Merton's model of optimal portfolio in a Black-Scholes Market driven by a fractional Brownian motion with short-range dependence

Analysis of Fractals, Image Compression, Entropy Encoding, Karhunen-Loève Transforms

“…In financial mathematics, certain models use a stochastic differential and integral equation named after Black and Scholes, called the Black-Scholes equation [APS06], [HK06], [JT06], [Jum05], [Mey06], [RM06]. In its simplest form it is…”

Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black–Scholes equations

“…The basic Merton's model of optimal portfolio in a Black-Scholes market driven by fractional Brownian motion is defined by the dynamics (Jumarie, 2005c) …”

“…(7.15) (v) So to solve our optimal portfolio problem, we shall have to consider each of the dynamics above, one at a time, and at first glance this could be done by directly duplicating our preceding approach (Jumarie, 2005c). (vi) The reader could be puzzled by the fact that we have several possible solutions to the same equation, but this should not be surprising at all, and on the contrary, it is quite right so.…”

Optimal control problem with an integral equation as the control object

“…To avoid the use of stochastic calculus in the optimal portfolio selection problem, Jumarie [6] proposed to use the state moment as a new state variable of the investors' budget equation. The link between fractional variational calculus, the Hamilton-Jacoby equation and fractional dynamic programming was established in [7].…”

“…As in [6], we transform the stochastic problem to a non-stochastic one, and obtain an integral object equation. So, our goal is to find a solution to the problem with an integral object's equation (which corresponds to a stochastic process with short-range or long-range dependence), subject to control constraints.…”