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Asymptotic analysis of forward performance processes in incomplete markets and their ill-posed HJB equations

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  • Mykhaylo Shkolnikov
  • Ronnie Sircar
  • Thaleia Zariphopoulou

Abstract

We consider the problem of optimal portfolio selection under forward investment performance criteria in an incomplete market. The dynamics of the prices of the traded assets depend on a pair of stochastic factors, namely, a slow factor (e.g. a macroeconomic indicator) and a fast factor (e.g. stochastic volatility). We analyze the associated forward performance SPDE and provide explicit formulae for the leading order and first order correction terms for the forward investment process and the optimal feedback portfolios. They both depend on the investor's initial preferences and the dynamically changing investment opportunities. The leading order terms resemble their time-monotone counterparts, but with the appropriate stochastic time changes resulting from averaging phenomena. The first-order terms compile the reaction of the investor to both the changes in the market input and his recent performance. Our analysis is based on an expansion of the underlying ill-posed HJB equation, and it is justified by means of an appropriate remainder estimate.

Suggested Citation

  • Mykhaylo Shkolnikov & Ronnie Sircar & Thaleia Zariphopoulou, 2015. "Asymptotic analysis of forward performance processes in incomplete markets and their ill-posed HJB equations," Papers 1504.03209, arXiv.org, revised Sep 2015.
    • Handle: RePEc:arx:papers:1504.03209
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    References listed on IDEAS

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    1. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    2. M. Musiela & T. Zariphopoulou, 2009. "Portfolio choice under dynamic investment performance criteria," Quantitative Finance, Taylor & Francis Journals, vol. 9(2), pages 161-170.
    3. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-257, August.
    4. M. Musiela & T. Zariphopoulou, 2011. "Initial Investment Choice And Optimal Future Allocations Under Time-Monotone Performance Criteria," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 14(01), pages 61-81.
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    Cited by:

    1. Gechun Liang & Thaleia Zariphopoulou, 2015. "Representation of homothetic forward performance processes in stochastic factor models via ergodic and infinite horizon BSDE," Papers 1511.04863, arXiv.org, revised Nov 2016.
    2. Bahman Angoshtari & Thaleia Zariphopoulou & Xun Yu Zhou, 2016. "Predictable Forward Performance Processes: The Binomial Case," Papers 1611.04494, arXiv.org, revised Mar 2019.

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