Ivancevic_Applied-Diff-Geom

1030 Applied Differential Geometry: A Modern Introductionand early exercise features, integrals of the type (6.43) cannot be analyticallysolved. As a consequence, we demand two things from a path integralframework: a very quick way to estimate the transition probability associatedto a stochastic process (6.37) and a clever choice of the integrationpoints with which evaluate the integrals (6.43). In particular, our aim isto develop an efficient calculation of the probability distribution withoutlosing information on the path followed by the asset price during its timeevolution.Transition ProbabilityThe probability distribution function related to a SDE verifies theChapman–Kolmogorov equation [Paul and Baschnagel (1999)]∫p(z ′′ |z ′ ) = dzp(z ′′ |z)p(z|z ′ ), (6.44)which states that the probability (density) of a transition from the valuez ′ (at time t ′ ) to the value z ′′ (at time t ′′ ) is the ‘summation’ over all thepossible intermediate values z of the probability of separate and consequenttransitions z ′ → z, z → z ′′ . As a consequence, if we consider a finite timeinterval [t ′ , t ′′ ] and we apply a time slicing, by considering n+1 subintervalsof length ∆t = (t ′′ − t ′ )/n + 1, we can write, by iteration of (6.44)p(z ′′ |z ′ ) =∫ +∞−∞· · ·∫ +∞−∞dz 1 · · · dz n p(z ′′ |z n )p(z n |z n−1 ) · · · p(z 1 |z ′ ),which, thanks to (6.38), can be written as [Montagna et. al. (2002)]∫ +∞· · · (6.45)−∞∫ +∞· · ·−∞{1dz 1 · · · dz n √(2πσ2 ∆t) exp n+1− 1n+1∑2σ 2 ∆tk=1[z k − (z k−1 + A∆t)] 2 }.In the limit n → ∞, ∆t → 0 such that (n + 1)∆t = (t ′′ − t ′ ) (infinitesequence of infinitesimal time steps), the expression (6.45), as explicitlyshown in [Bennati et. al. (1999)], exhibits a Lagrangian structure and it ispossible to express the transition probability in the path integral formalismas a convolution of the form [Bennati et. al. (1999)]∫{p(z ′′ , t ′′ |z ′ , t ′ ) = D[σ −1˜z] ∫ }t′′exp − L(˜z(τ), ˙˜z(τ); τ)dτ ,Ct ′