Invited 01

## Conformal Invariance and Related Topics (Organizer: Hao Wu)

Conference
9:30 PM — 10:00 PM KST
Local
Jul 21 Wed, 8:30 AM — 9:00 AM EDT

### Asymptotics of determinants of discrete Laplacians

Konstantin Izyurov (University of Helsinki)

4
The zeta-regularized determinants of Laplace-Beltrami operators play an important role in analysis and mathematical physics. We show that for Euclidean surfaces with conical singularities that are glued of finitely many equal equilateral triangles or squares, these determinants appear in the asymptotic expansions of the determinants of discrete Laplacians, as the mesh size of a lattice discretization of the surface tends to zero. This establishes a particular case of a conjecture by Cardy and Peschel on the behavior of partition functions of critical lattice models, and their relation to partition functions of underlying Conformal field theories. Joint work with Mikhail Khristoforov.

### On Loewner evolutions with jumps

Eveliina Peltola (Rheinische Friedrich-Wilhelms-Universität Bonn)

4
I discuss the behavior of Loewner evolutions driven by a Levy process. Schramm's celebrated version (Schramm-Loewner evolution), driven by standard Brownian motion, has been a great success for describing critical interfaces in statistical physics. Loewner evolutions with other random drivers have been proposed, for instance, as candidates for finding extremal multifractal spectra, and some tree-like growth processes in statistical physics. Questions on how the Loewner trace behaves, e.g., whether it is generated by a (discontinuous) curve, whether it is locally connected, tree-like, or forest-like, have been partially answered in the symmetric alpha-stable case. We shall consider the case of general Levy drivers.

Joint work with Anne Schreuder (Cambridge).

### Extremal distance and conformal radius of a CLE_4 loop

Titus Lupu (Centre National de la Recherche Scientifique / Sorbonne Université)

3
Consider CLE_4 in the unit disk and let be the loop of the CLE_4 surrounding the origin. Schramm, Sheffield and Wilson determined the law of the conformal radius seen from the origin of the domain surrounded by this loop. We complement their result by determining the law of the extremal distance between the loop and the boundary of the unit disk. More surprisingly, we also compute the joint law of these conformal radius and extremal distance. This law involves first and last hitting times of a one-dimensional Brownian motion. Similar techniques also allow us to determine joint laws of some extremal distances in a critical Brownian loop-soup cluster. This is a joint work with Juhan Aru (EPFL) and Avelio Sep_lveda (Universit_ Lyon 1 Claude Bernard).

### Q&A for Invited Session 01

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This talk does not have an abstract.

###### Session Chair

Hao Wu (Yau Mathematical Sciences Center, Tsinghua University)

Invited 14

## Optimal Transport (Organizer: Philippe Rigollet)

Conference
9:30 PM — 10:00 PM KST
Local
Jul 21 Wed, 8:30 AM — 9:00 AM EDT

### Density estimation and conditional simulation using triangular transport

Youssef Marzouk (Massachusetts Institute of Technology)

3
Triangular transformations of measures, such as the Knothe-Rosenblatt rearrangement, underlie many new computational approaches for density estimation and conditional simulation. This talk discusses two aspects of such constructions. First, is the problem of estimating a triangular transformation given a sample from a distribution of interest—and hence, transport-driven density estimation. We present a general functional framework for representing monotone triangular maps between distributions, and analyze properties of maximum likelihood estimation in this framework. We demonstrate that the associated optimization problem is smooth and, under appropriate conditions, has no spurious local minima. This result provides a foundation for a greedy semi-parametric estimation procedure. Second, we discuss a conditional simulation method that employs a specific composition of maps, derived from the Knothe-Rosenblatt rearrangement, to push forward a joint distribution to any desired conditional. We show that this composed-map approach reduces variability in conditional density estimates and reduces the bias associated with any approximate map representation. Moreover, this approach motivates alternative estimation objectives that focus on the removal of dependence. For context, and as a pointer to an interesting application domain, we elucidate links between conditional simulation with composed maps and the ensemble Kalman filter.

### Estimation of Wasserstein distances in the spiked transport model

Jonathan Niles-Weed (Courant Institute of Mathematical Sciences, New York University)

4
We propose a new statistical model, the spiked transport model, which formalizes the assumption that two probability distributions differ only on a low-dimensional subspace. We study the minimax rate of estimation for the Wasserstein distance under this model and show that this low-dimensional structure can be exploited to avoid the curse of dimensionality. As a byproduct of our minimax analysis, we establish a lower bound showing that, in the absence of such structure, the plug-in estimator is nearly rate-optimal for estimating the Wasserstein distance in high dimension. We also give evidence for a statistical-computational gap and conjecture that any computationally efficient estimator is bound to suffer from the curse of dimensionality.

### Statistical estimation of barycenters in metric spaces and the space of probability measures

Quentin Paris (National Research University Higher School of Economics)

2
The talk presents rates of convergence for empirical barycenters over a large class of geodesic spaces with curvature bounds in the sense of Alexandrov. We show that parametric rates of convergence are achievable under natural conditions that characterise the bi-extendibility of geodesics emanating from a barycenter. We show that our results apply to infinite-dimensional spaces such as the 2-Wasserstein space, where bi-extendibility of geodesics translates into regularity of Kantorovich potentials

### Q&A for Invited Session 14

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This talk does not have an abstract.

###### Session Chair

Philippe Rigollet (Massachusetts Institute of Technology)

Invited 21

## Probabilistic Theory of Mean Field Games (Organizer: Xin Guo)

Conference
9:30 PM — 10:00 PM KST
Local
Jul 21 Wed, 8:30 AM — 9:00 AM EDT

### Portfolio liquidation games with self-exciting order flow

Ulrich Horst (Humboldt University Berlin)

2
We analyze novel portfolio liquidation games with self-exciting order flow. Both the $N$-player game and the mean-field game are considered. We assume that players' trading activities have an impact on the dynamics of future market order arrivals thereby generating an additional transient price impact. Given the strategies of her competitors each player solves a mean-field control problem. We characterize open-loop Nash equilibria in both games in terms of a novel mean-field FBSDE system with unknown terminal condition. Under a weak interaction condition we prove that the FBSDE systems have unique solutions. Using a novel sufficient maximum principle that does not require convexity of the cost function we finally prove that the solution of the FBSDE systems do indeed provide existence and uniqueness of open-loop Nash equilibria.

This is joint work with Guanxing Fu and Xiaonyu Xia.

### A mean-field game approach to equilibrium pricing in renewable energy certificate markets

Sebastian Jaimungal (University of Toronto)

3
Solar Renewable Energy Certificate (SREC) markets are a market-based system that incentivizes solar energy generation. A regulatory body imposes a lower bound on the amount of energy each regulated firm must generate via solar means, providing them with a tradeable certificate for each MWh generated. Firms seek to navigate the market optimally by modulating their SREC generation and trading rates. As such, the SREC market can be viewed as a stochastic game, where agents interact through the SREC price. We study this stochastic game by solving the mean-field game (MFG) limit with sub-populations of heterogeneous agents. Market participants optimize costs accounting for trading frictions, cost of generation, non-linear non-compliance costs, and generation uncertainty. Moreover, we endogenize SREC price through market clearing. We characterize firms' optimal controls as the solution of McKean-Vlasov (MV) FBSDEs and determine the equilibrium SREC price. We establish the existence and uniqueness of a solution to this MV-FBSDE, and prove that the MFG strategies form an $\epsilon$-Nash equilibrium for the finite player game. Finally, we develop a numerical scheme for solving the MV-FBSDEs and conduct a simulation study.

### Entropic optimal transport

Marcel Nutz (Columbia University)

1
Applied optimal transport is flourishing after computational advances have enabled its use in real-world problems with large data sets. Entropic regularization is a key method to approximate optimal transport in high dimensions while retaining feasible computational complexity. In this talk we discuss the convergence of entropic optimal transport to the unregularized counterpart as the regularization parameter vanishes, as well as the stability of entropic optimal transport with respect to its marginals.

Based on joint works with Espen Bernton (Columbia), Promit Ghosal (MIT), Johannes Wiesel (Columbia).

###### Session Chair

Xin Guo (University of California, Berkeley)

Invited 35

## Stochastic Analysis in Mathematical Finance and Insurance (Organizer: Marie Kratz)

Conference
9:30 PM — 10:00 PM KST
Local
Jul 21 Wed, 8:30 AM — 9:00 AM EDT

### From signature based models in finance to affine and polynomial processes and back

Christa Cuchiero (University of Vienna)

1
Modern universal classes of dynamic processes, based on neural networks or signature methods, have recently entered the field of stochastic modeling, in particular in Mathematical Finance. This has opened the door to more data-driven and thus more robust model selection mechanisms, while first principles like no arbitrage still apply. We focus here on signature based models, i.e. (possibly Levy driven) stochastic processes whose characteristics are linear functions of an underlying process' signature and present methods how to learn these characteristics from data. From a more theoretical point of view, we show how these new models can be embedded in the framework of affine and polynomial processes, which have been -- due to their tractability -- the dominating process class prior to the new era of highly overparametrized dynamic models. Indeed, we prove that generic classes of models can be viewed as infinite dimensional affine processes, which in this setup coincide with polynomial processes. A key ingredient to establish this result is again the signature process. This then allows to get power series expansions for expected values of analytic functions of the process' marginals.

The talk is based on joint works with Guido Gazzani, Francesca Primavera, Sara-Svaluto-Ferro and Josef Teichmann.

### Optimal dividends with capital injections at a level-dependent cost

Ronnie Loeffen (University of Manchester)

2
Assume the capital or surplus of an insurance company evolves randomly over time as in the Cramér-Lundberg model but where in addition the company has the possibility to pay out dividends to shareholders and to inject capital at a cost from shareholders. We impose that when the resulting surplus becomes negative the company has to decide whether to inject capital to get to a positive surplus level in order for the company to survive or to let ruin occur. The objective is to find the combined dividends and capital injections strategy that maximises the expected paid out dividends minus cost of injected capital, discounted at a constant rate, until ruin. Such optimal dividends and capital injections problems have been studied before but in the cae where the cost of capital (injections) is constant whereas we consider the setting where the cost of capital is level-dependent in the sense that it is higher when the surplus is below 0 than when it is above 0. We investigate optimality of a 3-parameter strategy with parameters -r < 0 < c < b where dividends are paid out to keep the surplus below b, capital injections are made in order to keep the surplus above c unless capital drops below the level -r in which case the company decides to let ruin occur.

This is joint work with Zbigniew Palmowski.

### Exponential Lévy-type change-point models in mathematical finance

Lioudmila Vostrikova (University of Angers)

1

### Q&A for Invited Session 35

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This talk does not have an abstract.

###### Session Chair

Marie Kratz (ESSEC Business School, CREAR)

Invited 40

## KSS Invited Session: Nonparametric and Semi-parametric Approaches in Survival Analysis (Organizer: Woncheol Jang)

Conference
9:30 PM — 10:00 PM KST
Local
Jul 21 Wed, 8:30 AM — 9:00 AM EDT

### Smoothed quantile regression for censored residual lifetime

Sangwook Kang (Yonsei University)

6
We consider a regression modeling of the quantiles of residual lifetime at a specific time given a set of covariates. For estimation of regression parameters, we propose an induced smoothed version of the existing non-smooth estimating equations approaches. The proposed estimating equations are smooth in regression parameters, so solutions can be readily obtained via standard numerical algorithms. Moreover, smoothness in the proposed estimating equations enables one to obtain a closed form expression of the robust sandwich-type covariance estimator of regression estimators. To handle data under right censoring, inverse probabilities of censoring are incorporated as weights. Consistency and asymptotic normality of the proposed estimator are established. Extensive simulation studies are conducted to verify performances of the proposed estimator under various finite samples settings. We apply the proposed method to dental study data evaluating the longevity of dental restorations.

### Superefficient estimation of future conditional hazards based on marker information

Enno Mammen (Heidelberg University)

5
We introduce a new concept for forecasting future events based on marker information. The model is based on a nonparametric approach with counting processes featuring so-called high quality markers. Despite the model having nonparametric parts we show that we attain a parametric rate of uniform consistency and uniform asymptotic normality. In usual nonparametric scenarios reaching such a fast convergence rate is not possible, so one can say that our approach is superefficient. We then use these theoretical results to construct simultaneous confidence bands directly for the hazard rate.

### On a semiparametric estimation method for AFT mixture cure models

Ingrid Van Keilegom (Katholieke Universiteit Leuven)

4
When studying survival data in the presence of right censoring, it often happens that a certain proportion of the individuals under study do not experience the event of interest and are considered as cured. The mixture cure model is one of the common models that take this feature into account. It depends on a model for the conditional probability of being cured (called the incidence) and a model for the conditional survival function of the uncured individuals (called the latency). This work considers a logistic model for the incidence and a semiparametric accelerated failure time model for the latency part. The estimation of this model is obtained via the maximization of the semiparametric likelihood, in which the unknown error density is replaced by a kernel estimator based on the Kaplan-Meier estimator of the error distribution. Asymptotic theory for consistency and asymptotic normality of the parameter estimators is provided. Moreover, the proposed estimation method is compared with several competitors. Finally, the new method is applied to data coming from a cancer clinical trial.

### Q&A for Invited Session 40

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This talk does not have an abstract.

###### Session Chair

Woncheol Jang (Seoul National University)