Student Probability & Related Fields Seminar

The project implements the method of pathwise derivatives in MC using adjoint algorithmic differentiation in the context of credit related instruments. Given a portfolio of names with specified hazard rates and pairwise default correlations, we want to estimate the default correlation risk of basket default products. Whereas the traditional method of bumping produces reasonable results as the size of the bump becomes small, the main focus is to obtain the same degree of accuracy while considerably improving the time efficiency of the calculations.

Let M be a compact, smooth Riemannian manifold. Varadhan proved that tlog*p_t(x,y), where p is heat kernel, converges to -E(x,y) where E(x,y) is an energy function defined by distance between x and y. Malliavin and Stroock showed that under some condition, the hessian is asymptotic to -1/t times the variance of a random variable as t goes to zero and Neel more recently described a method to compute the hessian as simply an integral over the set of midpoints of a minimal geodesic from x to y. I will determine such set and compute the hessian of several symmetric spaces such as sphere, projective spaces and Lie Groups.

The calibration of the local volatility model is based on the Dupire equation. However, this equation becomes difficult once the default term is introduced. I will present the original calibration technique as well as explain how to handle the difficulty caused by the default term.

Optimal transport was originally studied round the time of WWII and dealt with the best way of moving materials to different factories. Times have changed and now it has many applications in geometry and probability. I will give basic definitions and provide an introduction to the method of duality used, among others, to obtain bounds on quantities of interest.

Let $U \subseteq \C$ be a bounded domain with smooth boundary and let $F$ be an instance of the continuum Gaussian free field on $U$ with respect to the Dirichlet inner product $\int_U \nabla f(x) \cdot \nabla g(x) dx$. The set $T_U(a)$ of $a$-thick points of $F$ consists of those $z \in U$ such that the average of $F$ on a disk of radius $r$ centered at $z$ has growth $\sqrt{a/\pi} \log \tfrac{1}{r}$ as $r \to 0$. We show that for each $0 \leq a \leq 2$ the Hausdorff dimension of $T_U(a)$ is almost surely $2-a$ and that $T_U(a)$ is invariant under conformal transformations in an appropriate sense. The notion of a thick point is connected to the Liouville quantum gravity measure with parameter $\gamma$ given formally by $\Gamma(dz) = e^{\gamma F(z)} dz$ considered by Duplantier and Sheffield. (Joint work with Xiaoyu Hu and Yuval Peres)

I will describe the progress made in the so-called Kac's model, which is a continuous state space, discrete time Markov chain on either the unit (n-1)-sphere or SO(n). The problem originated from the pair-wise particle collision problem in physics and the unrealistic assumption that total momentum is not conserved (but total energy is). I will prove the best possible L^2 Wasserstein convergence rate of this Markov chain to stationarity and if time permit, indicate briefly the total variation convergence starting from an L^2 initial distribution. A model in which total momentum is conserved will also be outlined and results stated in that case. I would also like to sketch some generalizations I proved along the same line as Olliviera's paper on L^2 Wasserstein convergence rate.