Overview: We 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).
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.
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.
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.
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.