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

Mykhaylo Shkolnikov, Ronnie Sircar, Thaleia Zariphopoulou

Research output: Contribution to journalArticlepeer-review

17 Scopus citations


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 supplemented by an appropriate convergence result.

Original languageEnglish (US)
Pages (from-to)588-618
Number of pages31
JournalSIAM Journal on Financial Mathematics
Issue number1
StatePublished - 2016
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Finance
  • Applied Mathematics


  • Factor models
  • Forward performance processes
  • HJB equations
  • Incomplete markets
  • Multiscale asymptotic analysis
  • Optimal investment
  • Stochastic volatility models


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