On-line workshop

• Nicolas Perkowski, Infinite regularization by noise: It is a classical yet surprising result that noise can have a regularizing effect on differential equations. For example, adding a Brownian motion to an ODE with a bounded and measurable vector field leads to a well-posed equation with Lipschitz continuous flow. While the equation without noise may have none or many solutions. Classical proofs are based on stochastic analysis and on the link between Brownian motion and heat equation. In that argument it is not obvious which property of the noise gives the regularization. A more recent approach by Catellier and Gubinelli leads to a pathwise understanding of regularization. I will present a simplified version of their approach and use it to construct "infinitely regularizing" paths: after adding them to an ODE we have a unique solution and an infinitely smooth flow - even if the vector field is only a tempered distribution. This is joint work with Fabian Harang.

• Goncalo dos Reis, Numerical methods for McKean Vlasov SDE: the super-measure case: We present a fully probabilistic Euler scheme for the simulation of McKean-Vlasov Stochastic Differential Equations (MV-SDEs) with drifts of super-linear growth in the measure component (also in the spatial component, and random initial condition). We develop a split-step (implicit) scheme attaining an almost 1/2 root-mean-square error rate in stepsize and addresses a gap in the literature regarding efficient numerical methods and their convergence rate for this class of McKean Vlasov SDEs.

• Eva Verschueren, It Takes Two to Tango: Estimation of the Zero-Risk Premium Strike of a Call Option via Joint Physical and Pricing Density Modeling: It is generally said that out-of-the-money call options are expensive and one can ask the question from which moneyness level this is the case. Expensive actually means that the price one pays for the option is more than the discounted average payoff one receives. If so, the option bears a negative risk premium. The objective of this paper is to investigate the zero-risk premium moneyness level of a European call option, i.e., the strike where expectations on the option’s payoff in both the P- and Q-world are equal. To fully exploit the insights of the option market we deploy the Tilted Bilateral Gamma pricing model to jointly estimate the physical and pricing measure from option prices. We illustrate the proposed pricing strategy on the option surface of stock indices, assessing the stability and position of the zero-risk premium strike of a European call option. With small fluctuations around a slightly in-the-money level, on average, the zero-risk premium strike appears to follow a rather stable pattern over time.

• Jonas Tölle, Variability of functions, compositions and differential equations with BV-coefficients: in stochastic analysis, it is well-established to interpret a differential system in integrated form, a viewpoint conceptually strongly related to the distributional formulation of partial differential equations. However, there are many situations, where even the concept of the integral is subtle. Several powerful theories have emerged to treat these situations, such as rough path theory or the theory of regularity structures. On the other hand, these methods are usually applied to situations where the coefficient maps are smooth, and most of the existing methods break down completely if one allows discontinuities (provided that the forcing term is not too regular). In particular, this is the case if we admit general functions of bounded variation (BV-functions) as a possible choice of our nonlinear coefficients.

In this talk, we combine tools from fractional calculus and harmonic analysis, together with certain fine properties of BV-functions, allowing us to give a meaningful definition for (multidimensional) generalized Lebesgue-Stieltjes integrals for sufficiently regular Hölder functions. The key idea is that the unknown function should not spend too much time on the "bad" regions of the BV-coefficient maps. Our novel multiplicative composition estimate leads to a systematic way to quantify this in terms of potential theory of Riesz energies and the occupation measure of the unknown function. We discuss several consequences, and provide existence and uniqueness results for certain differential systems involving BV-coefficients.

While all the results are given in a purely analytic fashion, we give several examples related to stochastic differential equations, with focus on the fractional Brownian motion.

• Gerardo Barrera Vargas, Cutoff thermalization for Langevin dynamics driven by Lévy processes: The so-called cut-off phenomenon was identified in the study of Markov chain models of shuffling cards. By cutoff phenomenon one means abrupt convergence to equilibrium. Ever since, it has become a broad challenge to characterize when the cut-off phenomenon occurs. In the present talk, we study the profile of the convergence to equilibrium for a coercive Langevin dynamics subject to additive small noise of the Lévy type with finite p-moments. We prove that, as the magnitude of the noise tends to zero, the system exhibits cut-off phenomenon to its equilibrium in the Wasserstein distance of index p.

The talk is based in the following joint works with Michael A. Högele (Universidad de los Andes, Colombia) and Juan Carlos Pardo (Mathematics Research Center, CIMAT, Mexico):

1.-Cutoff thermalization for Ornstein-Uhlenbeck systems with small Lévy noise in the Wasserstein distance. JSP, 184-27, 2021,1-54.

2.-The cutoff phenomenon in Wasserstein distance for nonlinear stable Langevin systems with small Lévy noise. To appear in JDDE.

• Henri Elad Altman, Scaling limits of additive functionals of rough processes without self-similarity: Scaling limit results for additive functionals of rough stochastic processes in infinite volume will be discussed. In the case where the processes are self-similar the limit is well-known to be described by local times of the process, however no similar results seem to have been obtained in the case where self-similarity is broken. We propose techniques to tackle this more general problem. This presentation will be based on joint work with Khoa Lê (TU Berlin).

• Jiawei Li, Fluctuations of a nonlinear stochastic Heat equation: In this talk, I will introduce recent progress in the study of Gaussian fluctuations of stochastic heat equations. It is known that the rescaled fluctuation of the solution to SHEs converges weakly to the solution of the Edwards-Wilkinson equation. I will start with the results for linear SHEs and then move on to our results for SHEs with Lipschitz nonlinearity in dimensions three and higher. I will also mention the work in dimension two if time permits.