University of Waterloo Canada

Many classical fractal functions, such as the Weierstrass and Takagi-van der Waerden functions, admit a finite p-th variation along a natural sequence of partitions. They can thus serve as integrators in pathwise Itô calculus. Motivated by this observation, we
introduce a new class of stochastic processes, which we call Weierstrass bridges. They have continuous sample paths and arbitrarily low regularity and so provide a new example class of “rough” stochastic processes. We study some of their sample path properties
including p-th variation and moduli of continuity. This talk includes joint work with Xiyue Han and Zhenyuan Zhang.

This seminar is run jointly with OMI.

Throughout the Western world, defined benefit pension plans are disappearing, replaced by defined contribution (DC) plans. Retail investors are thus faced with managing investments over a thirty year accumulation period followed by a twenty year decumulation phase. Holders of DC plans are thus truly long term investors. We consider dynamic mean variance asset allocation strategies for long term investors. We derive the "embedding result" which converts the mean variance objective into a form suitable for dynamic programming using an intuitive approach. We then discuss a semi-Lagrangian technique for numerical solution of the optimal control problem via a Hamilton-Jacob-Bellman PDE. Parameters for the inflation adjusted return of a stock index and a risk free bond are determined by examining 89 years of US data. Extensive synthetic market tests, and resampled backtests of historical data, indicate that the multi-period mean variance strategy achieves approximately the same expected terminal wealth as a constant weight strategy, while reducing the probability of shortfall by a factor of two to three.