An application of path integral method to Merton and Vasicek stochastic models of interest rate is considered. Two approaches to a path integral construction are shown. The first approach consists in using Wieners measure with the following substitution of solutions of stochastic equations into the models. The second approach is realised by using transformation from Wieners measure to the integral measure related to the stochastic variables of Merton and Vasicek equations. The introduction of boundary conditions is considered in the second approach in order to remove incorrect time asymptotes from the classic Merton and Vasicek models of interest rates. By the example of Merton model with zero drift, a Dirichlet boundary condition is considered. A path integral representation of term structure of interest rate is obtained. The estimate of the obtained path integrals is performed, where it is shown that the time asymptote is limited.

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--Planck equation solution, corresponding to the stochastic equation. It was shown that these two approaches give equivalent results.

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.