Some illustrative examples based on onedimensional diffusion processes are then provided. By doob s theorem, a supermartingale indexed by the natural numbers decomposes uniquely into the difference of a martingale and an increasing, predictable process. Nonlinear decomposition of doobmeyers type for continuous g supermartingale with uniformly continuous coefficient shi, xuejun, jiang, long, and ji, ronglin, journal of applied. The target of this paper is to establish the bidask pricing frame work for the american contingent claims against risky assets with gasset price systems see \citechen20b on the financial market under knight uncertainty. Doob decomposition, 8 doobs htransform, 105 doobs inequality, 4 doobmeyer decomposition, 8 dynkins formula, 110 excursion theory, 81 exit distribution, 36 exit time, 36 distribution, 60 explosion, 39, 71, 102 fellers test, 102 feynmankac formula, 56 filtration, 3 generator, 31 girsanov formula, 69 harmonic oscillator, 59 heat equation, 53. A fair game that is stopped at a random time will remain fair to the end of the game if it is assumed that. The reader can also refer to the recent book of aksamit and jeanblanc 3 for a modern english. Rigorous mathematical finance relies strongly on two additional fields. In particular, a martingale is a sequence of random variables i. Optional decomposition for continuous semimartingales under arbitrary.
Mathematical finance extended from lectures of fall 2012. If b is an event, then, ib is a random variable with pfib 1 j ag pfb j ag, and pfib 0 j ag 1 pfb j ag so that pfb j ag efib j ag. Lectures on mathematical finance and related topics. The decomposition book line by michael roger is a new spin on an old concept. Optional decomposition for continuous semimartingales. First, the problems concerning regularity and the doobmeyer decomposition of quasimartingales arose naturally in the attempt to develop a stochastic integration theory. Research article nonlinear decomposition of doobmeyer s. Maxplus decomposition of supermartingales and convex. As an application, we show how the maxplus supermartingale decomposition allows, in particular, to solve the american optimal stopping problem without having to compute the option price. Decomposition of supermartingale into ui martingale and supermartingale. The theorem was proved by and is named for joseph l. S an 3 t so that mn t 2d n is a martingale and an t n is predictable with respect to f t t2d n.
In probability theory, a martingale is a model of a fair game where knowledge of past events never helps predict the mean of the future winnings. Supermartingales as radonnikodym densities 5 f t t 0. Part of the lecture notes in mathematics book series lnm, volume 1686. The second assumes that stock prices are a submartingale with constant expected returns. P, where denotes a nonempty set, fa sigma algebra on, f t t 0 a rightcontinuous ltration with f t ffor all t 0, and pa prob ability measure. It proves important results such as kolmogorovs strong law of large numbers and the threeseries theorem by. Introduction by doob s theorem, a supermartingale indexed by the natural numbers decomposes uniquely into the di erence of a martingale and an increasing, predictable process. On the submartingale supermartingale property of di usions in natural scale alexander gushchin mikhail urusovy mihail zervosz november, 2014 abstract kotani 5 has characterised the martingale property of a onedimensional di usion in natural scale in terms of the classi cation of its boundaries. On the submartingale supermartingale property of di. The strap of a horses harness that connects the girth to the noseband and is designed to prevent the horse from throwing back its.
The realization of its potential and the fundamental development of the subject, however, are due to j. A general doobmeyermertens decomposition for g supermartingale systems bruno bouchardy dylan possamaiz xiaolu tanx first version. In the theory of stochastic processes in discrete time, a part of the mathematical theory of probability, the doob decomposition theorem gives a unique decomposition of every adapted and integrable stochastic process as the sum of a martingale and a predictable process or drift starting at zero. Section 1 gives some examples of martingales, submartingales, and supermartingales. We also provide explicit formulas for the prices of defaultable claims and analyze the risk premiums that form in the market in anticipation of losses which occur at. Suppose we roll a pair of dice, but dont look immediately at the outcome.
We can use the existence of the doob decomposition to prove results about a submartingale or supermartingale x t even if we dont know how to compute it. In particular, gerber used the martingale approach to find the famous exponential bound, which is known as the lundberg inequality, for. The decomposition book is made with 100% postconsumer waste recycled paper, printed with soy ink, and made in the usa. G 0,t, the representation of the solution to gbsde and the approximation method via penalization. This book is a modern, lively and rigorous account which has doobs theory of martingales in discrete time as its main theme. We prove that a continuous supermartingale with uniformly continuous coeffcient on finite or infinite horizon, is a supersolution of the corresponding backward stochastic differential equation. For supermartingales, the same decomposition works, but now z t is nonincreasing. The exit measure of a supermartingale article pdf available in probability theory and related fields 212.
In mathematics specifically, in the theory of stochastic processes doobs martingale convergence theorems are a collection of results on the limits of supermartingales, named after the american mathematician joseph l. Martingale probability theory project gutenberg self. Supermartingale definition of supermartingale by the. If a gambler repeatedly plays a game with positive expectation, his payoff over time is a submartingale. It appears therefore that the decomposition of doob and meyer depends more on the structure of the index set than on the filtration. Pdf optional decomposition and lagrange multipliers. Some remarks on the optional decomposition theorem springerlink. A less general decomposition is based on a further uniform integrability condition for processes. Discretetime martingale march 20, 2002 the notion of martingale is fundamental in stochastic analysis. This book is the first one which presents not only main results in the mathematical finance but also these related topics with all proofs and in a selfcontained form.
July 2015 abstract weprovideageneraldoobmeyerdecompositionforgsupermartingale systems, which does not require any rightcontinuity on the system, nor thatthe. To solve our problem, we used the supermartingale approach, which generalizes the martingale approach introduced by gerber ger 73. Supermartingale decomposition with a general index set. The wellknown doobmeyer decomposition of a supermartingale as the difference of a martingale and an increasing process is extended in several ways for.
Yuliya mishura, olena ragulina, in ruin probabilities, 2016. Gdoobmeyer decomposition and its application in bidask. Supermartingaledecompositiontheoremunder gexpectation. We complement this result by establishing a necessary and sufficient condition for a onedimensional diffusion in natural scale to be a submartingale or a supermartingale. Martingale theory illustrates the history of mathematical probability. Regularity and the doobmeyer decomposition of abstract. It is a new nonlinear doobmeyer decomposition theorem for the supermartingale with continuous trajectory 1. Supermartingale decompositions, maxplus algebra, running supremum process, american options, optimal stopping, l.
Supermartingale with constant expectation is a martingale. Let v be a bounded below q supermartingale for all q e mx. Supermartingale synonyms, supermartingale pronunciation, supermartingale translation, english dictionary definition of supermartingale. Optional decomposition of supermartingales and hedging. If a gambler repeatedly plays a game with negative expectation, his payoff over time is a supermartingale.
Tis equivalent to another stochastic process y t,t. Probability theory is nowadays applied in a huge variety of fields including physics, engineering, biology, economics and the social sciences. M i, it follows that there exist a d dimensional predictable process a, and a predictable process ctaking values in the set of nonnegativede. Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. When new information decreases that ignorance, it changes our probabilities. As an application, we show how the maxplus supermartingale decomposition allows, in particular, to.
Generalised submartingale property proved through use of doob decomposition theorem. We then provide all the relevant characteristics of a default time i. Regularity and decomposition of twoparameter supermartingales. July 2015 abstract weprovideageneraldoobmeyerdecompositionforg supermartingale systems, which does not require any rightcontinuity on the system, nor thatthe. Kramkov, optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Prove that supermartingale with specific characteristics is a martingale.
Each book features creative imagery and a distinct aesthetic. Azumahoeffding inequality for sub and supermartingales. By the doob meyer decomposition, any supermartingale is locally the. Strong supermartingales and limits of non negative martingales. Strong supermartingales and limits of nonnegative martingales. Honest times, azemas supermartingale, additive decomposition, multiplica. Characterisation of honest times and optional semimartingales of. Chapter viii doobs decomposition of positive supermartingales. Informally, the martingale convergence theorem typically refers to the result that any supermartingale satisfying a certain boundedness. The interested reader is referred to the book by baccelli, cohen, olsder.
Another interesting application concerns the portfolio insurance. It is a new nonlinear doobmeyer decomposition theoremforthe supermartingale with continuous trajectory. Stock margins and the conditional probability of price reversals for example, the discussion of a 1979 article by jovanovic is presented at the level of a discussion for undergraduate economics majors, whereas elsewhere. If you have trouble downloading to your device, you can download on your personal computer. Doob decomposition doobi lahutus quadratic variation ruutvariatsioon. First, we prove gdoobymeyer decomposition for gsupermartingale. We prove that a continuous supermartingale with uniformly continuous coe cient on nite or in nite horizon, is a supersolution of the corresponding backward stochastic di erential equation. Kotani 2006 has characterised the martingale property of a onedimensional diffusion in natural scale in terms of the classification of its boundaries. The main ideas are to apply the uniformly continuous property of s. We are especially interested in the existence of cadlag modifications and in the doobmeyer decomposition of such quasimartingales.
The new hexagon patterns to download the pdf, click on the project name. Nonlinear decomposition of doobmeyers type for continuous g supermartingale with uniformly continuous coefficient shi, xuejun, jiang, long, and ji, ronglin, journal of applied mathematics, 2014 generalized dynkin games and doubly reflected bsdes with jumps dumitrescu, roxana, quenez, marieclaire, and sulem, agnes, electronic journal of. Submartingale article about submartingale by the free. The introduction of the potentials of increasing processes suggests writing the doob decomposition of a finite positive supermartingale in a slightly different way.
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