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THE SCHEDULE FOR SPRING'23
March 25, 3 p.m.-4:30 p.m. (Moscow time)
Serguei PERGAMENCHTCHIKOV ,
LMRS, University of Rouen, France
Optimal investment and consumption for spread financial markets
In this talk we develop stochastic optimal control methods for spread financial markets defined by the Ornstein–Uhlenbeck (OU) processes. To this end, we study the Hamilton–Jacobi–Bellman (HJB) equation using the Feynman–Kac (FK) probability representation. We show an existence and uniqueness theorem for the classical solution of the HJB equation which is a quasi-linear partial derivative equation of parabolic type. Then we show a special verification theorem and, as a consequence, construct optimal consumption/investment strategies for power utility functions. Moreover, using fixed point tools we study the numeric approximation for the HJB solution and we establish the convergence rate which, as it turns out in this case, is super geometric, i.e., more rapid than any geometric one.
March 18, 5 p.m.-6:30 p.m. (Moscow time)
Tahir CHOULLI ,
Professor, Faculty of Science - Mathematics & Statistical Sciences, University of Alberta
Reflected BSDEs arising from pricing and hedging in models under random horizon
We consider the model given by the quadruplet: complete probability space, a filtration F defined on this space and representing the flow of information available to all agents throughout time, and an arbitrary random time. This random time, which might not be observable via F, represents default time of a firm in credit risk and death time of an insured in life insurance where mortality and/or longevity risks pose serious challenges. Thus, the flow G that incorporates all the information, whether it is public or not, follows from the progressive enlargement of F with τ. In this setting, our analysis yields the family of reflected backward stochastic differential equations (RBSDE). We focus on answering the following questions:
1. What are the sufficient minimal conditions on the data that guarantee the existence of the solution of the G-RBSDE in Lp (p > 1)?
2. How this RBSDE can be explicitly connected to an RBSDE in F?
3. How can we estimate –in norm– the solution in terms of the data-triplet? What are the adequate norms and adequate spaces for both the solution and the data-triplet?
In my talk, I will answer all these questions deeply and beyond. Importantly, I will prove that for any random time, having a positive Azéma supermartingale, there exists a positive discount factor.
March 11, 3 p.m.-4:30 p.m. (Moscow time)
Rainer BUCKDAHN ,
Université de Bretagne Occidentale, Brest
Mean field stochastic control under sublinear expectation
The talk is devoted to the study of Pontryagin's stochastic maximum principle for a mean-field optimal control problem under Peng's sublinear G-expectation. The dynamics of the controlled state process is given by a SDE driven by a G-Brownian motion, whose coefficients depend on the control, the controlled state process but also on its law under the G-expectation. Also the associated cost functional is of mean-field type. Under the assumption of a convex control state space we study Pontryagin's stochastic maximum principle, which gives a necessary optimality condition for control process. Under additional convexity assumptions on the Hamiltonian it is shown that this necessary condition is also a sufficient one. The main difficulty which we had to overcome in our work consists in the differentiation of the G-expectation of parametrised random variables. As particularly delicate turns out to handle with the G-expectation of a function of the controlled state process inside the running cost of the cost functional. For this we had to study a measurable selection theorem for set-valued functions whose values are subsets of the representing set of probability measures for the G-expectation.
The talk is based on a recent joint work with Juan Li and Bowen He (Shandong University, Weihai, China).
March 4, 3 p.m.-4:30 p.m. (Moscow time)
Paolo GUASONI ,
Dublin City University
Investing on behalf of a firm, a trader can feign personal skill by committing fraud that with high probability remains undetected and generates small gains, but that with low probability bankrupts the firm, offsetting ostensible gains. Honesty requires enough skin in the game: if two traders with isoelastic preferences operate in continuous-time and one of them is honest, the other is honest as long as the respective fraction of capital is above an endogenous fraud threshold that depends on the trader’s preferences and skill. If both traders can cheat, they reach a Nash equilibrium in which the fraud threshold of each of them is lower than if the other one were honest. More skill, higher risk aversion, longer horizons, and greater volatility all lead to honesty on a wider range of capital allocations between the traders.
February 25, 3 p.m.-4:30 p.m. (Moscow time)
Evgeny BURNAEV ,
Full Professor, Head of Applied AI center, Skoltech
Neural Optimal Transport
Solving optimal transport (OT) problems with neural networks has become widespread in machine learning. The majority of existing methods compute the OT cost and use it as the loss function to update the generator in generative models (Wasserstein GANs). In this presentation, I will discuss the absolutely different and recently appeared direction - methods to compute the OT plan (map) and use it as the generative model itself. Recent advances in this field demonstrate that they provide comparable performance to WGANs. At the same time, these methods have a wide range of superior theoretical and practical properties.
The presentation will be mainly based on our recent paper "Neural Optimal Transport" (ICLR, 2023). I am going to present a neural algorithm to compute OT plans (maps) for weak & strong transport costs. For this, I will discuss important theoretical properties of the duality of OT problems that make it possible to develop efficient practical learning algorithms. Besides, I will prove that neural networks actually can approximate transport maps between probability distributions arbitrarily well. Practically, I will demonstrate the performance of the algorithm on the problems of unpaired image-to-image style transfer and image super-resolution.
Neural Optimal Transport (ICLR, 2023)
Unpaired Image Super-Resolution with Optimal Transport Maps (preprint)
Generative Modeling with Optimal Transport Maps (ICLR, 2022)
Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark (NeurIPS, 2021)
December 17, 3 p.m.- 4:30 p.m. (Moscow time)
Alexander LYKOV ,
On market regimes and adaptive portfolio models
In many areas of finance, the problem of heteroscedasticity is raised. Generally financial time series are not homogeneous. Particularly in portfolio management tasks. To construct more realistic and useful models we should consider time dependent or (and) state dependent models for allocation algorithms. In the talk we will discuss one state dependent model for asset allocation which admits a close solution and can be used to test some hypothesis analytically. We will find some unexpected features of that model in realistic settings. Also, we will discuss different approaches to market regime identifications.
December 10, 3 p.m.- 4:30 p.m. (Moscow time)
Elena BOGUSLAVSKAYA ,
Brunel University London, Department of Mathematical Sciences; PhD University of Amsterdam
Appell integral transforms and martingales: yet another way to construct martingales.
In this talk we show an unconventional way to construct a martingale reaching a particular boundary condition at a particular time.
All we should know is the cumulants of the underlying process. This approach is especially effective with polynomial functions in boundary conditions.
December 3, 7 p.m.- 8:30 p.m. (Moscow time)
Çağın ARARAT ,
Ph.D.; Assistant Professor, Department of Industrial Engineering, Bilkent University
Dynamic mean-variance problem: recovering time-consistency
The dynamic mean-variance problem is a well-studied optimization problem that is known to be time-inconsistent. The main source of time-inconsistency is that the family of conditional variance functionals indexed by time fails to be recursive. We consider the mean-variance problem in a discrete-time setting and study an auxiliary dynamic vector optimization problem whose objective function consists of the conditional mean and conditional second moment. We show that the vector optimization problem satisfies a set-valued dynamic programming principle and is time-consistent in a generalized sense. Moreover, its weighted sum scalarizations are closely related to the mean-variance problem through simple nonlinear transformations. This is at the cost of using stochastic and time-varying weights in the mean-variance problem. We also discuss the relationship between our results and some recent results in the literature that discuss the use of time-varying weights under special dynamics. Finally, in a finite probability space, we propose a computational procedure that relies on convex vector optimization and convex projection problems, and we use this procedure to calculate time-consistent solutions in concrete market models. Joint work with Seyit Emre Düzoylum (UC Santa Barbara).
November 26, 3 p.m.- 4:30 p.m. (Moscow time)
Yuri KABANOV ,
Chairman of the Board, Scientific Director of the Foundation, Dr. Sci. in Phys. and Maths, Professor, Member of Academia Europaea
Ruin theory with random interest rates
The lecture will outline the mathematical aspects of the modern ruin, which assumes that the insurance company invests its reserve in a risky asset paying a random interest rate.
In particular, we will present a reduction of the ruin problem to the study of asymptotical behavior of the tail of distribution of the sum of series of increasing products of independent random variables. This reduction makes it possible to obtain the asymptoticsof the ruin probability using the Kesten-Goldie theory. The main attention will be paid to models in which the interest rate is a random process, the characteristics of which depend on the state of the economy. In mathematical terminology, they are called hidden Markov models; in financial terminology, they are referred to as models with stochastic volatility. We concentrate our efforts on understanding "technical" issues of various approaches.
November 19, 3 p.m.- 4:30 p.m. (Moscow time)
Mikhail URUSOV ,
Reducing Obizhaeva-Wang type trade execution problems to LQ stochastic control problems
We start with a stochastic control problem where the control process is a process of finite variation with jumps and acts as integrator both in the state dynamics and in the target functional. Problems of such type arise in the stream of literature on optimal trade execution pioneered by Obizhaeva and Wang (models with finite resilience). We consider a general framework where the price impact and the resilience are stochastic processes. Both are allowed to have diffusive components. First we continuously extend the problem from processes of finite variation to progressively measurable processes. Then we reduce the extended problem to a linear quadratic (LQ) stochastic control problem. Using the well developed theory on LQ problems we describe the solution to the obtained LQ problem and trace it back up to the solution to the (extended) initial trade execution problem. Finally, we discuss several examples. Among other things the examples show the Obizhaeva-Wang model with random terminal and moving targets, the necessity to extend the initial trade execution problem to a reasonably large class of progressively measurable processes (it is necessary to go beyond semimartingales!) and the effects of diffusive components in the price impact process and/or in the resilience process. This is a joint work with Julia Ackermann and Thomas Kruse.
November 12, 3 p.m.- 4:30 p.m. (Moscow time)
Miklós Rásonyi ,
Professor, Renyi Institute, Hungarian Academy of Sciences
Strongly risk-averse investors in mean-reverting market
October 29, 3 p.m.- 4:30 p.m. (Moscow time)
Saïd HAMADENE ,
Professor, Laboratoire Manceau de Mathématiques (LMM), Le Mans Université
Mean-field doubly reflected backward SDEs and zero-sum Dynkin games
We study mean-field doubly reflected BSDEs. First, using the fixed-point method, we show existence and uniqueness of the solution when the data which define the BSDE are $p$-integrable with $p=1$ or $p>1$. The two cases are treated separately. Next by penalization we show also the existence of the solution. The two do not cover the same set of assumptions.
October 22, 3 p.m.- 4:30 p.m. (Moscow time)
Dean Fantazzini ,
Crypto-Coins and Credit Risk: Modelling and Forecasting Their Probability of Death
This paper examined a set of over two thousand crypto-coins observed between 2015 and 2020 to estimate their credit risk by computing their probability of death. We employed different definitions of dead coins, ranging from academic literature to professional practice; alternative forecasting models, ranging from credit scoring models to machine learning and time-series-based models; and different forecasting horizons. We found that the choice of the coin-death definition affected the set of the best forecasting models to compute the probability of death. However, this choice was not critical, and the best models turned out to be the same in most cases. In general, we found that the cauchit and the zero-price-probability (ZPP) based on the random walk or the Markov Switching-GARCH(1,1) were the best models for newly established coins, whereas credit-scoring models and machine-learning methods using lagged trading volumes and online searches were better choices for older coins. These results also held after a set of robustness checks that considered different time samples and the coins’ market capitalization.
October 15, 3 p.m.- 4:30 p.m. (Moscow time)
Alexander Veretennikov ,
On SDE with reflection
An introduction to SDEs with reflection along with Ito's formula with local times will be presented in this talk. Weak and strong solutions as well as weak and strong uniqueness problems will be addressed.
October 8, 3 p.m.- 4:30 p.m. (Moscow time)
Mikhail Zhitlukhin ,
Senior Researcher, Steklov Institute of Mathematics (Russian Academy of Sciences)
Growth optimal strategies in a mean-field market
We consider a model of a financial market which consists of a large agent (the market) and a small agent (an individual investor), who invest money in dividend paying stock. Stock prices are determined by the actions of the large agent, and the small agent is a price-taker. The goal of the work is to find a strategy of the large agent which does not allow a small agent to achieve long-term growth of wealth greater than that of the large agent. The motivation for studying this problem arises from the known empirical fact that it is not possible "to beat" the market in the long run. If one assumes that this fact is true, it can be used to describe long-term behavior of the market.
October 4, 3 p.m.- 4:30 p.m. (Moscow time)
Arnak S. Dalalyan ,
Professor of Statistics at ENSAE Paris, Director of CREST
Approximate sampling from smooth and log-concave densities
Sampling from various kinds of distributions is an issue of paramount importance in statistics since it is often the key ingredient for constructing estimators, test procedures or confidence intervals.
In many situations, the exact sampling from a given distribution is impossible or computationally expensive and, therefore, one needs to resort to approximate sampling strategies. Hence, developing theory providing meaningful nonasymptotic guarantees for the approximate sampling procedures is of high importance, especially in the high-dimensional problems.
This talk reviews some recent progress in this direction by considering the problem of sampling from a distribution having a smooth and log-concave density defined on \(\RR^p\), for some integer \(p>0\). We present nonasymptotic bounds for the error of approximating the target distribution by the one obtained by the Langevin Monte Carlo method and its variants. We illustrate the effectiveness of the established guarantees with various experiments.
Underlying our analysis are insights from the theory of continuous-time diffusion processes, which may be of interest beyond the framework of log-concave densities.