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Register a new userAsymptotic analysis of forward performance processes in incomplete markets and their ill-posed HJB equations
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Added by
Mykhaylo Shkolnikov
Publication date
2015
fields
Financial
and research's language is
English
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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 investors 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.
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We consider the problem of optimal portfolio selection under forward investment performance criteria in an incomplete market. Given multiple traded assets, the prices of which depend on multiple observable stochastic factors, we construct a large class of forward performance processes with power-utility initial data, as well as the corresponding optimal portfolios. This is done by solving the associated non-linear parabolic partial differential equations (PDEs) posed in the wrong time direction, for stock-factor correlation matrices with eigenvalue equality (EVE) structure, which we introduce here. Along the way we establish on domains an explicit form of the generalized Widders theorem of Nadtochiy and Tehranchi [NT15, Theorem 3.12] and rely hereby on the Laplace inversion in time of the solutions to suitable linear parabolic PDEs posed in the right time direction.
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