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Руководителем семинара является д.ф.-м.н., профессор, научный директор Фонда Кабанов Юрий Михайлович.
РАСПИСАНИЕ НА ВЕСЕННИЙ СЕМЕСТР'23
6 мая, 15:00-16:30
Евгений ФАЙНБЕРГ ,
Stony Brook University, USA
Sequential Optimization of CVaR
This talk describes theoretical foundations and algorithms for optimizing Conditional Value at Risk (CVaR) for a discounted total-cost Markov Decision Process (MDP) with finite state and action sets. This CVaR optimization problem can be reformulated as a Robust MDP(RMDP) with compact state spaces. States in these RMDPs are the original states of the problems augmented with tail risk levels, and the Decision Maker (DM) knows only the tail risk level at the initial state and time. Thus, in order to find an optimal policy following this approach, the DM needs to solve an RMDP with incomplete state observations because after the first move the DM observes the states of the system, but the tail risk levels are unknown. We show that for the CVaR optimization problem the corresponding RMDP can be solved by using the methods of convex analysis. The talk presents an algorithm for computing and implementing an optimal CVaR policy by using the value function for the version of this RMDP with completely observable tail risk levels at all states. This algorithm and the major results of the talk cover a more general problem of optimizing sum of a mean and CVaR for possibly different cost functions. This talk is based on a joint paper with Rui Ding.
22 апреля, 15:00-16:30
Юрий Михайлович КАБАНОВ ,
д.ф.-м.н., профессор, Председатель Совета, Научный директор Фонда, Член Academia Europaea
О свойствах безарбитражности
В докладе будут представлены некоторые результаты теории арбитража формулируемые в терминах процессов стоимости портфелей.
15 апреля, 15:00-16:30
Костас КАРДАРАС ,
Professor in Statistics, London School of Economics
Portfolio choice under taxation and expected market time constraint
We consider the problem of choosing an investment strategy that will maximise utility over distributions, under capital gains tax and constraints on the expected liquidation date. We show that the problem can be decomposed in two separate ones. The first involves choosing an optimal target distribution, while the second involves optimally realising this distribution via an investment strategy and stopping time. For the latter step, a variant of the Azema-Yor solution to the Skorokhod embedding problem is utilised, and its description is given very precisely in terms of the first time that the wealth of the growth optimal portfolio, properly taxed, crosses a moving stochastic (depending on its infimum) level.
8 апреля, 17:00-18:30
Ростислав БЕРЕЗОВСКИЙ ,
Liquidity provision in CLMMs
In this talk we shall work with digital assets and decentralized financial services. The blockchain technology creates an opportunity to avoid traditional exchanges and perform trading with the help of specialised programs called smart contracts which effectively implement a decentralized exchange. We shall study in detail the popular types of decentralized exchanges and consider the optimal control problem of a liquidity provider in a decentralized exchange pool with concentrated liquidity.
1 апреля, 15:00-16:30
Юрий КУТОЯНЦ ,
Le Mans University, France
Adaptive Kalman-Bucy Filter. Ergodic Case.
We consider a linear partially observed system of SDEs. The coefficients of this system depend on some finite - dimensional unknown parameter. We study the problems of this parameter estimation and the construction of adaptive Kalman-Bucy filtration equations in the asymptotic of large sam- ples. The properties of the MLE and BE are described. The adaptive filter is constructed in two steps. First we propose a method of moments prelim- inary estimator using observations on a relatively small interval of observa- tions. Then this estimator is used for construction of One-step MLE-process. Finally the last estimator allows us to construct an adaptive recurrent filter.
25 марта, 15:00-16:30
Сергей ПЕРГАМЕНЩИКОВ ,
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.
18 марта, 17:00-18:30
Тахир ШУЛИ ,
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.
11 марта, 15:00-16:30
Райнер БУКДАН ,
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).
4 марта, 15:00-16:30
Паоло ГУАСОНИ ,
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.
25 февраля, 15:00-16:30
Евгений Владимирович БУРНАЕВ ,
Профессор, рук. Центра Прикладного ИИ Сколтеха
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)
17 декабря, 15:00-16:30
Александр Андреевич ЛЫКОВ ,
Научный сотрудник кафедры теории вероятностей МГУ, кандидат физико-математических наук
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.
10 декабря, 15:00-16:30
Елена Владимировна БОГУСЛАВСКАЯ ,
Интегральные преобразования Аппеля и мартингалы: еще один способ построения мартингалов
В этом докладе мы покажем нетрадиционный способ построения мартингала, достигающего определенного граничного условия в заданный момент времени.
Все, что мы должны знать — это кумулянты основного процесса. Этот подход особенно эффективен для полиномиальных функций в граничных условиях.
3 декабря, 19:00-20:30
Чагын АРАРАТ ,
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).
26 ноября, 15:00-16:30
Юрий Михайлович КАБАНОВ ,
д.ф.-м.н., профессор, Председатель Совета, Научный директор Фонда,Член Academia Europaea
Теория разорения со случайными процентными ставками
В докладе будут изложены математические аспекты современной теории разорения, в которой предполагается, что резерв страховой компании инвестирован в рисковый актив со случайной процентной ставкой.
В частности, будет изложена редукция задачи о разорении к асимптотике хвоста распределения случайной величины представимого в виде суммы ряда из нарастающих произведений независимых случайных величин. Эта редукция позволяет получить асимптотику вероятности разорения применением теории Кестена-Голди. Основное внимание будет уделено моделям, в которых процентная ставка является случайным процессом, характеристики которого зависят от состояния экономики. В математической терминологии - это модели со скрытыми марковскими процессами, в финансовой - модели со стохастической волатильностью.
19 ноября, 15:00-16:30
Михаил Александрович УРУСОВ ,
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.
12 ноября, 15:00-16:30
Миклош РАСОНИ ,
Strongly risk-averse investors in mean-reverting market
Саид ХАМАДЕН ,
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.
22 октября, 15:00-16:30
Деан ФАНТАЦЦИНИ ,
Доктор экономических наук, профессор кафедры эконометрики и математических методов экономики МШЭ МГУ им. М.В. Ломоносова
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.
15 октября, 15:00-16:30
Александр ВЕРЕТЕННИКОВ ,
On SDE with reflection
08 октября, 15:00-16:30
Михаил ЖИТЛУХИН ,
к.ф.-м.н., старший научный сотрудник МИАН им. В.А. Стеклова
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.
04 октября, 15:00-16:30
Арнак ДАЛАЛЯН ,
Профессор Департамента статистики ENSAE Paris, Директор 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.
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