Application of the path integral method to some stochastic models of financial engineering

Path integral method for stochastic equations of financial engineering

“…Direct computation in (2) consists of solution of stochastic equation (1) for r(τ ) and it's substitution into (2) [17]. It is understood that the solution can be found only for the simple cases.…”

“…The received formula is convenient in application because it allows receiving a path integral by direct substitution of respective values of the stochastic equation. Based on (17) one can receive transition probabilities density for stochastic processes which are used in financial engineering. Examples for Brownian geometric motion and Cox-Ingersoll-Ross (CIR) process [16] are given in Appendix B.…”

“…As we already noted, the Ito rule was used to obtaine formulas (17) for transition probability density. As it is known [2] the Fokker-Plank equation for transition probability density corresponding to stochastic equation ( 1) is…”

“…Path integral (50) is defined in space of "velocities" ν(τ ) [4,8,17,19] where connection with "coordinate" r(τ ) is defined in (49). Path integrals ( 16) and (50) give us equivalent solutions of the problem.…”