On the parabolic equation for portfolio problems

Volume 122 / 2020

Dariusz Zawisza Banach Center Publications 122 (2020), 287-302 MSC: Primary 91G80; Secondary 93E20, 35K58, 60A38, 91B25. DOI: 10.4064/bc122-16


We consider a semilinear equation linked to the finite horizon consumption-investment problem under stochastic factor framework, prove it admits a classical solution and provide all obligatory estimates to successfully apply a verification reasoning. The paper covers the standard time additive utility, as well as the recursive utility framework. We extend existing results by considering more general factor dynamics including a nontrivial diffusion part and a stochastic correlation between assets and factors. In addition, this is the first paper which compromise many other optimization problems in finance, for example those related to the indifference pricing or the quadratic hedging problem. The extension of the result to the stochastic differential utility and robust portfolio optimization is provided as well. The essence of our paper lays in using improved stochastic methods to prove gradient estimates for suitable HJB equations with restricted control space.


  • Dariusz ZawiszaInstitute of Mathematics
    Faculty of Mathematics and Computer Science
    Jagiellonian University
    Łojasiewicza 6, 30-348 Kraków, Poland

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