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An exactly solvable multiple stochastic optimal stopping problem
- Hidekazu Yoshioka^{1}Email author
https://doi.org/10.1186/s13662-018-1626-7
© The Author(s) 2018
Received: 17 October 2017
Accepted: 2 May 2018
Published: 8 May 2018
Abstract
A new kind of multiple stochastic optimal stopping problem is formulated and its associated recursive variational inequalities are derived. We show that these variational inequalities can be solved exactly in a cascading manner. The relevance of the present problem in analyzing animal migration, which is an ecologically important problem, is also briefly discussed.
Keywords
- Multiple optimal stopping
- Geometric Brownian motion
- Dynamic programming principle
- Variational inequality
1 Introduction
Stochastic optimal stopping models are useful mathematical tools for analyzing decision-making processes in the fields of financial [1, 2], environment [3, 4], and ecology [5–7]. Multiple optimal stopping problems based on stochastic differential equations (SDEs) are among the ones that have been analyzed most in detail because of their rich mathematical structures [8, 9]. Exactly solvable multiple optimal stopping models are useful from both theoretical and practical point of views [10, 11].
The main difference between the present model and the existing models [8–11] is that the former has an ecological background, while the latter have the financial backgrounds. In addition, the performance indices to be maximized or minimized have different functional forms with each other. The resulting VIs have different forms as well. The main contribution of this paper is the derivation of an exact solution to the cascading system of VIs and its ecological implications.
2 Variational inequalities
The recursive equations (6) and (7) are later utilized to show that the present multiple optimal stopping problem results in a cascading system of VIs that can be solved in a cascading manner from \(i = M\) to \(i = 1\). The value function Φ is then obtained: \(\Phi ( z ) = \Phi_{1} ( z )\).
The VI (9) is subject to the boundary condition \(\Phi_{i} ( 0 ) = 0\).
3 Main result
Theorem 1
Proof of Theorem 1
Uniqueness of the solution \(\Phi_{M}\) to VI (9) in a viscosity sense follows from an infinite horizon counterpart of Theorem 3.1 [14]. Regularity conditions \(\Phi_{M} \in C^{1} ( 0, + \infty ) \cap C^{0} ( [ 0, + \infty ) )\) and \(\Phi_{M} \in C^{2} ( 0,\bar{z}_{M} ) \cap C^{2} ( \bar{z}_{M}, + \infty )\) directly follow from the form of \(\Phi_{M}\). Hence, \(\Phi_{M} = \Phi_{M} ( z )\) is twice continuously differentiable almost everywhere for \(z > 0\).
Uniqueness of the solution \(\Phi_{i} \in C^{1} ( 0, + \infty ) \cap C^{0} ( [ 0, + \infty ) )\) is then proven as follows. In addition, \(\Phi_{i} = \Phi_{i} ( z )\) is identified with \(\Phi_{i + 1} = \Phi_{i + 1} ( z )\) for \(z > \bar{z}_{i}\) by the construction, meaning that \(\Phi_{i}\) is twice continuously differentiable except at the \(M - i + 1\) points \(\bar{z}_{i},\bar{z}_{i + 1},\ldots,\bar{z}_{M}\). Furthermore, uniqueness of the solution \(\Phi_{i}\) to VI (9) in a viscosity sense follows from an infinite horizon counterpart of Theorem 3.1 [14]. Therefore, by the induction, it is shown that the statement of the theorem is true if we can construct a sequence \(0 < \bar{z}_{1} < \bar{z}_{2} <\cdots < \bar{z}_{M} < + \infty\). This issue is not encountered for \(M = 1\) where the problem involves a single optimal stopping time, since we have \(0 < \bar{z}_{1} < + \infty\).
Actually, (17) and (18) with \(i = 1\) show that the left-hand side of (17) is increasing with respect to \(q_{1}\), while the right-hand side of (17) is independent of \(q_{1}\). Therefore, we can choose a sufficiently small \(q_{1} > 0\) such that \(B_{1} ( k_{1} - 1 + \beta_{1} ) < f_{1} ( \bar{z}_{2} )\); namely, \(0 < \bar{z}_{1} < \bar{z}_{2}\). We then have \(0 < \bar{z}_{1} < \bar{z}_{2} < \bar{z}_{3} < + \infty\). The proof for \(M \ge 3\) is essentially the same. □
The following proposition is proven in an essentially similar way with Theorem 1.
Proposition 2
The second equation of (29) shows that \(\bar{z}_{i}\) is expressed as a monotonically increasing and unbounded function of \(q_{i} / q_{i + 1}\), implying that \(\bar{z}_{M - 1} < \bar{z}_{M}\) if \(q_{M}\) is sufficiently larger than \(q_{M - 1}\). Similarly, we have \(\bar{z}_{M - 2} < \bar{z}_{M - 1}\) if \(q_{M - 1}\) is sufficiently larger than \(q_{M - 2}\). We can choose larger \(q_{M}\) if necessary. Since M is bounded, we can construct a sequence \(0 < q_{1} < q_{2} <\cdots < q_{M} < + \infty\) such that \(0 < \bar{z}_{1} < \bar{z}_{2} <\cdots< \bar{z}_{M} < + \infty\).
An immediate consequence of Theorem 1 and Proposition 2 is the next proposition.
Proposition 3
\(\Phi_{1}\) is the value function Φ under the assumption of Theorem 1 or that of Proposition 2.
Remark 4
A numerical example of Theorem 1 is provided. Set the following parameter values: \(\delta_{1} = 6.5\), \(\delta_{2} = 4\), \(r_{1} = 3\), \(r_{2} = 2\), \(\sigma_{1} = 0.5\), \(\sigma_{2} = 0.2\), \(\beta_{1} = 0.8\), \(\beta_{2} = 0.5\), \(\beta_{3} = 0.1\), \(q_{1} = 0.4\), and \(q_{2} = 0.9\). In this case, the growth rate \(r_{i}\) of the animal population increases while its fluctuation \(\sigma_{i}\) decreases as i increases, which is an ecologically reasonable situation. Based on these parameter values, we have \(k_{1} = 2.074 > k_{2} = 1.981\), \(\lambda_{1} = 5.920\), and \(\lambda_{2} = 3.005\). These given and calculated constants comply with the assumption of Theorem 1. Furthermore, we have \(A_{1} = 1.393\), \(A_{2} = 1.224\), \(B_{1} = 0.338\), \(B_{2} = 0.599\), \(B_{3} = 1.111\), \(\bar{z}_{1} = 0.145\), and \(\bar{z}_{2} = 0.469\). The obtained results satisfy \(0 < \bar{z}_{1} < \bar{z}_{2}\) and \(A_{1},A_{2} > 0\), which comply with the results of Theorem 1.
Remark 5
Remark 6
The present multiple optimal stopping problem is a simple theoretical model for migration of animals, migratory fishes in particular [16]. A single optimal stopping problem for analyzing animal migration between two habitats has been discussed in Yoshioka and Yaegashi [7] from a numerical viewpoint. Assume that there are \(M + 1\) habitats, which are denoted \(H_{0},H_{1},\ldots,H_{M}\) where \(H_{0}\) is the initial habitat and \(H_{M}\) is the final habitat: the goal of the migration. The stochastic process \(Z_{t}\) represents the biomass of an animal population at the time t. The stopping time \(\tau_{i}\) represents the time to move from \(H_{i - 1}\) to \(H_{i}\). The objective of the animal population is to choose the sequence of stopping times \(\tau_{i}\) (\(1 \le i \le M\)), so that the sum of the cumulative profit gain in each habitat \(H_{i}\) (\(0 \le i \le M - 1\)) and the terminal wealth gained at the goal of migration \(H_{M}\), namely the performance index \(J_{\tau} \) in (2), is maximized.
Meaning of the variables and parameters of the present multiple optimal stopping problem in analyzing fish migration
Parameter | Meaning |
---|---|
\(r_{i}\) | Deterministic growth rate of the population biomass in the habitat \(H_{i}\) |
\(\sigma_{i}\) | Stochasticity involved in the growth of the population biomass in the habitat \(H_{i}\) |
\(\delta_{i}\) | Discount factor of the cumulative profit in the habitat \(H_{i}\) |
\(q_{i}\) | Quality of the habitat \(H_{i}\) |
\(\beta_{i}\) | Sensitivity of the profit gained in the habitat \(H_{i}\) on the population biomass |
The assumptions \(\delta_{i} > r_{i}\) and \(\lambda_{i} > 0\) for \(1 \le i \le M\) and \(k_{i} > k_{i + 1}\) for \(1 \le i \le M - 1\) when \(M \ge 2\) actually have ecological meanings for the animal migration problems. The conditions \(\delta_{i} > r_{i}\) and \(\lambda_{i} > 0\) can be restated as that \(\delta_{i} > 0\) is sufficiently large, implying that the habitat quality degrades as the time elapses. This is in accordance with the fact that animal migration is often driven by seasonal changes of habitat quality. The remaining condition \(k_{i} > k_{i + 1}\) is satisfied if \(\delta_{i}\) is sufficiently larger than \(\delta_{i + 1}\). For the animal migration, this condition to the situation where degradation of the habitat quality is critical for the earlier period of the animal life history.
4 Conclusions
This paper focused on a solvable multiple optimal stopping problem related to animal migration. An extension of the present problem is to consider a refraction \(\tau_{i + 1} - \tau_{i} \ge \mu_{i} > 0\), which leads to a different system of VIs, and consequently different value functions and optimal stopping criteria. Solvability of the problem with a refraction is currently under investigation for more realistic mathematical modelling of animal migration. There exist recent studies on optimization models of ecological and biological systems involving delays [18–20]. These systems are clearly more complicated than the system focused on in this paper. To examine the applicability of the present methodology, to extend these models will be a quite interesting topic. Applicability of the present formalism to real animal migration, which is based on a mixed control problem like those in Koo et al. [21] and Lee and Shin [22], is also currently in progress.
Declarations
Acknowledgements
JSPS Research Grant No. 17K15345 and a grant to Shimane University Fisheries Management Research Center from the Ministry of Land, Infrastructure, Transport and Tourism of Japan support this research. The author thanks the two Reviewers for their valuable comments and suggestions. The author also thanks Mr. Yuta Yaegashi of Graduate School of Agriculture, Kyoto University for his careful check on Grammar and Style of the manuscript.
Availability of data and materials
All the data are provided in the manuscript.
Funding
JSPS Research Grant No. 17K15345 supports this research. A grant to Shimane University Fisheries Management Research Center from the Ministry of Land, Infrastructure, Transport and Tourism of Japan.
Authors’ contributions
HY carried out mathematical analysis and wrote the manuscript. Author read and approved the final manuscript.
Competing interests
The author declares that he has 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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