Abstract
We use the dynamic programming principle method to obtain the Hamilton-Jacobi-Bellman (HJB) equation for the value function, and solve the optimal portfolio problem explicitly in a Black-Scholes type of market driven by fractional Brownian motion with Hurst parameter [see formula in PDF]. The results are compared with the corresponding well-known results in the standard Black-Scholes market [see formula in PDF]. As an application of our proposed model, two optimal problems are discussed and solved, analytically.
Highlights
The idea of replacing Brownian motion with another Gaussian process in the usual financial models has been around for some time
Rogers[1] showed that arbitrage is possible when the risky asset has a log-normal price driven by a fractional Brownian motion if stochastic integrals are defined using pointwise products
Using the white noise approach it is clear that stochastic integrals should be defined using Wick products
Summary
The idea of replacing Brownian motion with another Gaussian process in the usual financial models has been around for some time. When the factors are strongly independent, a Wick product reduces to a pointwise product, and in the Brownian motion case white noise integral reduces to the usual Itô integral. | | 0 k=0 where f (t, ω) : R R is Skorohod integrable, denotes the Wick product We call these fractional Itô integral because these integrals share many of the properties of the classical Itô integrals. We consider the method of fractional HJB equation This equation is a partial differential equation. As an application of this derivation, two optimal problems have discussed and solved by the method of fractional HJB equation
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
