Area distribution and the average shape of a Lévy bridge

Abstract

We consider a one dimensional Lévy bridge x_B of length n and index 0 < α< 2, i.e. a Lévy random walk constrained to start and end at the origin after n time steps, x_B(0) = x_B(n)=0. We compute the distribution P_B(A,n) of the area A = \sum_{m=1}^n x_B(m) under such a Lévy bridge and show that, for large n, it has the scaling form P_B(A,n) \sim n^{-1-1/α} F_α(A/n^{1+1/α}), with the asymptotic behavior F_α(Y) \sim Y^{-2(1+α)} for large Y. For α=1, we obtain an explicit expression of F_1(Y) in terms of elementary functions. We also compute the average profile < \tilde x_B (m) > at time m of a Lévy bridge with fixed area A. For large n and large m and A, one finds the scaling form < \tilde x_B(m) > = n^{1/α} H_α({m}/{n},{A}/{n^{1+1/α}}), where at variance with Brownian bridge, H_α(X,Y) is a non trivial function of the rescaled time m/n and rescaled area Y = A/n^{1+1/α}. Our analytical results are verified by numerical simulations.