1}\big[\nu _{0}(ds,dv)-\varPi _{0}(dv)ds\big]\\{} & \displaystyle \hspace{1em}+{\int _{0}^{t}}\int _{\mathbb{R}}c\big(X_{n}(s-),v\big)\mathbf{1}_{|c(X_{n}(s-),v)|>1}\nu _{0}(ds,dv).\end{array}\](25)
\[ \displaystyle \mathbf{E}FY_{n+1}(t)\leqslant Q{\int _{0}^{t}}\mathbf{E}F\big(Y_{n}(s)\big)ds\](26)
\[ \displaystyle \mathbf{E}\big(Y_{n+1}{(t)}^{2}\wedge 1\big)\leqslant \tilde{Q}{\int _{0}^{t}}\mathbf{E}\big(Y_{n}{(s)}^{2}\wedge 1\big)ds\leqslant \frac{{\tilde{Q}}^{n}}{n!},\hspace{1em}t\in [0;1],\](27)
\[ \displaystyle \underset{t\in [0;1]}{\sup }\mathbf{E}{\rho }^{2}\big(X_{n+1}(t),X_{n}(t)\big)=0,\hspace{1em}n\geqslant 1.\](28)
\[ \displaystyle \underset{n\to \infty }{\lim }\underset{t\in [0;1]}{\sup }\mathbf{E}{\rho }^{2}\big(X_{n}(t),X(t)\big)=0.\]The proof of the Markov property for X 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. □
3 Examples
3.1 The Gamma-type process
Let us consider Gamma process with the Lévy measure $\varPi (du)=\frac{\gamma {e}^{-\lambda u}}{u}du$, $y>0$, $\gamma ,\lambda >0$. Here parameter γ can be interpreted as a rate of jump arrivals, and λ 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.
Proposition 3.
-
1. Let $\varPi _{j}$, $j=1,2$ be two Gamma measures with the same parameter λ and different parameters $0<\gamma _{1}<\gamma _{2}$. Then there exists a constant $D>0$ such that the following bound holds true
-
2. Let $\varPi _{j}$, $j=1,2$ be two Gamma measures with the same parameter γ and different parameters $0<\lambda _{1}<\lambda _{2}$. Then there exists a constant $\tilde{D}>0$ such that the following bound holds true
For the proof we refer the reader to Appendix B.
We call a Gamma-type process a Lévy-type process with a characteristic triplet $(0,0,\varPi (x,du))$ where a Lévy kernel is $\varPi (x,du)=\frac{\gamma (x){e}^{-\lambda (x)u}}{u}du$, $u>0$. If we assume that only the size of jumps varies, i.e. the parameter γ is fixed, then the corresponding SDE can be easily written in the form:
where ${X}^{1}$ denotes the Gamma process with the same parameter γ and $\lambda (x)=1$. Meanwhile the case where the intensity of jumps $\gamma =\gamma (x)$ 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 $\gamma (x)$ 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 $c(x,v)=\varphi (\frac{1}{v\gamma (x)})/\lambda $.
3.2 The α-stable like process
Recall that for an α-stable process, its Lévy measure has the form
where $\alpha \in (0,2)$ and ${\lambda }^{-},{\lambda }^{+}\geqslant 0$. Analogously to Proposition 2 in [3] one can obtain the following
(31)
\[ \begin{array}{r@{\hskip0pt}l}\displaystyle \varPi (du)& \displaystyle =\big(\alpha {\lambda }^{-}|u{|}^{-\alpha -1}\mathbf{1}_{(-\infty ;0)}(u)+\alpha {\lambda }^{+}{u}^{-\alpha -1}\mathbf{1}_{(0,\infty )}(u)\big)\hspace{0.1667em}du,\end{array}\]Proposition 4.
-
1. Let $\varPi _{j}$, $j=1,2$ be two α-stable measures with the same shape parameter α and different scale parameters ${\lambda _{1,2}^{\pm }}\in [a,b]$, $a,b>0$. In this case
(32)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle T(\varPi _{1},\varPi _{2})& \displaystyle \leqslant \bigg(\frac{2}{2-\alpha }\bigg)\big({\big|{\big({\lambda _{1}^{-}}\big)}^{1/\alpha }-{\big({\lambda _{2}^{-}}\big)}^{1/\alpha }\big|}^{\alpha }\\{} & \displaystyle \hspace{1em}+{\big|{\big({\lambda _{1}^{+}}\big)}^{1/\alpha }-{\big({\lambda _{2}^{+}}\big)}^{1/\alpha }\big|}^{\alpha }\big).\end{array}\] -
2. Let $\varPi _{j}$, $j=1,2$ be two α-stable measures with the same scale parameters ${\lambda }^{+}$, ${\lambda }^{-}$, but different shape parameters $\alpha _{1,2}\in [c,d]\subset (0;2)$. In this case there exists a constant $D>0$ such that the estimate is as follows
We call as an α-stable like process, a Lévy-type process with a characteristic triplet $(0,0,\varPi (x,du))$, where a Lévy kernel is
Note that even if the functions α, ${\lambda }^{\pm }$ are Lipschitz continuous itself, then by estimates (32) and (33) we observe that the kernel $\varPi (x,du)$ is not Lipschitz continuous in transportation distance, hence we can not use Theorem 1 to construct the required Lévy-type process as a strong solution to the SDE (3). Nevertheless, we are still able to use Theorem 1: to do that, we represent the kernel $\varPi (x,du)$ as a sum of the Lipschitz continuous kernel and the one of a bounded total jump intensity.
(34)
\[ \begin{array}{r@{\hskip0pt}l}\displaystyle \varPi (x,du)& \displaystyle =\alpha (x){\lambda }^{-}(x)|u{|}^{-\alpha (x)-1}\mathbf{1}_{u<0}\hspace{0.1667em}du+\alpha (x){\lambda }^{+}(x){u}^{-\alpha (x)-1}\mathbf{1}_{u>0}\hspace{0.1667em}du.\end{array}\]Consider the damped α-stable Lévy measure
Proposition 5.
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1. Let $\varPi _{j}$, $j=1,2$ be two damped α-stable measures with the same shape parameter α and different scale parameters ${\lambda _{1,2}^{\pm }}\in [a,b]$, $a,b>0$. In this case there exists some constant $D_{1}>0$ such that
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2. Let $\varPi _{j}$, $j=1,2$ be two damped α-stable measures with the same scale parameter ${\lambda }^{+}$, but different shape parameters $\alpha _{1,2}\in [c,d]\subset (0;2)$. In this case there exists some constant $D_{2}>0$ such that
The proof is given in Appendix B.
Now we can rewrite the kernel (34) as follows:
Hence, if the functions ${\lambda }^{-}(x)$ and ${\lambda }^{+}(x)$ are Lipschitz continuous and bounded, and the function $\alpha (x)$ is Lipschitz continuous and takes its values in a segment $[c,d]\subset (0,2)$, then the kernel $\varPi (x,du)$ admits a decomposition such that all the conditions of Theorem 1 are verified. Correspondingly, the α-stable like process with the Lévy kernel (34) can be obtained as a solution to the SDE
