Random Walks, Feynamn Path Integrals, and Multiple Spacetime Dimensions of Quantum Finance

Volker Ziemann

Osamu Tsuchiya

Claudio CORIANO', Belal Ehsan Baaquie, Srikant Marakani

Antonio Sciarretta

In this paper, I propose a realistic interpretation (RI) of quantum mechanics, that is, an interpretation according to which a particle follows a definite path in spacetime. The path is not deterministic but it is rather a random walk. However, the probability of each step of the walk is found to depend from some average properties of the particle that can be interpreted as its propensity to have a certain macroscopic momentum and energy. The proposed interpretation requires spacetime to be discrete. Prediction of standard quantum mechanics coincide with predictions of large ensembles of particles in the RI.

Wei-Shih Yang, Chaobin Liu, Kai Zhang

We consider general quantum random walks in a d -dimensional half-space. We first obtain a path integral formula for general quantum random walks in a d -dimensional space. Our path integral formula is valid for general quantum random walks on Cayley graphs as well. Then the path integral formula is applied to obtain the scaling limit of the exit distribution, the expectation of exit time and the asymptotic behaviour of the exit probabilities, for general quantum random walks in a half-space under some conditions on amplitude functions. The conditions are shown to be satisfied by both the Hadamard and Grover quantum random walks in two-dimensional half-spaces. For the two-dimensional case, we show that the critical exponent for the scaling limit of the hitting distribution is 1 as the lattice spacing tends to zero, i.e. the natural magnitude of the hitting position is of order O (1) if the lattice spacing is set to be 1/ n . We also show that the rate of convergence of the total hitting probability has lower bound n −2 and upper bound n −2+ for any > 0. For a quantum random walk with a fixed starting point, we show that the probability of hitting times at the hyperplane decays faster than that of the classical random walk. In both one and two dimensions, given the event of a hit, the conditional expectation of hitting times is finite, in contrast to being infinite for the classical case. In the one-dimensional case, we also obtain an exact order of the probability distribution of the hitting time at 0.

Raymond Lee