In this talk, we consider the coefficients of the Fourier series obtained by exponentiating a logarithmically correlated holomorphic function on the open unit disc, whose Taylor coefficients are independent complex Gaussian variables, the variance of the coefficient of degree k being theta/k where theta > 0 is an inverse temperature parameter. In joint articles with Paquette, Simm and Vu, we show a randomized version of the central limit theorem in the subcritical phase theta < 1, the random variance being related to the Gaussian multiplicative chaos on the unit circle. We also deduce, from results on the holomorphic multiplicative chaos, other results on the coefficients of the characteristic polynomial of the Circular Beta Ensemble, where the parameter beta is equal to 2/theta. In particular, we show that the central coefficient of the characteristic polynomial of the Circular Unitary Ensembles tends to zero in probability, answering a question asked in an article by Diaconis and Gamburd.

Ever wondered how ideas from physics can used in real-world scenarios? Come to this talk to understand what is an option and how they are traded in markets. I will recall some basic notions of stochastic calculus and derive the Black-Scholes (BS) equation for plain vanilla options. The BS equation can be solved using standard path integral techniques, that also allow to price more exotic derivatives. Finally, I will discuss whether the assumptions behind Black-Scholes dynamics are reasonable in real-world markets (spoiler: they're not), volatility smiles and term structures of the implied volatility.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

A consequence of the works of Costello and Lurie is that the Hochschild chain complex of a Calabi-Yau category admits the structure of a framed E_2 algebra (the genus zero operations). I will describe a new algebraic point of view on these operations which admits generalizations to the setting of relative
Calabi-Yau structures, which do not seem to fit into the framework of TQFTs. In particular, we obtain a generalization of string topology to manifolds with boundary, as well as interesting operations on Hochschild homology of Fano varieties. This is joint work with Chris Brav.

We introduce a non-probabilistic, path-by-path framework for continuous-time, path-dependent portfolio allocation. Extending the self-financing concept recently introduced in Chiu & Cont (2023), we characterize self-financing portfolio allocation strategies through a path-dependent PDE and provide explicit solutions for the portfolio value in generic markets, including price paths that are not necessarily continuous or exhibit variation of any order.

As an application, we extend an aggregating algorithm of Vovk and the universal algorithm of Cover to continuous-time meta-algorithms that combine multiple strategies into a single strategy, respectively tracking the best individual and the best convex combination of strategies. This work extends Cover’s theorem to continuous-time without probability.

An explicit first-order drift-randomized Milstein scheme for a regime switching stochastic differential equation is proposed and its bi-stability and rate of strong convergence are investigated for a non-differentiable drift coefficient. Precisely, drift is Lipschitz continuous while diffusion along with its derivative is Lipschitz continuous. Further, we explore the significance of evaluating Brownian trajectories at every switching time of the underlying Markov chain in achieving the convergence rate 1 of the proposed scheme. In this context, possible variants of the scheme, namely modified randomized and reduced randomized schemes, are considered and their convergence rates are shown to be 1/2. Numerical experiments are performed to illustrate the convergence rates of these schemes along with their corresponding non-randomized versions. Further, it is illustrated that the half-order non-randomized reduced and modified schemes outperform the classical Euler scheme.