Path integral method for stochastic equations of financial engineering

Abstract

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.

Call warrants pricing formula under mixed-fractional Brownian motion with Merton jump-diffusion

“…This model captures long-memory phenomenon and discontinuous behavior in the logarithmic returns. Future work may include utilizing the path integral method [26,27] to model the pricing of warrants.…”

Fractional Brownian motion in financial engineering models

“…As it is known, the transition probability density for mentioned stochastic differential equation can be defined based on the probabilistic measure of the specified process. It is understood that both approaches must give the same result as it takes place in the case of Brownian motion [19]. In this paper, a solution of the Fokker-Planck equation for the transition probability density of fBm in the form of path integral was obtained.…”

“…, n}, and J({r i }) denotes a Jacobian of variable substitution according to equation (19). The approach of calculating J({r i }) is given in works [19,24,25]. Transition probability density we obtain according to formula ( 9)…”

“…Based on measure (25), let us build a measure for stochastic process r(τ ) that is given by a stochastic equation (16). For this, we shall use the approach for Brownian motion given in [19,25]. In particular, equation ( 16) we shall write in an integral form…”

A Statistical Dependence Framework Based on a Multivariate Normal Copula Function and Stochastic Differential Equations for Multivariate Data in Forestry

“…Forest science researchers often underestimate the utility of the stochastic model by stating that stochastic models are intellectually preferable to and aesthetically superior to deterministic models [7]. Stochastic differential equations are distinguished by their wide applicability in various scientific areas, such as economics, forestry, and medicine [8][9][10][11][12][13]. When it comes to data analysis, it is a multidisciplinary topic that has well survived a period of intense research where new concepts have led to exciting changes and applications for solving problems in various areas of science.…”