Filtration Processes Introduction. Filtration is a process that removes particles from suspension in water. Removal takes place by a number of mechanisms that include straining, flocculation, sedimentation and surface capture. Filters can be categorised by the main method of capture, i.e. exclusion of particles at the surface of the filter media i.e. straining, or deposition within the media i.
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Local Martingales and Filtration Shrinkage Hans F ollmer Philip Protter y March 30, 2010 Abstract A general theory is developed for the projection of martingale re-lated processes onto smaller ltrations, to which they are not even adapted. Martingales, supermartingales, and semimartingales retain.
A strict local martingale is a local martingale that is not a martingale. We investigate how such a process might arise from a true martingale as a result of an enlargement of the filtration and a change of measure. We study and implement a particular type of enlargement, initial expansion of filtration, for stochastic volatility models with.
Strict Local Martingales via Filtration Enlargement. By Aditi Dandapani and Philip Protter. Get PDF (256 KB) Abstract. A strict local martingale is a local martingale that is not a martingale. We investigate how such a process might arise from a true martingale as a result of an enlargement of the filtration. We study and implement a particular type of enlargement, initial expansion of.
Suppose that F t is a filtration, and G t is the filtration generated by F t and a countable set of disjoint measurable sets. Then, every F t-semimartingale is also a G t-semimartingale. (Protter 2004, p. 53) Semimartingale decompositions. By definition, every semimartingale is a sum of a local martingale and a finite variation process. However, this decomposition is not unique. Continuous.
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Notes on Elementary Martingale Theory by John B. Walsh 1 Conditional Expectations 1.1 Motivation Probability is a measure of ignorance. When new information decreases that ignorance, it changes our probabilities. Suppose we roll a pair of dice, but don’t look immediately at the outcome. The result is there for anyone to see, but if we haven’t yet looked, as far as we are concerned, the.
Martingales are typically used to shortcut a proof or calculation, sometimes in the formulation of models. A discrete-time stochastic process is a martingale with respect to a filtration provided: Note this does not contain nor is contained as a special case of a Markov process. Martingales don't have to be Markov processes and Markov processes don't have to be martingales. The continuous-time.
Martingale representation in progressive enlargement by the reference filtration of a semimartingale: a note on the multidimensional case. . Fd, where, fixed i in (1,.,d), Fi is the reference filtration of a real martingale Mi, which enjoys the Fi-predictable representation property. A second application falls into the framework of credit risk modeling and in particular into the study of.
Then we elaborate our martingale representation results, which state that any martingale in the large filtration stopped at the random time can be decomposed into orthogonal local martingales (i.e., local martingales whose product remains a local martingale). This constitutes our first principal contribution, while our second contribution consists in evaluating various defaultable securities.
Conditional expectations, filtration and martingales: Lecture 9: Filtration and martingales (PDF) 10: Martingales and stopping times I: Lecture 10: Martingales I (PDF) 11: Martingales and stopping times II. Martingale convergence theorem. Lecture 11: Martingales II (PDF) Additional materials: Martingale convergence theorem (PDF) 12.
CONDITIONAL EXPECTATION AND MARTINGALES 1. INTRODUCTION Martingales play a role in stochastic processes roughly similar to that played by conserved quantities in dynamical systems. Unlike a conserved quantity in dynamics, which remains constant in time, a martingale’s value can change; however, its expectation remains constant in time. More important, the expectation of a martingale is.
In this paper, we obtain stability results for martingale representations in a very general framework. More specifically, we consider a sequence of martingales each adapted to its own filtration, and a sequence of random variables measurable with respect to those filtrations. We assume that the terminal values of the martingales and the associated filtrations converge in the extended sense.
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Martingale Methods in Statistics Eric V. Slud Mathematics Department University of Maryland, College Park c January, 2003.
Conditional expectations (like, e.g., discounted prices in financial applications) are martingales under an appropriate filtration and probability measure. When the information flow arrives in a punctual way, a reasonable assumption is to suppose the latter to have piecewise constant sample paths between the random times of information updates. Providing a way to find and construct piecewise.
The main application of martingales will be to recover in an elegant way the previous results on gambling processes of Chap. 2. Before that, let us state many recent applications of stochastic modeling are relying on the notion of martingale. In financial mathematics for example, the notion of martingale is used to characterize the fairness and.
First of all we show that the predictable representation property for a square-integrable semi-martingale X does not transfer from the reference filtration F to a larger filtration G when the.