L6

Zhenru Wang
Title: Multi-Index Monte Carlo Estimators for a Class of Zakai SPDEs
Abstract:   
We first propose a space-time Multi-Index Monte Carlo (MIMC) estimator for a one-dimensional parabolic SPDE of Zakai type. We compare the computational cost required for a prescribed accuracy with the Multilevel Monte Carlo (MLMC) method of Giles and Reisinger (2012). Then we extend the estimator to a two-dimensional variant of SPDE. The theoretical analysis shows the benefit of using MIMC in high dimensional problems over MLMC methods. Numerical tests confirm these finding empirically.

Vadim Kaushansky
Title: An extended structural default model with jump risk
Abstact:
We consider a structural default model in an interconnected banking network as in Itkin and Lipton (2015), where there are mutual obligations between each pair of banks. We analyse the model numerically for the case of two banks with jumps in their asset value processes. Specifically, we develop a finite difference method for the resulting two-dimensional partial integro-differential equation, and study its stability and consistency. By applying this method, we compute joint and marginal survival probabilities, as well as prices of credit default swaps (CDS) and first-to-default swaps (FTD), Credit and Debt Value Adjustments (CVA and DVA).

 

We will introduce variants of the optimal transport problem, namely martingale optimal transport problem and subharmonic martingale transport problem. Their motivation is partly from mathematical finance. We will see that in dimension greater than one, the additional constraints imply interesting and deep mathematical subtlety on the attainment of dual problem, and it also affects heavily on the geometry of optimal solutions. If time permits, we will introduce still another variant of the martingale transport problem, called the multi-martingale optimal transport problem.

In practice, stochastic decision problems are often based on statistical estimates of probabilities. We all know that statistical error may be significant, but it is often not so clear how to incorporate it into our decision making. In this informal talk, we will look at one approach to this problem, based on the theory of nonlinear expectations. We will consider the large-sample theory of these estimators, and also connections to `robust statistics' in the sense of Huber.

A Steiner Triple System on a set X is a collection T of 3-element subsets of X such that every pair of elements of X is contained in exactly one of the triples in T. An example considered by Plücker in 1835 is the affine plane of order three, which consists of 12 triples on a set of 9 points. Plücker observed that a necessary condition for the existence of a Steiner Triple System on a set with n elements is that n be congruent to 1 or 3 mod 6. In 1846, Kirkman showed that this necessary condition is also sufficient. In 1974, Wilson conjectured an approximate formula for the number of such systems. We will outline a proof of this
conjecture, and a more general estimate for the number of Steiner systems. Our main tool is the technique of Randomised Algebraic Construction, which
we introduced to resolve a question of Steiner from 1853 on the existence of designs.

In the theory of random graphs, the behaviour of the typical largest component was studied a lot. The initial results on G(n,m), the random graph on n vertices and m edges, are due to Erdős and Rényi. Recently, similar results for planar graphs were obtained by Kang and Łuczak.

In the first part of the talk, we will extend these results on the size of the largest component further to graphs embeddable on the orientable surface S_g of genus g>0 and see how the asymptotic number and properties of cubic graphs embeddable on S_g are used to obtain those results. Then we will go through the main steps necessary to obtain the asymptotic number of cubic graphs and point out the main differences to the corresponding results for planar graphs. In the end we will give a short outlook to graphs embeddable on surfaces with non-constant genus, especially which results generalise and which problems are still open.