Given a geometric Brownian motion wealth process, log-Normal lower bound is constructed for the returns of regular investing schedule. The distribution parameters this are computed recursively. For dollar cost averaging (equal amounts in equal time intervals), closed form. A lump sum (single amount at 0) schedule described which achieves terminal that matches indicated by bound. Results applied...

We consider a 2D stochastic wave equation driven by Gaussian noise, which is temporally white and spatially colored described the Riesz kernel. Our first main result functional central limit theorem for spatial average of solution. And we also establish quantitative marginal rate convergence total-variation distance. A fundamental ingredient in our proofs pointwise Lp-estimate Malliavin derivat...