Eigenvalues of stochastic Hamiltonian systems driven by Poisson process with boundary conditions
- Haiyang Wang^{1} and
- Zhen Wu^{2}Email author
Received: 13 June 2017
Accepted: 23 October 2017
Published: 9 November 2017
Abstract
In this paper, we study an eigenvalue problem for stochastic Hamiltonian systems driven by a Brownian motion and Poisson process with boundary conditions. By means of dual transformation and generalized Riccati equation systems, we prove the existence of eigenvalues and construct the corresponding eigenfunctions. Moreover, a specific numerical example is considered to illustrate the phenomenon of statistic periodicity for eigenfunctions of stochastic Hamiltonian systems.
Keywords
1 Introduction
The backward stochastic differential equations driven by Poisson process (BSDEP) were first introduced and studied by Tang and Li [1]. Later, Situ Rong [2] proved the existence and uniqueness of solutions to BSDEP with non-Lipschitz coefficients. Barles et al. [3] adopted the BSDEP to provide a probabilistic interpretation for a system of parabolic integro-partial differential equations. Then, the fully coupled forward-backward stochastic differential equations driven by Poisson process (FBSDEP) were deeply investigated by Wu [4, 5], etc. Precisely, in [4], the author established the well-posedness for FBSDEP under the so-called ‘monotone assumptions’ via the continuation method; while in [5], the author discussed the BSDEP and FBSDEP with stopping time duration.
The stochastic Hamiltonian systems are proposed in the optimal control theory as a necessary condition for optimality, called the stochastic maximum principle. The fundamental works related to this topic include Bismut [6], Bensoussan [7], Peng [8] and so on. Due to the discontinuity of stock prices and other common ‘random jump’ phenomena in reality, the stochastic Hamiltonian systems driven by Poisson process, which is a special kind of FBSDEP, are very suitable for us to study the stochastic optimal control problems with random jumps. Wu and Wang [9] discussed a linear quadratic stochastic optimization problem with random jumps, and furthermore associated its Hamiltonian system with a generalized Riccati equation system to give the linear feedback optimality for this problem.
However, all literature works above only concern the uniqueness of solutions to FBSDEP as well as stochastic Hamiltonian systems. Peng [10] studied a kind of eigenvalue problem for stochastic Hamiltonian systems driven by Brownian motion with boundary conditions. In this paper, we extend that problem to stochastic Hamiltonian systems driven by Poisson process within the formulation of FBSDEP established by Wu [4]. Generally speaking, for a class of stochastic Hamiltonian systems driven by Poisson process parameterized by \(\lambda \in\mathbb{R}\) which always admit the trivial solution \((x_{t},y_{t},z_{t},k_{t})\equiv(0,0,0,0)\) for all λ, our problem is to find some real numbers \(\lambda_{i}\), \(i=1,2,\ldots \) , such that the corresponding Hamiltonian system has multi-solutions. Here, \(\lambda_{i}\) are called eigenvalues and the corresponding nontrivial solutions are called eigenfunctions of this class of stochastic Hamiltonian systems. Inspired by the method of dual transformation introduced in [11] and the relationship between stochastic Hamiltonian systems driven by Poisson process and generalized Riccati equation systems given in [9], we obtain the existence of eigenvalues and construct the eigenfunctions explicitly. We also provide some sufficient conditions for the existence of multi-solutions to FBSDEP. Moreover, it follows from the construction of eigenfunctions that they keep the ‘statistic periodicity’ property as that in the deterministic case and also in the stochastic case with Brownian motion. On the other hand, for any real number λ larger than 1, we establish a family of stochastic Hamiltonian systems whose eigenvalue systems contain λ and give the corresponding eigenfunctions. A numerical example is presented to illustrate the theoretical result as well as the phenomenon of statistic periodicity for eigenfunctions.
The rest of this paper is organized as follows. In Section 2, we first recall the formulation of general FBSDEP and then formulate the eigenvalue problem for stochastic Hamiltonian systems driven by Poisson process. The main results are given by two theorems in Section 3: one is the existence of eigenvalues and eigenfunctions for an arbitrarily dimensional case; the other is a more concrete conclusion for a one-dimensional case. To prove our results, we introduce the dual transformation for stochastic Hamiltonian systems and establish the relationship between stochastic Hamiltonian systems driven by Poisson process and a kind of Riccati equation systems in Section 4. Thus, the proofs of two theorems above are completed in Section 5. In Section 6, we discuss the ‘statistic periodicity’ property for eigenfunctions from another viewpoint and present a numerical example to show our theoretical result vividly. The last section is devoted to concluding the novelty of this paper.
2 Formulation
Assumption 2.1
- (i)
For any \(\xi=(x,y,z,k)\in\mathbb{R}^{4n}\), \(f(\xi,\cdot)\in \mathcal{M}^{2}(\mathbb{R}^{4n})\);
- (ii)There exists a constant \(C>0\) such that$$\begin{aligned}& \bigl\vert f(\xi,t)-f\bigl(\xi',t\bigr) \bigr\vert \le C \bigl\vert \xi-\xi' \bigr\vert ,\quad \forall\xi,\xi'\in \mathbb {R}^{4n}, \\& \bigl\vert \Psi(x)-\Psi\bigl(x'\bigr) \bigr\vert \le C \bigl\vert x-x' \bigr\vert , \quad \forall x,x'\in \mathbb{R}^{n}; \end{aligned}$$
- (iii)There exists a constant \(\alpha>0\) such that$$\begin{aligned}& \bigl\langle f(\xi,t)-f\bigl(\xi',t\bigr),\xi-\xi' \bigr\rangle \le-\alpha \bigl\vert \xi-\xi ' \bigr\vert ^{2},\quad \forall\xi,\xi'\in\mathbb{R}^{4n}, \\& \bigl\langle \Psi(x)-\Psi\bigl(x'\bigr),x-x'\bigr\rangle \ge0,\quad \forall x,x'\in\mathbb{R}^{n}. \end{aligned}$$
We have the following from Theorem 3.1 of Wu [4].
Theorem 2.2
Let Assumption 2.1 hold. Then FBSDEP (1) admits a unique solution \((x_{t},y_{t},z_{t},k_{t})\in\mathcal{M}^{2}(\mathbb{R}^{n})\times\mathcal {M}^{2}(\mathbb{R}^{n})\times F_{N}^{2}(\mathbb{R}^{n})\times F_{N}^{2}(\mathbb{R}^{n})\).
Remark 2.3
The result above is slightly stronger than that of Wu [4]. In fact, Theorem 2.2 can be proved by the same arguments of Wu [4] since the martingale representation theorem from [1] guarantees that the component \(z_{t}\) belongs to \(F_{N}^{2}(\mathbb{R}^{n})\).
Definition 2.4
\(\lambda\in\mathbb{R}\) is called an eigenvalue of stochastic Hamiltonian system (7) if (7) corresponding to λ admits nontrivial solutions \((x_{t},y_{t},z_{t},k_{t})\in\mathcal{M}^{2}(\mathbb {R}^{n})\times\mathcal{M}^{2}(\mathbb{R}^{n})\times F_{N}^{2}(\mathbb{R}^{n})\times F_{N}^{2}(\mathbb{R}^{n})\). These solutions are the eigenfunctions corresponding to λ. The linear subspace of \(\mathcal {M}^{2}(\mathbb{R}^{n})\times\mathcal{M}^{2}(\mathbb{R}^{n})\times F_{N}^{2}(\mathbb {R}^{n})\times F_{N}^{2}(\mathbb{R}^{n})\) consisting all eigenfunctions corresponding to eigenvalue λ is the eigenfunction subspace corresponding to λ.
3 Main results
Theorem 3.1
Let condition (4) hold. Then the system of all eigenvalues of stochastic Hamiltonian system (8) has at least one element \(\lambda>0\), which is the smallest eigenvalue. Moreover, the dimension of the eigenfunction subspace corresponding to λ is no more than n.
Theorem 3.2
For \(n=1\), let condition (4) hold, and assume \(H_{23}=-H_{33}H_{13}\), \(H_{24}=-H_{44}H_{14}\). Then the system of all eigenvalues of stochastic Hamiltonian system (9) is a strictly increasing real number sequence \(\{\lambda_{i}\}_{i\ge1}\) with \(\lambda _{i}\) going to infinity as \(i\rightarrow\infty\). Moreover, the dimension of the eigenfunction subspace corresponding to each \(\lambda_{i}\) is 1.
4 Dual transformation of stochastic Hamiltonian systems with Poisson process
Inspired by the method given in Peng [11], we introduce the dual transformation of stochastic Hamiltonian systems driven by Poisson process. We shall see that this dual transformation is a powerful tool for solving the eigenvalue problem.
4.1 General case
Suppose that the stochastic Hamiltonian system (2) admits a solution \((x_{t},y_{t},z_{t},k_{t})\). Now, we exchange the role of \(x_{t}\) and \(y_{t}\), i.e., define \((\tilde{x}_{t},\tilde{y}_{t})=(y_{t},x_{t})\) to see whether \((\tilde{x}_{t},\tilde{y}_{t})\) will still satisfy some stochastic Hamiltonian system. In addition, we assume
Assumption 4.1
h and Φ are both \(\mathcal{C}^{2}\) functions. Moreover, for all \((x,y)\in\mathbb{R}^{2n}\), \(h(x,y,\cdot,\cdot)\) is concave and \(\Phi (\cdot)\) is convex.
4.2 Linear case
4.3 Generalized Riccati equation systems
The Riccati equations are widely applied to investigate the linear-quadratic optimal control problems, e.g., Wonham [12], Bismut [6], Peng [8], Wu and Wang [9] and so on. Inspired by [9], we shall reformulate the Riccati equations in a general form.
Remark 4.2
- (i)
For any given constant \(\gamma<1\) and \(K\in D^{\gamma}_{F}=\{ K\in\mathcal{S}^{n}: K\ge\gamma H^{-1}_{33}\vee\gamma H^{-1}_{44}\}\), \((I_{n}-K H_{33})^{-1}\) and \((I_{n}-K H_{44})^{-1}\) exist and are both uniformly bounded in \(D^{\gamma}_{F}\);
- (ii)For any \(K\in\mathcal{S}^{n}_{+}\), \(F_{0}(K)\) and \(F_{1}(K)\) are both bounded and monotone. More precisely, we haveand$$0\le F_{0}(K)\le-H^{-1}_{33},\qquad 0\le F_{1}(K)\le-H^{-1}_{44},\quad \forall K\ge0, $$$$F_{0}(K_{1})\ge F_{0}(K_{2}),\qquad F_{1}(K_{1})\ge F_{1}(K_{2}), \quad \forall K_{1}\ge K_{2}\ge0. $$
The following lemma shows the relationship between generalized Riccati equation system (14) and linear stochastic Hamiltonian system (12).
Lemma 4.3
Proof
It can be easily verified that (19) is a solution of linear stochastic Hamiltonian system (12) with boundary condition (18) by applying Itô’s formula to \(K(t)x_{t}\).
On the other hand, under the weaker condition (21) instead of (15), again by the similar arguments for proving Theorem 2.1 of [4], we can still show that \((\hat{y}_{t},z'_{t},k'_{t})\equiv (0,0,0)\). Thus, it follows from the forward SDE in (12) that \(\{ x_{t}\}\) is the solution of (20). So we have \((x_{t},y_{t},z_{t},k_{t})\equiv(x(t), K(t)x(t), M(t)x(t-), L(t)x(t-))\). The proof is completed. □
Remark 4.4
Lemma 4.5
The proof of Lemma 4.5 is very similar to that of Lemma 8.2 in [10]. So we just omit it.
5 Proof of main results
We shall complete the proofs of Theorems 3.1 and 3.2 in this section. It will be seen that the features of eigenvalues and corresponding eigenfunctions are dominated by the blow-up times of solutions to related Riccati equation systems of the linear stochastic Hamiltonian systems (8) and (9).
5.1 Proof of Theorem 3.1
Lemma 5.1
Proof
As for the case where \(\rho\in(-\infty, 0)\), thanks to Lemma 4.5, we can prove all conclusions by the very similar method introduced in Lemmas 5.1 and 5.2 of [10]. So we just omit it. □
With the results above in hand, now we can give the proof of Theorem 3.1 as follows.
Proof of Theorem 3.1
According to Lemma 5.1, there exists a unique \(\rho^{1}<0\) such that the blow-up time of the corresponding Riccati equation (28) is \(t_{\rho^{1}}=0\). Then, by the very similar arguments for the proof of Theorem 3.1 in [10], we can prove that \(\lambda^{1}=1-\rho^{1}\) is the smallest eigenvalue of linear stochastic Hamiltonian system (8), and the dimension of the eigenfunction subspace corresponding to \(\lambda_{1}\) is no more than n. This completes the proof. □
5.2 Proof of Theorem 3.2
Lemma 5.2
This lemma can be proved by the same arguments as Lemma 5.1. So we just omit it. Very similarly, we can get the following properties of blow-up time \(\tilde{t}_{\rho}\) of (30) with the terminal condition \(\tilde{K}(T)=0\).
Lemma 5.3
Thus, just by the same arguments as the proof of Theorem 3.2 in [10], we can find a sequence of eigenvalues \(\lambda_{1}<\lambda _{2}<\lambda_{3}<\cdots\) and construct corresponding eigenfunctions, of which the dimension with respect to each \(\lambda_{i}\) is 1. So we omit the details for the proof of Theorem 3.2.
6 Statistic periodicity
It follows from the proof of Theorem 3.2 that the eigenfunctions of linear stochastic Hamiltonian system (9) own the ‘statistic periodicity’ property. Now, we observe this property for stochastic Hamiltonian systems from another viewpoint.
Proposition 6.1
Remark 6.2
- (i)
Comparing Figures 5 and 6 with Figures 3 and 4, we can see that the period of eigenfunctions corresponding to \(\lambda=3\) is shorter than that corresponding to \(\lambda=2\). In fact, it follows from Lemma 4.5 that the blow-up times \(t_{\rho}\) and \(\tilde{t}_{\rho}\) of (33) and (34) will rise when λ becomes larger. Thus, the period of eigenfunctions will decrease indeed as λ increases.
- (ii)
In the numerical example above, the eigenfunctions \(\{x_{t}\}\) and \(\{y_{t}\}\) are both continuous since \(H_{14}\) and \(H_{24}\) are assumed to be 0 and thus \(k_{t}\equiv0\) for \(t\in[0, T_{5}]\). We demonstrate this continuous case for simplicity and convenience to see the statistic periodicity of eigenfunctions visually. The general situation can be dealt with by the same method, and the corresponding eigenfunctions are with jumps. So we omit the details.
7 Conclusions
To our best knowledge, it is the first time to consider the eigenvalue problem for stochastic Hamiltonian systems driven by Poisson process with boundary conditions. Under certain conditions, we obtain the existence of eigenvalues and corresponding eigenfunctions by means of the dual transformation and generalized Riccati equation systems. From another viewpoint, for any real number \(\lambda>1\), we can establish a family of linear stochastic Hamiltonian systems whose eigenvalue systems contain λ and give the corresponding eigenfunctions explicitly. Moreover, a specific numerical example is studied to illustrate our theoretical results above and show the ‘statistic periodicity’ vividly for the eigenfunctions of stochastic Hamiltonian systems. Besides, the main results of this paper can help us to construct some examples of multi-solutions for FBSDEP.
On the other hand, as is shown in [10], our problem can also be formulated as an eigenvalue problem for a bounded and self-adjoint operator in a Hilbert space, and then investigated in a standard way by the theory of functional analysis. We leave the details to interested readers.
Declarations
Acknowledgements
The second author is supported by the Natural Science Foundation of China (61573217, 11125102, 11221061 and 61174092), the National High-Level personnel of special support program and the Chang Jiang Scholar Program of Chinese Education Ministry.
Authors’ contributions
All authors contributed equally to the manuscript, read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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
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