A Fokker–Plank equation of multiple variables corresponding to a system of SDE is considered. Solution for transition probability density is written in a form of path integral. It is shown that the proposed path integral brings a known result received by a different approach for Heston model. Differences of results based on path integral given in a number of papers were also pointed out.
An application of fractional Brownian motion (fBm) is considered in stochastic financial engineering models. For the known Fokker–Planck equation for the fBm case, a solution for transition probability density for the path integral method was built. It is shown that the mentioned solution does not result from the Gaussian unit of fBm with precise covariance. An expression for approximation of fBm covariance was found for which solutions are found based on the Gaussian measure of fBm and those found based on the known Fokker–Planck equation match.
The integral path method was applied to determine certain stochastic variables which occur in problems of financial engineering. A stochastic variable was defined by a stochastic equation where drift and volatility are functions of a stochastic variable. As a result, for transition probability density, a path integral was built by substituting variables Wiener's path integral (Wiener's measure). For the stochastic equation, Ito rule was applied in order to interpret a stochastic integral. The path integral for transition probability density was also found as a result of the Fokker--Plan