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Abstract, classic, and explicit turnpikes

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Abstract

Portfolio turnpikes state that as the investment horizon increases, optimal portfolios for generic utilities converge to those of isoelastic utilities. This paper proves three kinds of turnpikes. In a general semimartingale setting, the abstract turnpike states that optimal final payoffs and portfolios converge under their myopic probabilities. In diffusion models with several assets and a single state variable, the classic turnpike demonstrates that optimal portfolios converge under the physical probability. In the same setting, the explicit turnpike identifies the limit of finite-horizon optimal portfolios as a long-run myopic portfolio defined in terms of the solution of an ergodic HJB equation.

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Notes

  1. This interpretation underpins the literature on risk-sensitive control, introduced by Fleming and McEneaney [15], and applied to optimal portfolio choice by Bielecki et al. [4], Bielecki and Pliska [3], Fleming and Sheu [16, 17], Nagai and Peng [41, 42], among others.

  2. A subset A of Ω is N-negligible if there exists a sequence (B n ) n≥0 of subsets of Ω such that for all n≥0, we have \(B_{n} \in\mathcal{F}_{n}\) and \(\mathbb {P}[B_{n}]=0\), and A⊂⋃ n≥0 B n . This notion is introduced in [2, Definition 1.3.23] and [43]. Such a completion of \(\mathcal {F}_{0}\) ensures, for all T≥0, that the space \((\varOmega, \mathcal {F}_{T}, (\mathcal{F}_{t})_{0\leq t\leq T}, \mathbb {P})\) satisfies the usual conditions. Hence all references below on finite-horizon problems with completed filtration can be used in this paper.

  3. For any ϵ>0, there exists M ϵ such that U′(x)≤(1+ϵ)x p−1 for x≥M ϵ . Integrating the previous inequality on (M ϵ ,x) yields \(U(x) \leq(1+\epsilon)(x^{p}-M^{p}_{\varepsilon })/p + U(M_{\varepsilon })\), when x≥M ϵ and 0<p<1, from which the claim follows. The proof for the case p=0 is similar.

  4. These probabilities already appear in the work of Kramkov and Sîrbu [35–37] under the name of \(\bf R\).

  5. The notation \(\widetilde{\mathbb {Q}}\)-lim T→∞ is short for the limit in probability under \(\widetilde{\mathbb {Q}}\).

  6. Since R 0=0 by assumption, \(\mathbb {P}^{\xi}\) with ξ=(0,y) is denoted as \(\mathbb {P}^{y}\). The same convention applies to other probabilities introduced later.

  7. In the model (2.7) and (2.8), u 0,T depends on the initial value of the state variable Y 0=y. Hence u T is a function of y. Since Proposition 2.5 reduces the problem to the comparison of the optimal isoelastic finite-horizon portfolio with its long-run limit, the superscript 0 will be omitted in this section.

  8. Any y 0∈E suffices. This y 0 is chosen to align m with \(\hat{m}\).

  9. If a logarithmic investor is present (γ i =1 for some i), a constant is added to U(x), and the stated equivalence remains valid.

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Acknowledgements

We are grateful to the Associate Editor and two anonymous referees for carefully reading this paper and providing valuable suggestions, which greatly assisted us in improving this paper.

Paolo Guasoni is partially supported by the ERC (278295), NSF (DMS-0807994, DMS-1109047), SFI (07/MI/008, 07/SK/M1189, 08/SRC/FMC1389), and FP7 (RG-248896). Constantinos Kardaras is partially supported by the NSF DMS-0908461. Hao Xing is partially supported by an LSE STICERD grant.

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Authors and Affiliations

  1. School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland

    Paolo Guasoni

  2. Department of Mathematics and Statistics, Boston University, 111 Cummington St, Boston, MA, 02215, USA

    Paolo Guasoni

  3. Department of Statistics, London School of Economics and Political Science, 10 Houghton St, London, WC2A 2AE, UK

    Constantinos Kardaras & Hao Xing

  4. Department of Mathematical Sciences, Carnegie Mellon University, Wean Hall 6113, Pittsburgh, PA, 15213, USA

    Scott Robertson

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  1. Paolo Guasoni
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Correspondence to Hao Xing.

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Guasoni, P., Kardaras, C., Robertson, S. et al. Abstract, classic, and explicit turnpikes. Finance Stoch 18, 75–114 (2014). https://doi.org/10.1007/s00780-013-0216-5

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