Past Stochastic Analysis Seminar

Abstract: Consider a gaussian field f on R^2 and a level l. One can define a random coloring of the plane by coloring a point x in black if f(x)>-l and in white otherwise. The topology of this coloring is interesting in many respects. One can study the "small scale" topology by counting connected components with fixed topology, or study the "large scale" topology by considering black crossings of large rectangles. I will present results involving these quantities.

In semimartingale optimal transport problem, the functional to be minimized can be considered as a “stochastic action”, which is the expectationof a “stochastic Lagrangian” in terms of differential semimartingale characteristics. Therefore it would be natural to apply variational calculus approach to characterize the minimizers. R. Lassalle and A.B. Cruzeiro have used this approach to establish a stochastic Euler-Lagrangian condition for semimartingale optimal transport by perturbing the drift terms. Motivated by their work, we want to perform the same type of calculus for martingale optimal transport problem. In particular, instead of only considering perturbations in the drift terms, we try to find a nice variational family for volatility,and then obtain the stochastic Euler-Lagrangian condition for martingale laws. In the first part of this talk we will mention some basic results regarding the existence of minimizers in semimartingale optimal transport problem. In the second part, we will introduce Lassalle and Cruzeiro’s  work, and give a simple example related to this topic, where the variational family is induced by time-changes; and then we will introduce some potential problems that are needed to be solved.

We prove the quenched version of the central limit theorem for the displacement of a random walk in doubly stochastic random environment, under the $H_{-1}$-condition, with slightly stronger,  $L^{2+\epsilon}$ (rather than $L^2$) integrability condition on the stream tensor. On the way we extend Nash's moment bound to the non-reversible, divergence-free drift case.  

I will talk about R^n valued random processes driven by a "noise", which is generated by a deterministic dynamical system, randomness coming from the choice of the initial condition.

Such processes were considered by D.Kelly and I.Melbourne.I will present our joint work with I.Chevyrev, P.Friz, I.Melbourne and H.Zhang, where we consider the noise with long term memory. We prove convergence to solution of a stochastic differential equation which is, depending on the noise, driven by either a Brownian motion (optimizing the assumptions of Kelly-Melbourne) or a Lévy process.Our work is made possible by recent progress in rough path theory for càdlàg paths in p-variation topology.

We consider the problem of optimally hedging a portfolio of derivatives in a scenario based discrete-time market with transaction costs. Risk-preferences are specified in terms of a convex risk-measure. Such a framework has suffered from numerical intractability up until recently, but this has changed thanks to technological advances: using hedging strategies built from neural networks and machine learning optimization techniques, optimal hedging strategies can be approximated efficiently, as shown by the numerical study and some theoretical results presented in this talk (based on joint work with Hans Bühler, Ben Wood and Josef Teichmann).

Solutions to Rough Differential Equations (RDE) may be constructed by several means. Beyond the fixed point argument, several approaches rely on using approximations of solutions over short times (Davie, Friz & Victoir, Bailleul, ...). In this talk, we present a generic, unifying framework to consider approximations of flows, called almost flows, and flows through the non-linear sewing lemma. This framework unifies the approaches mentioned above and decouples the analytical part from the algebraic part (manipulation of iterated integrals) when studying RDE. Beyond this, flows are objects with their own properties.New results, such as existence of measurable flows when several solutions of the corresponding RDE exist, will also be presented.

From a joint work with Antoine Brault (U. Toulouse III, France).

In the context of randomly fluctuating surfaces in (1+1)-dimensions two Universality Classes have generally been considered, the Kardar-Parisi-Zhang and the Edwards-Wilkinson. Models within these classes exhibit universal fluctuations under 1:2:3 and 1:2:4 scaling respectively. Starting from a modification of the classical Ballistic Deposition model we will show that this picture is not exhaustive and another Universality Class, whose scaling exponents are 1:1:2, has to be taken into account. We will describe how it arises, briefly discuss its connections to KPZ and introduce a new stochastic process, the Brownian Castle, deeply connected to the Brownian Web, which should capture the large-scale behaviour of models within this Class. 

The classical Lorentz gas model introduced by Lorentz in 1905, studied further by Sinai in the 1960s, provides a rich source of examples of chaotic dynamical systems with strong stochastic properties (despite being entirely deterministic).  Central limit theorems and convergence to Brownian motion are well understood, both with standard n^{1/2} and nonstandard (n log n)^{1/2} diffusion rates.

In joint work with Paulo Varandas, we discuss examples with diffusion rate n^{1/a}, 1<a<2, and prove convergence to an a-stable Levy process.  This includes to the best of our knowledge the first natural examples where the M_2 Skorokhod topology is the appropriate one.



 

I will consider cost minimizing stopping time solutions to Skorokhod embedding problems, which deal with transporting a source probability measure to a given target measure through a stopped Brownian process. A PDE (free boundary problem) approach is used to address the problem in general dimensions with space-time inhomogeneous costs given by Lagrangian integrals along the paths.  An Eulerian---mass flow---formulation of the problem is introduced. Its dual is given by Hamilton-Jacobi-Bellman type variational inequalities.  Our key result is the existence (in a Sobolev class) of optimizers for this new dual problem, which in turn determines a free boundary, where the optimal Skorokhod transport drops the mass in space-time. This complements and provides a constructive PDE alternative to recent results of Beiglb\"ock, Cox, and Huesmann, and is a first step towards developing a general optimal mass transport theory involving mean field interactions and noise.

SLE curves are an important family of random curves in the plane. They share many similarites with solutions of SDE (in particular, with Brownian motion). Any quesion asked for the latter can be asked for the former. Inspired by that, Yizheng Yuan and I investigate the support for SLE curves. In this talk, I will explain our theorem with more motivation and idea. 

In this seminar we will look at an extension of the well known sewing lemma from rough path theory to fields on [0; 1]k. We will first introduce a framework suitable to study such fields, and then find a criterion for convergence of multiple Riemann type sums of a class of abstract integrands. A simple application of this extension is construct the Young integral for fields.Furthermore, we will discuss the use of this theorem to study integration of fields of lower regularity by using ideas familiar from rough path theory. Moreover, we will discuss difficulties we face by looking at “multi-parameter ODE's” both from an existence and uniqueness point of view.

The physics literature has for a long time posited a connection between the geometry of continuous random fields and discrete percolation models. Specifically the excursion sets of continuous fields are considered to be analogous to the open connected clusters of discrete models. Recent work has begun to formalise this relationship; many of the classic results of percolation (phase transition, RSW estimates etc) have been proven in the setting of smooth Gaussian fields. In the first part of this talk I will summarise these results. In the second I will focus on the number of excursion set components of Gaussian fields in large domains and discuss new results on the mean and variance of this quantity.

By using Malliavin calculus, Bismut type formulas are established for the Lions derivative of , where  0,  is a bounded measurable function,  and  solves a distribution dependent SDE with initial distribution . As applications, explicit estimates are derived for the Lions derivative and the total variational distance between distributions of   solutions with different initial data. Both degenerate and non-degenerate situations are considered. Due to the lack of the semi-group property  and the invalidity of the formula =  , essential difficulties are overcome in the study.

Joint work with Professor Feng-Yu Wang

Unlike standard bivariate diffusion models, the rough Bergomi model by Bayer, Friz, and Gatheral (2016) allows to parsimoniously recover key stylized facts of market implied volatility surfaces such as the exploding power-law behaviour of the at-the-money volatility skew as time to maturity goes to zero. However, falling into the class of so-called rough stochastic volatility models sparked by Alo`s, Leo ́n, and Vives (2007); Fukasawa (2011, 2017); Gatheral, Jaisson, and Rosenbaum (2018), its non-Markovianity poses serious mathematical and computational challenges. To date, calibrating rough Bergomi remained prohibitively expensive since standard calibration routines rely on the repetitive evaluation of the map from model parameters to Black-Scholes implied volatility, which in the case of rough Bergomi involves a costly Monte Carlo simulation (Bennedsen, Lunde, & Pakkanen, 2017; McCrickerd & Pakkanen, 2018; Bayer et al., 2016; Horvath, Jacquier, & Muguruza, 2017). In this paper, we resolve the issue by combining a standard Levenberg-Marquardt calibration routine with a neural network regression, replacing expensive MC simulations with cheap forward runs of a network trained to approximate the implied volatility map. Some numerical results show the prowess of this approach.

TBC

We study several important fine properties for the family of fractional Brownian motions with Hurst parameter H under the (p,r)-capacity on classical Wiener space introduced by Malliavin. We regard fractional Brownian motions as Wiener functionals via the integral representation discovered by Decreusefond and \"{U}st\"{u}nel, and show non differentiability, modulus of continuity, law of iterated Logarithm(LIL) and self-avoiding properties of fractional Brownian motion sample paths using Malliavin calculus as well as the tools developed in the previous work by Fukushima, Takeda and etc. for Brownian motion case.

We develop a method to study the implied volatility for exotic options and volatility derivatives with European payoffs such as VIX options. Our approach, based on Malliavin calculus techniques, allows us to describe the properties of the at-the-money implied volatility (ATMI) in terms of the Malliavin derivatives of the underlying process. More precisely, we study the short-time behaviour of the ATMI level and skew. As an application, we describe the short-term behavior of the ATMI of VIX and realized variance options in terms of the Hurst parameter of the model, and most importantly we describe the class of volatility processes that generate a positive skew for the VIX implied volatility. In addition, we find that our ATMI asymptotic formulae perform very well even for large maturities. Several numerical examples are provided to support our theoretical results.  

We will discuss the percolation model on the hexagonal grid. In 2001 S. Smirnov proved conformal invariance of its scaling limit through the use of a tricky auxiliary combinatorial construction.

We present a more conceptual approach, implying that the construction in question can be thought of as geometrically natural one.

The main goal of the talk is to make it believable that not all nice and useful objects in the field have been already found.

No background is required.

We present sharp gradient estimates for the solution of the filtering equation and report on its applications in a high order cubature method for the nonlinear filtering problem.

I will present a mathematical model for the genetic evolution of a population which is divided in discrete colonies along a linear habitat, and for which the population size of each colony is random and constant in time. I will show that, under reasonable assumptions on the distribution of the population sizes, over large spatial and temporal scales, this population can be described by the solution to a stochastic partial differential equation with constant coefficients. These coefficients describe the effective diffusion rate of genes within the population and its effective population density, which are both different from the mean population density and the mean diffusion rate of genes at the microscopic scale. To do this, I will present a duality technique and a new convergence result for coalescing random walks in a random environment.

In this paper, we investigate the limit properties of frequency of empirical averages when random variables are described by a set of probability measures and obtain a law of large numbers for upper-lower probabilities. Our result is an extension of the classical Kinchin's law of large numbers, but the proof is totally different.

keywords: Law of large numbers,capacity, non-additive probability, sub-linear expectation, indepence

paper by: Zengjing Chen School of Mathematics, Shandong University and Qingyang Liu Center for Economic Research, Shandong University

Based on classical invariant theory, I describe a complete set of elements of the signature that is invariant to the general linear group, rotations or permutations.

A geometric interpretation of some of these invariants will be given.

Joint work with Jeremy Reizenstein (Warwick).

We present an algebraic formulation for the flow of a differential equation driven by a path in a Lie group. The formulation is motivated by formal differential equations considered by Chen.

Recently, Deya, Gubinelli, Hofmanova and Tindel ('16) (also Bailleul-Gubinelli '15) have provided a general approach in order to obtain a priori estimates for rough partial differential equations of the form
(*)    du = Au dt + Bu dX
where X is a two-step rough path, A is a second order operator (elliptic), while B is first order. We will pursue the line of this work by presenting an L^p theory "à la Krylov" for generalized versions of (*). We will give an application of this theory by proving boundedness of solutions for a certain class

The celebrated Black-Scholes theory shows that one can get a risk-neutral option price through hedging. The Cameron-Martin-Girsanov theorem for diffusion processes plays a key role in this theory. We show that one can get some statistical arbitrage from a sequence of well-designed repeated trading at their prices according to the ergodic theorem for stationary process. Our result is based on both theoretical model and the real market data.