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Existence of solution for stochastic differential equations driven by GLévy process with discontinuous coefficients
Advances in Difference Equations volume 2017, Article number: 188 (2017)
Abstract
The existence theory for the vectorvalued stochastic differential equations driven by GBrownian motion and pure jump GLévy process (GSDEs) of the type \(dY_{t}=f(t,Y_{t})\, dt+g_{j,k}(t,Y_{t})\, d\langle B^{j},B^{k}\rangle _{t}+\sigma_{i}(t,Y_{t}) \, dB^{i}_{t}+\int _{R_{0}^{d}}K(t,Y_{t},z)L(dt,dz)\), \(t\in[0,T]\), with first two and last discontinuous coefficients, is established. It is shown that the GSDEs have more than one solution if the coefficients f, g, K are discontinuous functions. The upper and lower solution method is used.
Introduction
In recent years much effort has been made to develop the theory of sublinear expectations connected with the volatility uncertainty and the socalled GBrownian motion. GBrownian motion was introduced by Shige Peng in [1, 2] as a way to incorporate the unknown volatility into financial models. Its theory is tightly associated with the uncertainty problems involving an undominated family of probability measures. Soon other connections have been discovered, not only in the field of financial mathematics, but also in the theory of pathdependent partial differential equations or backward stochastic differential equations. Thus GBrownian motion and connected Gexpectation are attractive mathematical objects.
Returning, however, to the original problem of volatility uncertainty in the financial models, one feels that GBrownian motion is not sufficient to model the financial world, as both G and the standard Brownian motion share the same property, which makes them often unsuitable for modeling, namely, the continuity of paths. Therefore, it is not surprising that Hu and Peng [3] introduced the process with jumps, which they called GLévy process. Then Ren [4] introduced the representation of the sublinear expectation as an upperexpectation. In [5], the author concentrated on establishing the integration theory for GLévy process with finite activity, introduced the integral w.r.t. the jump measure associated with the pure jump GLévy process and gave the Itô formula for general GItô Lévy process.
Under the integration theory for GLévy process, Paczka [5] established the existence and uniqueness of solutions for the following stochastic differential equation driven by GBrownian motion and pure jump GLévy process with Lipschitz continuous coefficients:
where \(Y_{0}\in R^{n}\), \((\langle B^{j},B^{k}\rangle_{t})_{t\geq0}\) is the mutual variation process of the GBrownian motion \((B_{t})_{t\geq0}\), \(L(t,z)\) is pure jump GLévy process. For each \(x\in R^{n}\), the coefficients \(f(t,x),g_{j,k}(t,x),\sigma _{i}(t,x)\) are in the space \(\hat{M}_{G}^{2}(0,T; R^{n})\), \(K(t,x,z)\in\hat{H}_{G}^{2}([0,T]\times R_{0}^{d};R^{n} )\) (which will be introduced in Section 2). A process \(Y_{t}\) belonging to \(\hat {M}_{G}^{2}(0,T;R^{n})\) and satisfying GSDE (1.1) is said to be its solution.
Motivated by the importance of discontinuous functions, Faizullah and Piao [6] established the existence of solutions for the stochastic differential equations driven by GBrownian motion with a discontinuous drift coefficient. Then Faizullah [7] developed the existence theory when the coefficient f or the coefficients f and g simultaneously are discontinuous functions. Motivated by the aforementioned works, in this paper, we consider equation (1.1) and assume that \(f(t,x)\), \(g(t,x) \) and \(K(t,x,z)\) are discontinuous for all \(x\in R^{n}\).
The rest of this paper is organized as follows. In Section 2, we introduce some preliminaries. In Section 3, the existence of solutions for GSDE (1.1) with simultaneous discontinuous coefficients f, g and K is developed.
Preliminaries
In this section, we introduce some notations and preliminary results in Gframework which are needed in the sequence. More details can be found in [5, 8–13].
Definition 2.1
Let Ω be a given set, and let \(\mathcal{H}\) be a linear space of realvalued functions defined on Ω. Moreover, if \(X_{i}\in \mathcal{H}\), \(i=1,2,\ldots,d\), then \(\varphi(X_{1},\ldots ,X_{d})\in\mathcal {H}\) for all \(\varphi\in C_{b,\mathrm{lip}}(R^{d})\), where \(C_{b,\mathrm{lip}}(R^{d})\) is the space of all bounded realvalued Lipschitz continuous functions. A sublinear expectation \(\mathbb{E}\) is a functional \(\mathbb {E}:\mathcal{H}\rightarrow R\) satisfying the following properties: for all \(X,Y\in\mathcal{H}\), we have:

(i)
Monotonicity: \(\mathbb{E}[X]\geq\mathbb{E}[Y]\) if \(X\geq Y\);

(ii)
Constant preserving: \(\mathbb{E}[C]=C\) for \(C\in R\);

(iii)
Subadditivity: \(\mathbb{E}[X+Y]\leq\mathbb {E}[X]+\mathbb{E}[Y]\);

(iv)
Positive homogeneity: \(\mathbb{E}[\lambda X]=\lambda \mathbb {E}[X]\) for \(\lambda\geq0\).
The triple \((\Omega,\mathcal{H},\mathbb{E})\) is called a sublinear expectation space. \(X\in\mathcal{H}\) is called a random variable in \((\Omega,\mathcal{H},\mathbb{E})\). We often call \(Y=(Y_{1},\ldots,Y_{d}), Y_{i}\in\mathcal{H}\) a ddimensional random vector in \((\Omega,\mathcal{H},\mathbb{E})\).
Definition 2.2
In a sublinear expectation space \((\Omega,\mathcal{H},\mathbb{E})\), an ndimensional random vector \(Y=(Y_{1},\ldots,Y_{n})\) is said to be independent from an mdimensional random vector \(X=(X_{1},\ldots,X_{m})\) if for each \(\varphi\in C_{b,\mathrm{lip}}(R^{m+ n})\),
Definition 2.3
Let \(X_{1}\), \(X_{2}\) be two ndimensional random vectors defined on sublinear expectation spaces \((\Omega_{1},\mathcal{H}_{1},\mathbb {E}_{1})\) and \((\Omega_{2},\mathcal{H}_{2},\mathbb{E}_{2})\), respectively. They are called identically distributed, denoted by \(X_{1}\overset{d}{=}X_{2}\), if
X̄ is said to be an independent copy of X if X̄ is identically distributed with X and independent of X.
Definition 2.4
GLévy process
Let \(X=(X_{t})_{t\geq0}\) be a ddimensional càdlàg process on a sublinear expectation space \((\Omega,\mathcal{H},\mathbb{E})\). We say that X is a Lévy process if:

(i)
\(X_{0}=0\),

(ii)
for each \(s,t\geq0\), the increment \(X_{t+s}X_{s}\) is independent of \((X_{t_{1}},\ldots, X_{t_{n}})\) for every \(n\in N\) and every partition \(0\leq t_{1}\leq t_{2}\leq\cdots\leq t_{n}\leq s\),

(iii)
the distribution of the increment \(X_{t+s}X_{s}\), \(s, t\geq0\) is stationary, i.e., does not depend on s.
Moreover, we say that a Lévy process X is a GLévy process if it satisfies additionally the following conditions:

(iv)
there is a 2ddimensional Lévy process \((X_{t}^{c},X_{t}^{d})_{t\geq0}\) such that for each \(t\geq0 \), \(X_{t}=X_{t}^{c}+X_{t}^{d}\),

(v)
processes \(X_{t}^{c}\) and \(X_{t}^{d}\) satisfy the following conditions:
$$ \lim_{t\downarrow0}\mathbb{E}\bigl[ \bigl\vert X_{t}^{c} \bigr\vert ^{3}\bigr]t^{1}=0; \qquad\mathbb{E}\bigl[ \bigl\vert X_{t}^{d} \bigr\vert \bigr]< Ct \quad\mbox{for all } t\geq0. $$
Peng and Hu noticed in their paper that each GLévy process might be characterized by a nonlocal operator \(G_{X}\).
Theorem 2.1
[3]
Let X be a GLévy process in \(R^{d}\). For every \(f\in C_{b}^{3}(R^{d})\) such that \(f(0)=0\), we put
The above limit exists. Moreover, \(G_{X}\) has the following LevyKhintchine representation:
where \(R_{0}^{d}:=R^{d}\backslash\{0\}\), \(\mathcal{U}\) is a subset \(\mathcal{U}\subset\mathcal{V}\times R^{d}\times R^{d\times d}\) and \(\mathcal{V}\) is a set of all Borel measures on \((R_{0}^{d}, \mathcal {B}(R_{0}^{d}))\). We know additionally that \(\mathcal{U}\) has the property
Theorem 2.2
[3]
Let X be a ddimensional GLévy process. For each \(\phi\in C_{b,\mathrm{lip}}(R^{d})\), define \(u(t,x):=\mathbb{E}[\phi (x+X_{t})]\). Then u is the unique viscosity solution of the following integroPDE:
with the initial condition \(u(0,x)=\phi(x)\).
Theorem 2.3
Let \(\mathcal{U}\) satisfy (2.1). Consider the canonical space \(\Omega :=\mathbb{D}_{0}(R^{+},R^{d})\) of all càdlàg functions taking values in \(R^{d}\) equipped with the Skorohod topology. Then there exists a sublinear expectation \(\hat{\mathbb{E}}\) on \(\mathbb {D}_{0}(R^{+},R^{d})\) such that the canonical process \((X_{t})_{t\geq 0}\) is a GLévy process satisfying LevyKhintchine representation with the same set \(\mathcal{U}\).
The proof might be found in [3]. We will give, however, the construction of \(\hat{\mathbb{E}}\) as it is important to understand it.
We denote \(\Omega_{T}:=\{w_{\cdot\wedge T}:w\in\Omega\}\). Put
where \(X_{t}(w)=w_{t}\) is the canonical process on the space \(\mathbb {D}_{0}(R^{+},R^{d})\) and \(L^{0}(\Omega)\) is the space of all random variables, which are measurable to the filtration generated by the canonical process. We also set
Firstly, consider the random variable \(\xi=\phi(X_{t+s}X_{s}), \phi \in C_{b,\mathrm{lip}}(R^{d})\). We define
where u is a unique viscosity solution of integroPDE (2.2) with the initial condition \(u(0,x)=\phi(x)\). For general
we set \(\hat{\mathbb{E}}[\xi]:=\phi_{n}\), where \(\phi_{n}\) is obtained via the following iterated procedure:
Lastly, we extend the definition of \(\hat{\mathbb{E}}\) on the completion of \(\operatorname{Lip}(\Omega_{T})\) (respectively \(\operatorname{Lip}(\Omega)\)) under the norm \(\\cdot\_{p}^{p}=\hat{\mathbb{E}}[\cdot^{p}]\), \(p\geq 1\). We denote such a completion by \(L_{G}^{p}(\Omega_{T})\) (or resp. \(L_{G}^{p}(\Omega)\)).
Let \(\mathcal{B}(\Omega)\) be the Borel σalgebra of Ω. It was proved in [4] that there exists a weakly compact probability measure family \(\mathcal{P}\) defined on \((\Omega, \mathcal{B}(\Omega ))\) such that
where \(E_{P}\) is the linear expectation with respect to P.
Definition 2.5
We define the capacity c associated with \(\hat{\mathbb{E}}\) by putting
We will say that a set \(A\in\mathcal{B}(\Omega)\) is polar if \(c(A)=0\). We say that a property holds quasisurely (q.s.) if it holds outside a polar set.
Remark 2.1
The condition (v) in Definition 2.4 implies that \(X^{c}\) is a ddimensional generalized GBrownian motion and the pure jump part \(X^{d}\) is of finite variation (see [3]). Moreover, \(X^{c}\) is just the ddimensional GBrownian motion \(B_{t}\) when \(p=0\) in (2.1). In this paper, we always let \(p=0\), i.e., the GLévy process X consists of GBrownian motion \(B_{t}\) and the pure jump part.
Let \(M_{G}^{0,p}(0,T)\) be the collection of processes of the following form: for a given partition \(\{t_{0},\ldots, t_{N}\}=\pi_{T}\) of \([0,T]\),
where \(\xi_{i}\in L_{G}^{p}(\Omega_{t_{i}})\), \(i=0,1,\ldots, N1\), \(p\geq1\). For each \(p\geq1\), denote by \(M_{G}^{p}(0,T)\) the completion of \(M_{G}^{0,p}(0,T)\) under the norm \(\\eta\_{M_{G}^{p}}:=(\hat {\mathbb{E}}[\int_{0}^{T}\eta_{t}^{p}\, dt])^{\frac{1}{p}}\).
For each \(\eta\in M_{G}^{p}(0,T)\), \(p\geq2\), the GItô integral \(\{\int_{0}^{t}\eta_{s}\, dB^{i}_{s}\}_{t\in[0,T]}\) is well defined. For each \(\eta_{s}^{j,k}\in M_{G}^{p}(0,T)\), \(p\geq1\), the integral \(\{ \int_{0}^{t}\eta_{s}^{j,k}\, d\langle B^{j},B^{k}\rangle_{s}\}_{t\in [0,T]}\) is well defined. \(i,j,k=1,\ldots,d\). See Peng [12] and Li et al. [13].
Lemma 2.1
[7]
Let \(\eta_{t}^{j,k},\zeta_{t}^{j,k}\in M_{G}^{1}(0,T)\). If \(\eta _{t}^{j,k}\leq\zeta_{t}^{j,k}\) for \(t\in[0,T]\), then
Assume that the GLévy process X has finite activity, i.e.,
Let \(X_{u}\) denote the left limit of X at point u, \(\Delta X_{u}=X_{u}X_{u}\), then we can define a random measure \(L(\cdot, \cdot)\) associated with the GLévy process X by putting
for any \(0< s< t<\infty\) and \(A\in\mathcal{B}(R_{0}^{d})\). The random measure is well defined and may be used to define the pathwise integral.
Let \(H_{G}^{S}([0,T]\times R_{0}^{d})\) be a space of all elementary random fields on \([0,T]\times R_{0}^{d}\) of the form
where \(0\leq t_{1}<\cdots<t_{n}\leq T\) is the partition of \([0,T]\), \(\{ \psi_{l}\}_{l=1}^{m}\subset C_{b,\mathrm{lip}}(R^{d})\) are functions with disjoint supports s.t. \(\psi_{l}(0)=0\) and \(F_{k,l}=\phi _{k,l}(X_{t_{1}},\ldots,X_{t_{k}}X_{t_{k1}})\), \(\phi_{k,l}\in C_{b,\mathrm{lip}}(R^{d\times k})\). We introduce the norm on this space
Definition 2.6
Let \(0\leq s< t\leq T\). The Itô integral of \(K\in H_{G}^{S}([0,T]\times R_{0}^{d})\) w.r.t. jump measure L is defined as
Lemma 2.2
For every \(K\in H_{G}^{S}([0,T]\times R_{0}^{d})\), we have that \(\int _{0}^{T}\int_{R_{0}^{d}}K(u,z)L(du,dz)\) is an element of \(L_{G}^{2}(\Omega_{T})\).
Let \(H_{G}^{p}([0,T]\times R_{0}^{d})\) denote the topological completion of \(H_{G}^{S}([0,T]\times R_{0}^{d})\) under the norm \(\ \cdot\_{H_{G}^{p}([0,T]\times R_{0}^{d})}\), \(p=1,2\). Then Itô integral can be continuously extended to the whole space \(H_{G}^{p}([0,T]\times R_{0}^{d})\), \(p=1,2\). Moreover, by Lemma 2.2 we know that the extended operator takes value in \(L_{G}^{2}(\Omega _{T})\), \(p=1,2\).
Lemma 2.3
Let \(K^{1}(t,z),K^{2}(t,z)\in H_{G}^{2}([0,T]\times R_{0}^{d})\). If \(K^{1}(t,z)\leq K^{2}(t,z)\) for \(t\in[0,T]\), then
Proof
Let \(K^{1}(t,z),K^{2}(t,z)\in H_{G}^{S}([0,T]\times R_{0}^{d})\), by Definition 2.6, the following holds:
For \(K(t,z)\in H_{G}^{p}([0,T]\times R_{0}^{d})\), the inequality still holds under a regular argument. □
To consider the solution of GSDEs, let us introduce the new norm on the integrands: for a process η, define
The completion of the space under this norm will be denoted as \(\hat {M}_{G}^{p}(0,T)\). Note that
thus appropriate integrals will be always well defined.
Similarly, we need to adjust the space of integrands for the jump measure. Let \(\hat{H}_{G}^{2}([0, T]\times R_{0}^{d})\) denote the completion of all \(H_{G}^{S}([0,T]\times R_{0}^{d})\) under the norm
We consider the following GSDE driven by ddimensional GBrownian motion B and the pure jump GLévy process L (in this paper we always use Einstein’s convention):
Let \(\hat{M}_{G}^{2}(0,T;R^{n})\) denote the space of \(R^{n}\)valued process and for each element belong to \(\hat{M}_{G}^{2}(0,T)\). We can define the space \(\hat{H}_{G}^{2}([0,T]\times R_{0}^{d};R^{n})\) in a similar way.
Theorem 2.4
[5]
Suppose that \(f(t,x)\), \(g_{j,k}(t,x)\), \(\sigma_{i}(t,x)\), \(K(t,x)\) are Lipschitz continuous w.r.t. x uniformly. For each \(x\in R^{n}\), \(f(\cdot,x),g_{j,k}(\cdot,x), \sigma_{i}(\cdot,x) \in\hat {M}_{G}^{2}(0,T;R^{n})\), \(K(\cdot,x,\cdot)\in\hat {H}_{G}^{2}([0,T]\times R_{0}^{d};R^{n})\). Then GSDE (2.3) with the initial condition \(Y_{0}\in R^{n}\) has a unique solution \(Y_{t}\in\hat {M}_{G}^{2}(0,T;R^{n})\).
Existence of solution for GSDEs with discontinuous coefficients
Definition 3.1
If, for any \(0 \leq s\leq t\), the process \(U_{t}\in\hat {M}_{G}^{2}(0,T;R^{n})\) satisfies the following inequality:
q.s., then it is said to be an upper solution of GSDE (2.3) on the interval \([0,T]\).
Definition 3.2
If, for any \(0\leq s\leq t\), the process \(L_{t}\in\hat {M}_{G}^{2}(0,T;R^{n})\) satisfies the following inequality:
q.s., then it is said to be a lower solution of GSDE (2.3) on the interval \([0,T]\).
Suppose that \(U_{t}\) and \(L_{t}\) are the respective upper and lower solutions of the GSDE
where \(f(\cdot,w),g_{j,k}(\cdot,w)\in\hat{M}_{G}^{2}(0,T;R^{n})\), \(K(\cdot,w,\cdot)\in\hat{H}_{G}^{2}([0,T]\times R_{0}^{d};R^{n})\) for \(w\in\Omega\), \(\sigma_{i}(\cdot,x)\in\hat {M}_{G}^{2}(0,T;R^{n})\) for each \(x\in R^{n}\) and \(\sigma_{i}(t,x)\) is Lipschitz continuous in x. Define two functions \(p,q:[0,T]\times R^{n}\times\Omega\rightarrow R^{n}\) by
and consider the following GSDE:
with a given constant initial condition \(Y_{0}\in R^{n}\), where
It is clear that f̃, \(\tilde{g}_{j,k}\), \(\tilde{\sigma }_{i}\), K̃ are Lipschitz continuous in x, thus the conditions of Theorem 2.4 are satisfied. Then GSDE (3.5) has a unique solution \(Y_{t}\in\hat{M}_{G}^{2}(0,T;R^{n})\).
Lemma 3.1
Suppose that \(U_{t}\) and \(L_{t}\) are the respective upper and lower solutions of GSDE (3.3) satisfying \(L_{t}\leq U_{t}\) for \(t\in [0,T]\). Then \(U_{t}\) and \(L_{t}\) are the upper and lower solutions of GSDE (3.5), respectively.
Proof
For \(0\leq s\leq t\), we have
where we have used \(p(t,U_{t},w)=U_{t}\) and \(q(t,U_{t},w)=0\). Therefore \(U_{t}\) is an upper solution of GSDE (3.5). One can show that \(L_{t}\) is a lower solution of GSDE (3.5) in a similar way as above. □
By Lemma 3.1 we know that if \(U_{t}\) and \(L_{t}\) are upper and lower solutions of GSDE (3.3), then they are the respective upper and lower solutions for GSDE (3.5). Suppose that \(Y_{t}\) is the solution of GSDE (3.5) such that
Since \(p(t,Y_{t},w)=Y_{t}\), \(q(t,Y_{t},w)=0\), then
which implies that \(Y_{t}\) is a solution of GSDE (3.3). Thus, if we can show that any solution \(Y_{t}\) of problem (3.5) does satisfy inequality (3.8), then \(Y_{t}\) is also the solution of GSDE (3.3).
Theorem 3.1
Suppose that

(i)
\(f(\cdot,w),g_{j,k}(\cdot,w)\in\hat{M}_{G}^{2}(0,T;R^{n})\), \(K(\cdot,w,\cdot)\in\hat{H}_{G}^{2}([0,T]\times R_{0}^{d};R^{n})\) for \(w\in\Omega\), \(\sigma_{i}(\cdot,x)\in\hat {M}_{G}^{2}(0,T;R^{n})\) for each \(x\in R^{n}\) and \(\sigma(t,x)\) is Lipschitz continuous in x;

(ii)
the respective upper and lower solutions \(U_{t}\) and \(L_{t}\) of GSDE (3.3) satisfy \(L_{t}< U_{t}\) for \(t\in[0,T]\);

(iii)
\(Y_{0}\in R^{n}\) is a given initial value with \(\hat {\mathbb {E}}[Y_{0}^{2}]<\infty\) and \(L_{0}< Y_{0}< U_{0}\).
Then there exists a unique solution \(Y_{t}\in\hat {M}_{G}^{2}(0,T;R^{n})\) of GSDE (3.3) such that \(L_{t}< Y_{t}< U_{t}\) for \(t\in[0,T]\), q.s.
Proof
We only need to prove that the solution \(Y_{t}\) of GSDE (3.5) does satisfy inequality (3.8). Assume that there exists an arbitrary interval \((t_{1},t_{2})\subset[0,T]\) such that \(Y_{t_{1}}=L_{t_{1}}\) and \(Y_{t}< L_{t}\) for \(t\in(t_{1},t_{2})\), then we have
Since \(Y_{t}\leq L_{t}\leq U_{t}\) for \(t\in(t_{1},t_{2})\), then \(p(t,L_{t},w)=L_{t}\) and \(p(t,Y_{t},w)=L_{t}\). Also \(q(t,L_{t},w)=0\) and \(q(t,Y_{t},w)=L_{t}Y_{t}\). Thus
which yields a contradiction. Thus \(Y_{t}\geq L_{t}\) for \(t\in[0,T]\). By using similar arguments as above, one can show that \(Y_{t}\leq U_{t}\) for \(t\in[0,T]\). Thus the proof is finished. □
Now we consider the following GSDE:
where \(f(t,x)\), \(g_{j,k}(t,x)\) and \(K(t,x,z)\) do not need to be Lipschitz continuous with respect to x, only \(\sigma_{i}(t,x)\) is Lipschitz continuous in x.
Theorem 3.2
Suppose that

(i)
for each \(x\in R^{n}\), \(f(\cdot,x), g_{j,k}(\cdot ,x),\sigma _{i}(\cdot,x)\in\hat{M}_{G}^{2}(0,T; R^{n})\), \(K(\cdot,x,\cdot)\in \hat{H}_{G}^{2}([0,T]\times R_{0}^{d}; R^{n})\);

(ii)
\(\sigma_{i}(t,x)\) is Lipschitz continuous in x, \(f(t,x)\), \(g_{j,k}(t,x)\) and \(K(t,x,z)\) are increasing in x;

(iii)
\(U_{t}\) and \(L_{t}\) are the respective upper and lower solutions of GSDE (3.12). Moreover, \(K(\cdot,U_{t},\cdot), K(\cdot ,L_{t},\cdot)\in\hat{H}_{G}^{2}([0,T]\times R_{0}^{d}; R^{n})\) and \(L_{t}\leq U_{t}\) for \(t\in[0,T]\).
Then there exists at least one solution \(Y_{t}\in\hat{M}_{G}^{2}(0,T; R^{n})\) of GSDE (3.12) such that \(L_{t}\leq Y_{t}\leq U_{t}\) for \(t\in[0,T]\), q.s.
Proof
Denote the order interval \([L,U]\) in \(\hat{M}_{G}^{2}(0,T; R^{n})\) by \(\mathcal{H}\), that is, \(\mathcal{H}=\{Y:Y \in\hat{M}_{G}^{2}(0,T; R^{n}) \mbox{ and } L_{t}\leq Y_{t}\leq U_{t}\}\) for \(t\in[0,T]\), which is closed and bounded. By using the monotone convergence theorem in [5], one can prove the convergence of a monotone sequence that belongs to \(\mathcal{H}\) in \(\hat{M}_{G}^{2}(0,T; R^{n})\). Thus \(\mathcal{H}\) is a regularly ordered metric space with the norm of \(\hat{M}_{G}^{2}(0,T; R^{n})\).
Since \(f(t,x)\), \(g_{j,k}(t,x)\) and \(K(t,x,z)\) are increasing in x, it is easy to see that for any process \(V\in\mathcal{H}\), \(U_{t}\) and \(L_{t}\) are the respective upper and lower solutions for the GSDE
Hence, by Theorem 3.1, for any \(Y_{0}\in R^{n}\) with \(\hat {\mathbb {E}}[Y_{0}^{2}]<\infty\), and \(L_{0}\leq Y_{0}\leq U_{0}\), GSDE (3.13) has a unique solution \(Y_{t}\in\hat{M}_{G}^{2}(0,T; R^{n})\) such that \(L_{t}\leq Y_{t}\leq U_{t}\) for \(t\in[0,T]\), q.s.
Define an operator \(F:\mathcal{H}\rightarrow\mathcal{H}\) by \(F(V)=Y\), where Y is the unique solution of GSDE (3.13). For all \(t\in[0,T]\), let \(V^{1}_{t}, V^{2}_{t} \in\mathcal{H}\) and \(V^{1}_{t}\leq V^{2}_{t}\) and define \(Y^{1}_{t}=F(V^{1}_{t})\), \(Y^{2}_{t}=F(V^{2}_{t})\). Since f, g, K are increasing functions, then
which implies that \(Y^{1}_{t}\) is a lower solution of the GSDE
However, this problem has an upper solution \(U_{t}\). Then the solution of GSDE (3.15) \(Y^{2}_{t}\) satisfies \(Y^{1}_{t}\leq Y^{2}_{t}\leq U_{t}\). Hence F is an increasing mapping and, by Theorem 3.3, it has a fixed point \(Y^{*}=F(Y^{*})\in\mathcal{H}\) such that \(L_{t}\leq Y^{*}_{t} \leq U_{t}\), q.s. and
Thus the proof is finished. □
Example 3.1
Consider the following scalar stochastic differential equation:
where the Heaviside function \(H:R\rightarrow R\) is defined by
This is an important function in science, and it is considered to be a fundamental function in engineering. The fractional part function \(\{x\} :R\rightarrow[0,1)\) has discontinuities at the integers and is defined by
where \([x]\) is the floor function. The importance of this function is clear from the sawtooth waves which are used in music and computer graphics.
Let \(U_{t}=U_{0}+\int_{0}^{t} du+\int_{0}^{t} d\langle B\rangle _{u}+\int_{0}^{t}dB_{u}+\int_{0}^{t}\int_{R_{0}}L(du,dz)\) for \(t\in [0,T]\). Then we have
where \(U_{s}=U_{0}+\int_{0}^{s}du+\int_{0}^{s}d\langle B\rangle _{u}+\int_{0}^{s}dB_{u}+\int_{0}^{s}\int_{R_{0}}L(du,dz)\) for \(0\leq s\leq t\leq T\). This implies that \(U_{t}\) is the upper solution of equation (3.17). In a similar way, one can show that \(L_{t}=L_{0}+\int _{0}^{t}dB_{u}\) is a lower solution of equation (3.17). Then, by Theorem 3.2, there exists at least one solution for equation (3.17).
For the following definition and theorem, see [14].
Definition 3.3
An ordered metric space M is called regularly (resp. fully regularly) ordered if each monotone and order (resp. metrically ) bounded ordinary sequence of M converges.
Theorem 3.3
If \([a,b]\) is a nonempty order interval in a regularly ordered metric space, then each increasing mapping \(F: [a,b]\rightarrow[a,b]\) has the least and the greatest fixed points.
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Acknowledgements
The work is supported in part by the TianYuan Special Funds of the National Natural Science Foundation of China (Grant No. 11526103), National Natural Science Foundation of China (Grant No. 11601203).
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Wang, B., Yuan, M. Existence of solution for stochastic differential equations driven by GLévy process with discontinuous coefficients. Adv Differ Equ 2017, 188 (2017). https://doi.org/10.1186/s136620171242y
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DOI: https://doi.org/10.1186/s136620171242y
MSC
 60H05
 60H10
 60H20
Keywords
 stochastic differential equations
 GLévy process
 upper and lower solution
 discontinuous coefficients