The on-line interactive Complexity Option Game allows players to test their intuition and knowledge of complexity and American options.
Its companion paper is “Pricing complexity options”, to appear in the journal Algorithmic Finance, joint with Math graduate Malihe Alikhani, and Shidler graduates Amir Pakravan and Babak Saadat.
The paper introduces a thought experiment: a financial derivative based on the complexity of a sequence of up and down ticks of a stock price.
The difficulty of succeeding in this game may be related to the phenomenon of bounded rationality, to be discussed by Lance Fortnow during the 11th International Conference on Computability, Complexity and Randomness, UH Manoa, January 4-8, 2016.
Event Sponsor
Mathematics, Manoa Campus
More Information
Bjørn Kjos-Hanssen, 9568595, bjoern.kjos-hanssen@hawaii.edu, http://math.hawaii.edu/wordpress/bjoern/software/web/complexity-option-game/
Speaker: Nicolas Monod (Ecole Poly Fed. de Lausanne)
Title: Cutting and pasting: Frankenstein’s method in group theory
Abstract:
We have known for a century that a ball can be decomposed into five pieces and these pieces rearranged so as to produce two balls of the same size as the original.
This apparent paradox has led von Neumann to the notion of amenability which is now much studied in many areas of mathematics.
However, the initial paradox has remained tied down to an elementary property of free groups of rotations for most of the 20th century. I will describe recent progress leading to new paradoxical groups.
Speaker: Pekka Koskela (University of Jyväskylä)
Title: A geometric characterization for planar Sobolev extension domains
Speaker: Peter Binev (USC)
Title: Data Assimilation in Reduced Modeling
Abstract: We consider the problem of optimal recovery of an element $u$ of a Hilbert space $\mathcal{H}$ from measurements of the form $\ell_j(u)$, $j=1, \dots,m$, where the $\ell_j$ are known linear functionals on $\mathcal{H}$. Problems of this type are well studied and usually are carried out under an assumption that $u$ belongs to a prescribed model class, typically a known compact subset of $\mathcal{H}$.
Motivated by reduced modeling for solving parametric partial differential equations,
we consider another setting where the additional information about $u$ is in the form of how well $u$ can be approximated by a certain known subspace $V_n$ of $\mathcal{H}$ of dimension $n$,
or more generally, in the form of how well $u$ can be approximated by each of a sequence of nested subspaces $V_0\subset V_1 \cdots \subset V_n$ with each $V_k$ of dimension $k$. A recovery algorithm for the one-space formulation was proposed in
[Y. Maday, A.T. Patera, J.D. Penn and M. Yano (2015), {\em A parametrized-background data-weak approach to variational data assimilation: Formulation, analysis, and application to acoustics}, Int. J. Numer. Meth. Engng, 102: 933-965].
We prove that their algorithm is optimal and show how the recovery problem for the one-space problem, has a simple formulation, if certain favorable bases are chosen to represent $V_n$ and the measurements. Our major contribution is to analyze the multi-space case. It is shown that, in this multi-space case, the set of all $u$ that satisfy the given information can be described as the intersection of a family of known ellipsoids in $\mathcal{H}$. It follows that a near optimal recovery algorithm in the multi-space problem is provided by identifying any point in this intersection.
It is easy to see that the accuracy of recovery of $u$ in the multi-space setting can be much better than in the one-space problems. Two iterative algorithms based on alternating projections are proposed for recovery in the multi-space problem and one of them is analyzed in detail. This analysis includes an a posteriori estimate for the performance of the iterates. These a posteriori estimates can serve both as a stopping criteria in the algorithm and also as a method to derive convergence rates.
Since the limit of the algorithm is a point in the intersection of the aforementioned ellipsoids, it provides a near optimal recovery for $u$.
This is a joint work with Albert Cohen, Wolfgang Dahmen, Ronald DeVore, Guergana Petrova, and Przemyslaw Wojtaszczyk. The results are available at [arXiv:1506.04770].