Tartaglia–Pascal triangle and Brownian motion in non-euclidean geometries: application to heat and Black–Scholes equations

The recent worldwide explosion of the financial market originating also from the Black–Scholes equation proved how fundamental could be Brownian motion to real life. Brownian motion is deeply rooted into discrete spaces that are well represented by a Tartaglia–Pascal triangle (TPt). Furthermore, this mapping can be extended to the case of the Schrödinger equation: one of the key equations of quantum mechanics. The connection arises from the asymptotic formula for the binomial coefficients and the normal probability density function. This paper shows how this mapping between the discrete spaces, represented through some forms of TPt, extends to Brownian motion in different geometries. One of the well-known cases is the heat equation; another one is the Black–Scholes equation that derives from geometric Brownian motion. It is shown that the TPt becomes a periodic structure for the Brownian motion on a circle. For the geometric Brownian motion, we get a scale deformed TPt the main effect being scale deformations into the corresponding Newton binomial formula. In the asymptotic limit, one recovers the known formula for the sum on a row of the TPt. This approach unveils discrete structures underlying Brownian motion on different geometries revealing a possible conjecture that, for a given stochastic motion, it is always possible to associate a discrete map such that a TPt is obtained. In a general case, outcomes of the elements of the triangle become real numbers.