publication

Fractional Gaussian fields on surfaces and graphs

Group Members: Tyler Campos, Andrew Gannon, Benjamin Hanzsek-Brill, Connor Marrs, Alexander Neuschotz, Trent Rabe and Ethan Winters. 

 

Mentors: Rachel Bailey, Fabrice Baudoin, Masha Gordina

Overview: We will study and simulate on computers the fractional Gaussian fields and their discretizations on surfaces like the two-dimensional sphere or two-dimensional torus. The study of the maxima of those processes will be done and conjectures formulated concerning limit laws. Particular attention will be paid to log-correlated fields (the so-called Gaussian free field).

 

Box-ball systems and RSK tableaux

Séminaire Lotharingien de Combinatoire (2021)

Sém. Lothar. Combin.  85B  (2021), Art. 14, 12 pp.

Proceedings of the 33rd Conference on Formal Power
Series and Algebraic Combinatorics 

Ben Drucker, Eli Garcia, Emily Gunawan, and Rose Silver

A box-ball system is a collection of discrete time states representing a permutation,
on which there is an action called a BBS move. After a finite number of BBS moves
the system decomposes into a collection of soliton states; these are weakly
increasing and invariant under BBS moves. The students proved that when this
collection of soliton states is a Young tableau or coincides with a partition of a type
described by Robinson-Schensted (RS), then it is an RS insertion tableau. They also
studied the number of steps required to reach this state.

Hedging by Sequential Regression in Generalized Discrete Models and the Follmer-Schweizer decomposition

Group Members

Sarah Boese, Tracy Cui, Sam Johnston

Supervisors

Gianmarco Molino, Olekisii Mostovyi

Overview

In practice, financial models are not exact — as in any field, modeling based on real data introduces some degree of error. However, we must consider the effect error has on the calculations and assumptions we make on the model.  In complete markets, optimal hedging strategies can be found for derivative securities; for example, the recursive hedging formula introduced in Steven Shreve’s “Stochastic Calculus for Finance I” gives an exact expression in the binomial asset model, and as a result the unique arbitrage-free price can be computed at any time for any derivative security.

In incomplete markets this cannot be accomplished; one possibility for computing optimal hedging strategies is the method of sequential regression.  We considered this in discrete-time; in the (complete) binomial model we showed that the strategy of sequential regression introduced by Follmer and Schweizer  is equivalent to Shreve’s recursive hedging formula, and in the (incomplete) trinomial model we both explicitly computed the optimal hedging strategy predicted by the Follmer-Schweizer decomposition and we showed that the strategy is stable under small perturbations.

Publication “Stability and asymptotic analysis of the Föllmer–Schweizer decomposition on a finite probability space” Involve, a Journal of Mathematics , v.13 , 2020 doi.org/10.2140/involve.2020.13.607

Presentation

Poster

The financial value of knowing the distribution of stock prices in discrete market models

The financial value of knowing the distribution of stock prices in discrete market models
Ayelet Amiran, Fabrice Baudoin, Skylyn Brock, Berend Coster, Ryan Craver, Ugonna Ezeaka, Phanuel Mariano and Mary Wishart
Vol. 12 (2019), No. 5, 883–899
DOI: 10.2140/involve.2019.12.883

arXiv:1808.03186

 

 

project page:

Financial Math: Portfolio Optimization and Dynamic Programming

Geodesic Interpolation on the Sierpinski Gasket

Group Members

Cory McCartanLaura LeGareCaitlin Davis.

Supervisors

Luke RogersSweta Pandey.

Overview

Geodesics (shortest paths) on manifolds such as planes and spheres are well understood.  Geodesics on fractal sets such as the Sierpinski Triangle are much more complicated.  We begin by constructing algorithms for building shortest paths and provide explicit formulas for computing their lengths.  We then turn to the question of interpolation along geodesics—given two subsets of the Sierpinski Triangle, we “slide” points in one set along geodesics to the other set.  We construct a measure along the interpolated sets which formalizes a notion of the interpolation of a distribution of mass, and we prove interesting self-similarity relations about this measure.

Publication: J. Fractal Geom. 8 (2021), 117-152 doi.org/10.4171/JFG/100 arXiv:1912.06698

Presentation

Poster

 

Gradients on Higher Dimensional Sierpinski Gaskets

Group Members

Luke Brown,  Giovanni E Ferrer SuarezKaruna Sangam.

Supervisors

Gamal MograbyDan KelleherLuke RogersSasha Teplyaev.

Overview

Laplacians have been well studied on post-critically finite (PCF) fractals. However, less is known about gradients on such fractals. Building on work by Teplyaev, we generalize results regarding the existence and continuity of the gradient on the standard Sierpinski Gasket to higher dimensional Sierpinski Gaskets. In particular, we find that, for functions with a continuous Laplacian, the gradient must be defined almost everywhere, and specify a set of points for which it is defined. Furthermore, we provide a counterexample on higher-dimensional Sierpinski gaskets where the Laplacian is continuous but the gradient is not defined everywhere. We conjecture that Hölder continuity of the Laplacian is a condition strong enough to guarantee that the gradient exists at each point.

Publication: arXiv:1908.10539  Fractals Vol. 28, No. 06, 2050108 (2020)

doi.org/10.1142/S0218348X2050108X

Presentation

Poster