Due to the Covid-19 epidemics, this research school could not take place in residence at CIRM as originally intended.
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KEY WORDS
Risk Modeling, Computational Finance, Simulation Techniques
DESCRIPTION
The focus lies on applications in numerical integration, risk modelling, computational finance, simulation techniques, and computational geometry.
Speakers will introduce basic terminology and methods and then discuss fundamental applications. The research school is open to advanced master students in mathematics, statistics, and computer science as well as to PhD students. |
SCIENTIFIC COMMITTEE
ORGANIZING COMMITTEE
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PRE-RECORDED MINI COURSES
Quasi-Monte Carlo Methods and Applications: Introduction
Robert Tichy (TU Graz / Aix-Marseille Université)
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Brief Introduction of Quasi-Monte Carlo Methods and their applications
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Stochastic Differential Equation
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Gunther Leobacher (KFU Graz)
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Gunther Leobacher (KFU Graz)
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Abstract. In the first part, we give a short introduction to Quasi-Monte Carlo (QMC) methods, starting with classical results and touching more recent developments. We also talk about applications and what to take care of when using QMC instead of classical Monte Carlo.
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Abstract. In the first part, we briefly recall the theory of stochastic differential equations (SDEs) and present Maruyama's classical theorem on strong convergence of the Euler-Maruyama method, for which both drift and diffusion coefficient of the SDE need to be Lipschitz continuous. In the second part we show how the classical result can be used also for SDEs with drift that may be discontinuous and diffusion that may be degenerate. In that context I will present a concept of (multidimensional) piecewise Lipschitz drift where the set of discontinuities is a sufficiently smooth hypersurface in the multi-dimensional euclidean space. We discuss geometric properties of the set of discontinuities that are needed to transfer the convergence result from the Lipschitz case to the piecewise Lipschitz case.
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PDMPs in risk theory and QMC integration
Stefan Thonhauser (TU Graz)
Abstract : These talks will give an overview on the usage of piecewise deterministic Markov processes for risk theoretic modeling and the application of QMC integration in this framework. This class of processes includes several common risk models and their generalizations. In this field, many objects of interest such as ruin probabilities, penalty functions or expected dividend payments are typically studied by means of associated integro-differential equations. Unfortunately, only particular parameter constellations allow for closed form solutions such that in general one needs to rely on numerical methods. Instead of studying these associated integro-differential equations, we adapt the problem in a way that allows us to apply deterministic numerical integration algorithms such as QMC rules.
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Optimizing dividends and capital injections limited by bankrupcy, and practical approximations for the Cramér-Lundberg process
Florin Avram (Université de Pau et des Pays de l'Adour)
Slides Abstract. The recent papers Gajek-Kucinsky(2017), Avram-Goreac-LiWu(2020) investigated the control problem of optimizing dividends when limiting capital injections by bankruptcy is taken into consideration. The first paper works under the spectrally negative Levy model. |
Slides
Abstract: I present the basics and numerical result of two (or three) concrete applications of quasi-Monte-Carlo methods in financial engineering. The applications are in : derivative pricing, in portfolio selection, and in credit risk management. |
Number sequences for simulation
Giray Ökten (Florida State University)
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Lecture 1
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Lecture 2
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Abstract. An overview of randomness, pseudorandom sequences, low-discrepancy sequences, numerical integration, Koksma-Hlawka inequality, uniform point sets and error bounds, and randomized quasi-Monte Carlo methods will be given.
Derivative pricing: simulation from non-uniform distributions
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Slides
Abstract. The models of Bachelier and Samuelson will be introduced. Methods for generating number sequences from non-uniform distributions, such as inverse transformation and acceptance rejection, as well as generation of stochastic processes will be discussed. Applications to pricing options via rendomized quasi-Monte Carlo methods will be presented. |
LIVE TALKS & DISCUSSION SESSIONS
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Lecture 1
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Lecture 2
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Lecture 3
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Abstract: Estimation of conditional quantiles is required for many purposes, in particular when the conditional mean is not sufficient to describe the impact of covariates on the dependent variable. For example, one may estimate the quantile of one financial index (e.g. WisdomTree Japan Hedged Equity Fund) knowing financial indices from other countries. It is also required to estimate conditional quantiles in Quantile Oriented Sensitivity Analysis (QOSA). QOSA indices are relevant in order to quantify uncertainty on quantiles, for example in insurance operational risk contexts. We shall present several view points on conditional quantile estimation: quantile regression and improvements, Kernel based estimation, random forest estimation. We shall focus on applications to QOSA.
The coordinate sampler: a non-reversible Gibbs-like MCMC sampler.
Christian P. Robert (Université Paris-Dauphine)
Christian P. Robert (Université Paris-Dauphine)
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Abstract. In this talk, we derive a novel non-reversible, continuous-time Markov chain Monte Carlo (MCMC) sampler, called Coordinate Sampler, based on a piecewise deterministic Markov process (PDMP), which can be seen as a variant of the Zigzag sampler. In addition to proving a theoretical validation for this new sampling algorithm, we show that the Markov chain it
induces exhibits geometrical ergodicity convergence, for distributions whose tails decay at least as fast as an exponential distribution and at most as fast as a Gaussian distribution. Several numerical examples highlight that our coordinate sampler is more efficient than the Zigzag sampler, in terms of effective sample size. [This is joint work with Wu Changye, ref. arXiv:1809.03388] |
Probabilistic models for selfish blockchain mining
Hansjoerg Albrecher (Université de Lausanne)
NB: There is no recording for this talk.
Hansjoerg Albrecher (Université de Lausanne)
NB: There is no recording for this talk.
