Existence of solution for stochastic differential equations driven by G-Lévy process with discontinuous coefficients
- Bingjun Wang^{1, 2}Email author and
- Mingxia Yuan^{3}
https://doi.org/10.1186/s13662-017-1242-y
© The Author(s) 2017
Received: 6 March 2017
Accepted: 13 June 2017
Published: 30 June 2017
Abstract
The existence theory for the vector-valued stochastic differential equations driven by G-Brownian motion and pure jump G-Lévy process (G-SDEs) 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 G-SDEs have more than one solution if the coefficients f, g, K are discontinuous functions. The upper and lower solution method is used.
Keywords
stochastic differential equations G-Lévy process upper and lower solution discontinuous coefficientsMSC
60H05 60H10 60H201 Introduction
In recent years much effort has been made to develop the theory of sublinear expectations connected with the volatility uncertainty and the so-called G-Brownian motion. G-Brownian 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 path-dependent partial differential equations or backward stochastic differential equations. Thus G-Brownian motion and connected G-expectation are attractive mathematical objects.
Returning, however, to the original problem of volatility uncertainty in the financial models, one feels that G-Brownian 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 G-Lévy process. Then Ren [4] introduced the representation of the sublinear expectation as an upper-expectation. In [5], the author concentrated on establishing the integration theory for G-Lévy process with finite activity, introduced the integral w.r.t. the jump measure associated with the pure jump G-Lévy process and gave the Itô formula for general G-Itô Lévy process.
Motivated by the importance of discontinuous functions, Faizullah and Piao [6] established the existence of solutions for the stochastic differential equations driven by G-Brownian 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 G-SDE (1.1) with simultaneous discontinuous coefficients f, g and K is developed.
2 Preliminaries
In this section, we introduce some notations and preliminary results in G-framework which are needed in the sequence. More details can be found in [5, 8–13].
Definition 2.1
- (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)
Sub-additivity: \(\mathbb{E}[X+Y]\leq\mathbb {E}[X]+\mathbb{E}[Y]\);
- (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 d-dimensional random vector in \((\Omega,\mathcal{H},\mathbb{E})\).
Definition 2.2
Definition 2.3
Definition 2.4
G-Lévy process
- (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.
- (iv)
there is a 2d-dimensional 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 G-Lévy process might be characterized by a non-local operator \(G_{X}\).
Theorem 2.1
[3]
Theorem 2.2
[3]
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 G-Lévy process satisfying Levy-Khintchine 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.
Definition 2.5
Remark 2.1
The condition (v) in Definition 2.4 implies that \(X^{c}\) is a d-dimensional generalized G-Brownian motion and the pure jump part \(X^{d}\) is of finite variation (see [3]). Moreover, \(X^{c}\) is just the d-dimensional G-Brownian motion \(B_{t}\) when \(p=0\) in (2.1). In this paper, we always let \(p=0\), i.e., the G-Lévy process X consists of G-Brownian motion \(B_{t}\) and the pure jump part.
For each \(\eta\in M_{G}^{p}(0,T)\), \(p\geq2\), the G-Itô 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]
Definition 2.6
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
Proof
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 G-SDE (2.3) with the initial condition \(Y_{0}\in R^{n}\) has a unique solution \(Y_{t}\in\hat {M}_{G}^{2}(0,T;R^{n})\).
3 Existence of solution for G-SDEs with discontinuous coefficients
Definition 3.1
Definition 3.2
Lemma 3.1
Suppose that \(U_{t}\) and \(L_{t}\) are the respective upper and lower solutions of G-SDE (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 G-SDE (3.5), respectively.
Proof
Theorem 3.1
- (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 G-SDE (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 \(L_{0}< Y_{0}< U_{0}\).
Proof
Theorem 3.2
- (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 G-SDE (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 \(t\in[0,T]\).
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 \(\hat{M}_{G}^{2}(0,T; R^{n})\).
Example 3.1
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 non-empty 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.
Declarations
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).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Peng, S: Filtration consistent nonlinear expectations and evaluations of contingent claims. Acta Math. Appl. Sinica (Engl. Ser.) 20, 1-24 (2004) MathSciNetMATHGoogle Scholar
- Peng, S: G-Expectations, G-Brownian motion and related stochastic calculus of Itô’s type. In: Benth, FE, Di Nunno, G, Lindstrom, T, Øksendal, B, Zhang, T (eds.) Proceedings of the 2005 Abel Symposium, pp. 541-567. Springer, Berlin (2006). Preprint version in: arXiv:math/0601035v2 Google Scholar
- Hu, M, Peng, S: G-Lévy processes under sublinear expectations. arXiv:0911.3533v1
- Ren, L: On representation theorem of sublinear expectation related to G-Lévy processes and paths of G-Lévy processes. Stat. Probab. Lett. 83, 1301-1310 (2013) View ArticleMATHGoogle Scholar
- Paczka, K: Itô calculus and jump diffusions for G-Lévy processes. arXiv:1211.2973v3
- Faizullah, F, Piao, D: Existence of solutions for G-SDEs with upper and lower solutions in the reverse order. Int. J. Phys. Sci. 7, 432-439 (2012) Google Scholar
- Faizullah, F: Existence of solutions for stochastic differential equations under G-Brownian motion with discontinuous coefficients. Z. Naturforsch. A 67, 692-698 (2012) Google Scholar
- Paczka, K: G-Martingale representation in the G-Lévy setting. arXiv:1404.2121v1
- Denis, L, Hu, M, Peng, S: Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion paths. Potential Anal. 34, 139-161 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Soner, M, Touzi, N, Zhang, J: Martingale representation theorem for the G-expectation. Stoch. Process. Appl. 121, 265-287 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Peng, S: Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stoch. Process. Appl. 118, 2223-2253 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Peng, S: Nonlinear expectations and stochastic calculus under uncertainly. arXiv:1002.4546v1 (2010)
- Li, X, Peng, S: Stopping times and related Itô’s calculus with G-Brownian motion. Stoch. Process. Appl. 121, 1492-1508 (2011) View ArticleMATHGoogle Scholar
- Heikkila, S, Hu, S: On fixed points of multifunctions in ordered spaces. Appl. Anal. 51, 115-127 (1993) MathSciNetView ArticleMATHGoogle Scholar