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Coupling, transportation metrics and applications to approximate counting
Kluvancová, Rozálie ; Prokešová, Michaela (advisor) ; Swart, Jan (referee)
An important property of discrete-time Markov chains with finite state space is the rate of convergence of the marginal distribution of the chain to the stationary distribution (i.e. mixing rate). If we construct a coupling of two Markov chains with the same transition matrix, where one starts from a stationary distribution and the other from a fixed state, we can use it to estimate the mixing rate. The main goal of this thesis is to describe how we can construct such a coupling using the transportation metric, and to apply this method to approximate counting of all proper colorings of the graph. 1
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Order book dynamics
Peržina, Vít ; Swart, Jan (advisor) ; Večeř, Jan (referee)
Main goal of this thesis is improvement of an order book model so that it behaved more realistically, based on a model developed by J. Plačková in her diploma thesis in 2011. We consider this simple model for evolution of order book in which limit orders of unit size arrive according to independent Poisson processes. Frequency of buy limit orders below resp. sell limit orders above a given price level is described by demand and supply functions. Buy (resp. sell) limit orders that arrive with price above (resp. below) the current ask (resp. bid) price are converted into market orders and cancellation of orders is not allowed. We extend this model by introducing market makers who place at the same time one buy and one sell limit order with current bid and ask prices. We show how introducing market makers reduces the spread that in the original model was unrealistically large and also show a method of calculating the precise rate of market makers needed to reduce the spread to zero. 1
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Volatility bursts and order book dynamics
Plačková, Jana ; Swart, Jan (advisor) ; Lachout, Petr (referee)
Title: Volatility bursts and order book dynamics Author: Jana Plačková Department: Department of Probability and Mathematical Statistics Supervisor: Dr. Jan M. Swart Supervisor's e-mail address: swart@utia.cas.cz Abstract: The presented paper studies the dynamics of supply and demand through the electronic order book. We describe and define the basic rules of the order book and its dynamics. We also define limit and market orders and describe the differences between them and how they influenced the evolution of ask, bid price and spread. Next part of the paper is dedicated to the de- scription and definition of volatility and its basic models. The brief overview about volatility clustering and its modeling by economists and physicists can be found in the following part. In the last part we introduce a simple model of order book in which we observe ask, bid price and spread. Then we study the empirical distribution of spread and try to find its probability distribu- tion. The volatility clustering is then observed through the relative returns of spread. In the last part we introduce some possible improvement of the model. Keywords: volatility clustering, order book, limit orders, market orders 1
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Mixing of Markov chains - lower bounds for mixing
Ditrich, Jakub ; Prokešová, Michaela (advisor) ; Swart, Jan (referee)
The focus of the thesis is the convergence of irreducible aperiodic homoge- neous Markov chains with a finite and discrete set of states. Specifically, lower bounds on the time needed for the chain's marginal probability distribution to be sufficiently close to the stationary distribution, so called mixing time. Multiple methods are introdu- ced, properly motivated and proven. Finally, each method is demonstrated on a suitable example. The result is an overview of three methods that can be used to derive lower bounds for mixing time. 1
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Absorption cascades of one-dimensional diffusions
Hudec, Tobiáš ; Swart, Jan (advisor)
It is known that the time until a birth and death process reaches certain level is distributed as a sum of independent exponential random variables. Diaconis, Miclo and Swart gave a probabilistic proof of this fact by coupling the birth and death process with a pure birth process such that the two processes reach the given level at the same time. We apply their techniques to find a one-dimensional diffusion and a pure birth process whose transition probabilities are related by an intertwining relation. From this we prove that the time to absorption of the diffusion has the same distribution as the time to explosion of the pure birth process, although we do not manage to couple them such that the two times are a. s. equal. This gives us a probabilistic proof of the known fact that the time to absorption of the diffusion is distributed as a sum of independent exponential random variables. We also find a coupling of a similar diffusion with the same pure birth process, which is now stopped at an arbitrary level. This allows us to interpret the diffusion as being initially reluctant to get absorbed, but later getting more and more compelled to get absorbed. 1
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The Stigler-Luckock model for a limit order book
Fornůsková, Monika ; Swart, Jan (advisor) ; Večeř, Jan (referee)
THE STIGLER-LUCKOCK MODEL FOR A LIMIT ORDER BOOK Abstract One of the types of modern-day markets are so-called order-driven markets whose core component is a database of all incoming buy and sell orders (order book). The main goal of this thesis is to extend the Stigler-Luckock model for order books to give a better insight into the price forming process and behaviour of the market participants themselves. The model introduced in this thesis focuses on a comparison of behaviour and various strategies of market makers who are sophisticated market participants profiting from extensive trading. The market is described using Markov chains, and the strategies are compared using Monte Carlo simulations and game theory. The results showed that market makers' orders should have small spread and large volumes. The final model compares two strategies in which market makers monitor their portfolio. In case of having more cash than asset (or vice versa), they shift prices of their orders to equalise the portfolio. The model recommends checking the market quite often, but acting conservatively, which means not changing prices that frequently and not jumping to conclusions just from a small imbalance in the portfolio.
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Absorption cascades of one-dimensional diffusions
Hudec, Tobiáš ; Swart, Jan (advisor) ; Maslowski, Bohdan (referee)
It is known that the time until a birth and death process reaches certain level is distributed as a sum of independent exponential random variables. Diaconis, Miclo and Swart gave a probabilistic proof of this fact by coupling the birth and death process with a pure birth process such that the two processes reach the given level at the same time. We apply their techniques to find a one-dimensional diffusion and a pure birth process whose transition probabilities are related by an intertwining relation. From this we prove that the time to absorption of the diffusion has the same distribution as the time to explosion of the pure birth process, although we do not manage to couple them such that the two times are a. s. equal. This gives us a probabilistic proof of the known fact that the time to absorption of the diffusion is distributed as a sum of independent exponential random variables. We also find a coupling of a similar diffusion with the same pure birth process, which is now stopped at an arbitrary level. This allows us to interpret the diffusion as being initially reluctant to get absorbed, but later getting more and more compelled to get absorbed. 1
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Absorption cascades of one-dimensional diffusions
Hudec, Tobiáš ; Swart, Jan (advisor)
It is known that the time until a birth and death process reaches certain level is distributed as a sum of independent exponential random variables. Diaconis, Miclo and Swart gave a probabilistic proof of this fact by coupling the birth and death process with a pure birth process such that the two processes reach the given level at the same time. We apply their techniques to find a one-dimensional diffusion and a pure birth process whose transition probabilities are related by an intertwining relation. From this we prove that the time to absorption of the diffusion has the same distribution as the time to explosion of the pure birth process, although we do not manage to couple them such that the two times are a. s. equal. This gives us a probabilistic proof of the known fact that the time to absorption of the diffusion is distributed as a sum of independent exponential random variables. We also find a coupling of a similar diffusion with the same pure birth process, which is now stopped at an arbitrary level. This allows us to interpret the diffusion as being initially reluctant to get absorbed, but later getting more and more compelled to get absorbed. 1
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Statistical inference for spatial and space-time Cox point processes
Dvořák, Jiří ; Prokešová, Michaela (advisor) ; Beneš, Viktor (referee) ; Swart, Jan (referee)
Fitting of parametric models to spatial and space-time point patterns has been a very active research area in the last few years. Concerning clustered patterns, the Cox point process is the model of choice. To avoid the computationally demanding maximum likelihood estimation or Bayesian inference, several estimation methods based on the moment properties of the processes in question were proposed in the literature. We give overview of the state-of-the-art moment estimation methods for stationary spatial Cox point processes and compare their performance in a simulation study. We also discuss generalization of such methods for inhomogeneous spatial point processes. In the core part of the thesis we focus on minimum contrast estimation for inhomogeneous space-time shot-noise Cox point processes and investigate the possibility to use projections to the spatial and temporal domain to estimate different parts of the model separately. We propose a step-wise estimation procedure based on projection processes and also a refined method which remedies the problem of possible cluster overlapping. We establish consistency and asymptotic normality of the estimators for both methods under the increasing window asymptotics and compare their performance on middle-sized observation windows by means of a simulation study....
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Volatility bursts and order book dynamics
Plačková, Jana ; Swart, Jan (advisor) ; Lachout, Petr (referee)
Title: Volatility bursts and order book dynamics Author: Jana Plačková Department: Department of Probability and Mathematical Statistics Supervisor: Dr. Jan M. Swart Supervisor's e-mail address: swart@utia.cas.cz Abstract: The presented paper studies the dynamics of supply and demand through the electronic order book. We describe and define the basic rules of the order book and its dynamics. We also define limit and market orders and describe the differences between them and how they influenced the evolution of ask, bid price and spread. Next part of the paper is dedicated to the de- scription and definition of volatility and its basic models. The brief overview about volatility clustering and its modeling by economists and physicists can be found in the following part. In the last part we introduce a simple model of order book in which we observe ask, bid price and spread. Then we study the empirical distribution of spread and try to find its probability distribu- tion. The volatility clustering is then observed through the relative returns of spread. In the last part we introduce some possible improvement of the model. Keywords: volatility clustering, order book, limit orders, market orders 1
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