- Research
- Open Access
Non-smooth analysis method in optimal investment-BSDE approach
- Helin Wu^{1},
- Yong Ren^{2} and
- Feng Hu^{3}Email author
https://doi.org/10.1186/s13662-018-1920-4
© The Author(s) 2018
- Received: 5 June 2018
- Accepted: 4 December 2018
- Published: 12 December 2018
Abstract
In this paper, the investment process is modeled by backward stochastic differential equation. We investigate a necessary condition for optimal investment problem by the method of non-smooth analysis. Furthermore, some applications of our result are given.
Keywords
- Backward stochastic differential equation (BSDE)
- Non-smooth analysis
- Optimal investment
MSC
- 60H10
- 93E20
- 49J52
1 Introduction
Single-period mean-variance portfolio selection was first introduced and studied by Markowitz [14]. Markowitz’s pioneer work [14] has laid down the foundation for modern financial portfolio theory. In 2000, Li and Ng [11] extended the Markowitz model to the dynamic setting. From then on, various continuous time Markowitz models were studied in much of the literature (see, e.g., [1, 12, 13, 21]).
Generally, there are two approaches, i.e. the forward (primal) approach and the backward (dual) approach, employed to solve the above problem in the continuous time case. The first approach is inspired by indefinite stochastic linear quadratic control (see, e.g., [3, 20]) and builds the relationship between the mean-variance problem and a family of indefinite stochastic linear quadratic optimal control problems (see, e.g., [12, 13, 21]). The second approach is first studied by Bielecki et al. in [1], which is the generalization of the well-known risk neutral computational approach in the discrete time case (see, e.g., [9, 17, 18]). The backward approach includes two steps: the first step is to compute the optimal terminal wealth; the second step is to compute the replicating portfolio strategy corresponding to the obtained optimal terminal wealth. As shown in [1], the optimal terminal wealth is first obtained using the Lagrange multiplier method and then the optimal replicating portfolio strategy is obtained by solving a backward stochastic differential equation (BSDE for short). Along this line, in 2008, Ji and Peng [10] used Ekeland’s variational principle to obtain a necessary condition for portfolio optimization problem by the backward approach. In this paper, we study optimal investment problem by the method of non-smooth analysis, which makes more general portfolio optimization problem be inside the scopes of our consideration.
This paper is organized as follows. In Sect. 2, we give some results about non-smooth analysis that are used in this paper. In Sect. 3, we study the optimal investment problem on wealth and portfolio processes. With the help of the method of non-smooth analysis, a necessary condition for an optimal objective is obtained, which generalizes the result of Ji and Peng [10]. In Sect. 4, we give some examples as the applications of our result.
2 Some results about non-smooth analysis
In this section, we give some results about non-smooth analysis that are used in this paper. The following definitions, lemmas and propositions can be found in [4, 5].
Suppose that X is a Banach space, and \(X^{*}\) is its dual space. Obviously, \(L^{2}(\mathcal{F}_{T})\) is a Banach space with the norm \(\Vert \xi \Vert :=(E[\xi^{2}])^{\frac{1}{2}}\) for any \(\xi\in L^{2}(\mathcal{F}_{T})\), and \(L^{2}(\mathcal{F}_{T})\) is also its dual space.
Definition 1
Remark 1
Lemma 1
(Exact penalization)
Suppose that f is a Lipschitz function with coefficient M defined on X, \(x^{*}\in C\subset X\) and f takes its minimum value at \(x^{*}\) on C. Then, for any \(\hat {M}\geq M\), \(g(x):=f(x)+\hat{M} d_{C}(x)\) attains minimum value at \(x_{0}\) on X. On the contrary, if \(\hat{M}>M\) and C is closed, then the minimum point of g, \(x_{0}\) must belong to C.
Remark 2
Definition 2
Proposition 1
Proposition 2
Remark 3
Definition 3
Lemma 2
A function f defined on X is strictly differentiable at \(x\in X\), then \(\partial^{o} f(x)=\{x^{*}\}\), where \(x^{*}\) can be obtained by (10).
Lemma 3
3 Maximum principle for optimal investment problem
The following proposition shows that \(\mathcal{E}_{0,T}^{g}\) is Lipschitz in \(L^{2}(\mathcal{F}_{T})\). For more details, see [2, 22, 23].
Proposition 3
Remark 4
The following theorem is our main result.
Theorem 1
In order to prove Theorem 1, we need the following lemma.
Lemma 4
Suppose that g satisfies (A.1) and (A.2). Then, for any \(\xi\in L^{2}(\mathcal{F}_{T})\), we have \(0\notin\partial^{o} \mathcal{E}_{0,T}^{g}(\xi)\).
Proof
We get a contradiction with \(h^{o}(\xi^{*}; \eta)\geq0\). The proof of Lemma 4 is complete. □
Proof of Theorem 1
From Remark 4, we know that \(\mathcal{E}_{0,T}^{g}\) is Lipschitz in \(L^{2}(\mathcal{F}_{T})\). By Definition 1, we can see that \(\partial^{o} \mathcal {E}_{0,T}^{g}(\xi)\) is not empty for any \(\xi\in L^{2}(\mathcal{F}_{T})\). Indeed, we have following result.
Lemma 5
Proof
In Ji and Peng [10], the authors assume that g is continuously differentiable in \((y,z)\). In this special case, we can get an explicit form of \(\partial^{o} \mathcal{E}_{0,T}^{g}\).
Lemma 6
Proof
By Theorem 1 and Lemma 6, we immediately obtain the following theorem.
Theorem 2
4 Some examples
In this section, we give three examples for our obtained result. In the sequel, suppose that g satisfies (A.1), (A.2), and is continuously differentiable in \((y,z)\).
We can extend Problem A to a more general framework.
In order to study the optimization problems A and B, we need the following lemma.
Lemma 7
If \(u:R\mapsto R\) has bounded continuous derivative \(u^{\prime}\), then the function \(\rho(\xi)=E[u(\xi)]\) for any \(\xi\in L^{2}(\mathcal {F}_{T})\) is strictly differentiable, and \(\partial^{o} \rho(\xi)=\{u^{\prime}(\xi)\}\).
By the Fubini theorem and the dominated convergence theorem, we can easily prove Lemma 7. So we omit it.
Theorem 3
Proof
Theorem 4
Proof
Case 1: \(\mathcal{E}_{0,T}^{g}(\xi^{*})-x>\max\{E[u(\xi ^{*})]-c,-\mathcal{E}_{0,T}^{g}(\xi^{*}) \}\). By Theorem 2, there exists a non-negative constant λ such that \(\varphi^{\prime}(\xi^{*})+\lambda q^{*}_{T}=0\) holds.
Case 2: \(-\mathcal{E}_{0,T}^{g}(\xi^{*})>\max\{E[u(\xi ^{*})]-c,\mathcal{E}_{0,T}^{g}(\xi^{*})-x \}\). By Theorem 2, there exists a non-negative constant λ such that \(\varphi^{\prime}(\xi^{*})-\lambda q^{*}_{T}=0\) holds.
Case 3: \(E[u(\xi^{*})]-c>\max\{\mathcal{E}_{0,T}^{g}(\xi ^{*})-x,-\mathcal{E}_{0,T}^{g}(\xi^{*}) \}\). By Theorem 2, there exists a non-negative constant λ such that \(\varphi^{\prime}(\xi^{*})+\lambda u^{\prime}(\xi^{*})=0\) holds.
Case 4: \(E[u(\xi^{*})]-c=\mathcal{E}_{0,T}^{g}(\xi^{*})-x>-\mathcal {E}_{0,T}^{g}(\xi^{*})\). By Theorem 2 and Lemma 3, there exist two constants \(\lambda\geq0\) and \(\alpha\in[0,1]\) such that \(\varphi^{\prime}(\xi^{*})+\lambda[\alpha u^{\prime}(\xi^{*})+(1-\alpha)q^{*}_{T} ]=0\) holds.
Case 5: \(E[u(\xi^{*})]-c=-\mathcal{E}_{0,T}^{g}(\xi^{*})>\mathcal {E}_{0,T}^{g}(\xi^{*})-x\). By Theorem 2 and Lemma 3, there exist two constants \(\lambda\geq0\) and \(\alpha\in[0,1]\) such that \(\varphi^{\prime}(\xi^{*})+\lambda[\alpha u^{\prime}(\xi^{*})+(\alpha-1)q^{*}_{T} ]=0\) holds.
Case 6: \(\mathcal{E}_{0,T}^{g}(\xi^{*})-x=-\mathcal{E}_{0,T}^{g}(\xi ^{*})>E[u(\xi^{*})]-c\). By Theorem 2 and Lemma 3, there exist two constants \(\lambda\geq0\) and \(\alpha\in[0,1]\) such that \(\varphi^{\prime}(\xi^{*})+\lambda(2\alpha-1)q^{*}_{T}=0\) holds.
Case 7: \(\mathcal{E}_{0,T}^{g}(\xi^{*})-x=-\mathcal{E}_{0,T}^{g}(\xi ^{*})=E[u(\xi^{*})]-c\). By Theorem 2 and Lemma 3, there exist three constants \(\lambda\geq0\) and \(a,\beta\in[0,1]\) satisfying \(\alpha+\beta \leq1\), such that \(\varphi^{\prime}(\xi^{*})+\lambda[(\alpha-\beta)q^{*}_{T}+(1-\alpha-\beta )u^{\prime}(\xi^{*}) ]=0\) holds. The proof of Theorem 4 is complete. □
In the classic investment problem, one often takes the variance as a risk measure and the mean-variance method is used in much of the literature. But by Delbaen [6], and Föllmer and Schied [8], such a kind of risk measure is not perfect. We often take \(\rho(\cdot)\) as a coherent or convex risk measure in (2). In Rosazza Gianin [19], the author proved that if g is a sub-additive and positively homogeneous function satisfying (A.1) and (A.2), define \(\rho(\xi):=\mathcal{E}_{0,T}^{g}(-\xi)\), for any \(\xi\in L^{2}(\mathcal{F}_{T})\), then \(\rho(\cdot)\) is a convex risk measure.
Theorem 5
The proof of Theorem 5 is very similar to that of Theorem 4. So we omit the proof.
Declarations
Acknowledgements
The authors would like to thank the anonymous referees for their constructive suggestions and valuable comments.
Funding
The work of Helin Wu is supported by the Scientific and Technological Research Program of Chongqing Municipal Education Commission (KJ1400922). The work of Yong Ren is supported by the National Natural Science Foundation of China (11871076). The work of Feng Hu is supported by the National Natural Science Foundation of China (11801307) and the Natural Science Foundation of Shandong Province (ZR2016JL002 and ZR2017MA012).
Authors’ contributions
The authors contributed equally and significantly in writing this article. All authors 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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