
Caitlin M. Davis
University of WisconsinMadison, USA

Laura A. LeGare
University of Notre Dame, USA

Cory W. McCartan
Harvard University, Cambridge, USA

Luke G. Rogers
University of Connecticut, Storrs, USA
 DOI 10.4171/JFG/100
 JFG VOL. 8, NO. 2 PP. 117–152
 2018 project page
publication
Fractional Gaussian fields on surfaces and graphs
Group Members: Tyler Campos, Andrew Gannon, Benjamin HanzsekBrill, 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 twodimensional sphere or twodimensional torus. The study of the maxima of those processes will be done and conjectures formulated concerning limit laws. Particular attention will be paid to logcorrelated fields (the socalled Gaussian free field).
Fair pricing and hedging under small perturbations of the numéraire on a finite probability space
William Busching, Delphine Hintz, Oleksii Mostovyi, Alexey Pozdnyakov
https://msp.org/soon/coming.php?jpath=involve
The Information Premium on a Finite Probability Space
Jake Koerner, Joo Seung Lee, Oleksii Mostovyi
Boxball 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 boxball 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 RobinsonSchensted (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 FollmerSchweizer 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 arbitragefree 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 discretetime; 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 FollmerSchweizer 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
A derivation of the BlackScholes option pricing model using a central limit theorem argument
Geodesic Interpolation on the Sierpinski Gasket
Group Members
Cory McCartan, Laura LeGare, Caitlin Davis.
Supervisors
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 selfsimilarity relations about this measure.
Publication: J. Fractal Geom. 8 (2021), 117152 doi.org/10.4171/JFG/100 arXiv:1912.06698
Presentation
Poster
Gradients on Higher Dimensional Sierpinski Gaskets
Group Members
Luke Brown, Giovanni E Ferrer Suarez, Karuna Sangam.
Supervisors
Gamal Mograby, Dan Kelleher, Luke Rogers, Sasha Teplyaev.
Overview
Laplacians have been well studied on postcritically 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 higherdimensional 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.