Levy Processes And Stochastic Calculus

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I have tried reading: -probability/probability-theory-and-stochastic-processes/levy-processes-and-stochastic-calculus-2nd-edition and it is very hard, I am not really able to get much out of it. Do you know about any lower level texts you can reccomend please, which contains stochastic calculus and the theory about Lévy processes

Since Lévy processes are used a lot in finance, there are several books on this topic (that is, Lévy processes and their applications in finance). For example \"Stochastic Calculus for Finance II\" by S. Shreve introduces stochastic integration and contains some material on Lévy processes. However, if you are less interested in applications, but more in the theory behind it, then this might not be your first choice.

It depends a little bit on your interests, but as you might know, stochastic processes and Itô-calculus is excessively used in quantitative finance. I can recommend some books which really explain the basics of stochastic integration and stochastic differential equations. However, these books have a (strong) focus towards financial applications.

We develop a stochastic calculus for processes which are built by convoluting a pure jump, zero expectation Lévy process with a Volterra-type kernel. This class of processes contains, for example, fractional Lévy processes as studied by Marquardt [Bernoulli 12 (2006) 1090--1126.] The integral which we introduce is a Skorokhod integral. Nonetheless, we avoid the technicalities from Malliavin calculus and white noise analysis and give an elementary definition based on expectations under change of measure. As a main result, we derive an Itô formula which separates the different contributions from the memory due to the convolution and from the jumps.

Stochastic simulation is increasingly important for understanding visualisation and estimation of of stochastic processes. These problems are also strongly motivated by problems in Statistics, Biology, Physics and many other areas. Warwick has a large and very active group working on problems in this area. Some of this work involves the study of algorithms (such as Markov chain Monte Carlo) which are prevalent in Computer Science and Statistics. Paul Jenkins, Adam Johansen, Wilfrid Kendall, Jere Koskela, Krzysztof Łatuszyński, Alex Mijatovic, Gareth Roberts, and Dario Spano are working on this topic.

Stochastic models of evolution seek to predict patterns of genetic diversity arising from evolutionary forces such as random mating, mutation, natural selection, and fluctuations in population size. They form the foundations of statistical inference methods for the ever-increasing volume of DNA sequence data, and are also drivers of research in probability and stochastic processes. Typical models come in pairs: one process describing the forward-in-time dynamics of population frequencies of genetic variants, and a reverse-time process of coalescing lineages which describes the common ancestry of a sample of individuals from the population. Prototypical examples include the Wright-Fisher diffusion and the Kingman coalescent, as well as more general (potentially measure-valued) jump diffusions and branching-coalescing random graphs. Probabilistic advances motivated by such models include so-called lookdown constructions in which the pair of processes are constructed on the same probability space, the notion of \"coming down from infinity\" where the ancestral process of a countably infinite number of lineages can either stay infinite or coalesce to a finite number of common ancestors, and separation of timescales phenomena in which \"fast\" dynamics occurring on the timescale of generations can be approximated by simple, mean-field processes on the much longer timescale of genetic evolution. Researchers at Warwick with interests in these phenomena include Paul Jenkins, Jere Koskela, and Dario Spano.

Research in stochastic partial differential equations (SPDEs) lies at the interface of analysis and probability theory. For example, in trying to understand how solutions of macroscopic partial differential equations can be approximated by scaled particle systems at the microscopic level, one of the main questions is about the structure of the fluctuations of the approximating system around its hydrodynamic limit. This limit can often be described by an SPDE of linear type with a Gaussian solution. However, if the space explored by the particle system lacks enough degrees of freedom, as in the case of the one-dimensional asymmetric simple exclusion process for instance, then the fluctuations become non-Gaussian and can only be described (if indeed they can be described at all) by a non-linear SPDE. The so-called current fluctuations, crucial for the understanding of the non-linearity, are linked with the distribution of eigenvalues of random matrices. Other problems are the limiting behaviour of systems with multiple temporal and spatial scales, the numerical analysis of approximations to SPDEs and the construction and analysis of MCMC methods in high dimensional spaces, which provide a probabilistic tool for understanding statistical sampling techniques. Recent developments include mean-field games and nonlinear Markov processes. Sigurd Assing, Vassili Kolokoltsov, Jon Warren and Nikos Zygouras are amongst those at Warwick who are involved in studies in this area.

The course covers: the Feynman-Kac formula and the Fokker-Plank equation, stochastic calculus with jumps, Levy processes and jump diffusion models in finance, Bellman's principle of dynamic programming and the Hamilton-Jacobi- Bellman equation, classical problems for optimal control in finance (Merton's problem, etc.), investment-consumption decisions with transaction costs, the connection between asset pricing and free-boundary problems for PDEs, optimal stopping problems and the exercise of American-style derivatives, capital structure and valuation of real options and corporate debt, exchange options, stochastic volatility models, and Dupire's formula. (Mathematical Finance courses are reserved for students enrolled in the Mathematical Finance program.)

In this course we shall study some techniques of stochastic integration beyond the Ito calculus, or the so called non-anticipating integration. Here \"anticipating\" and \"non-anticipating\" is referred to the information flow. We shall also introduce the concept of stochastic derivative.

These concepts and the calculus associated constitute a baggage of tools that has turned out to be powerful both in the development of stochastic theory, mathematical statistics, and also directly in applications. The applications we are focusing on are related to financial modelling and control. 59ce067264

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