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. Prove that supermartingale with specific characteristics is a martingale. 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 exit measure of a supermartingale article pdf available in probability theory and related fields 212. It appears therefore that the decomposition of doob and meyer depends more on the structure of the index set than on the filtration. The decomposition book is made with 100% postconsumer waste recycled paper, printed with soy ink, and made in the usa. Chapter viii doobs decomposition of positive supermartingales. Gdoobmeyer decomposition and its application in bidask. It is a new nonlinear doobmeyer decomposition theoremforthe supermartingale with continuous trajectory. Doob decomposition doobi lahutus quadratic variation ruutvariatsioon. Supermartingale synonyms, supermartingale pronunciation, supermartingale translation, english dictionary definition of supermartingale.
First, the problems concerning regularity and the doobmeyer decomposition of quasimartingales arose naturally in the attempt to develop a stochastic integration theory. A general doobmeyermertens decomposition for g supermartingale systems bruno bouchardy dylan possamaiz xiaolu tanx first version. Section 1 gives some examples of martingales, submartingales, and supermartingales. Regularity and the doobmeyer decomposition of abstract. Azumahoeffding inequality for sub and supermartingales. Suppose we roll a pair of dice, but dont look immediately at the outcome. Strong supermartingales and limits of non negative martingales. First, we prove gdoobymeyer decomposition for gsupermartingale. We are especially interested in the existence of cadlag modifications and in the doobmeyer decomposition of such quasimartingales. 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.
It proves important results such as kolmogorovs strong law of large numbers and the threeseries theorem by. 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. The second assumes that stock prices are a submartingale with constant expected returns. Nonlinear decomposition of doobmeyers type for continuous g supermartingale with uniformly continuous coefficient shi, xuejun, jiang, long, and ji, ronglin, journal of applied. Probability theory is nowadays applied in a huge variety of fields including physics, engineering, biology, economics and the social sciences. 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. July 2015 abstract weprovideageneraldoobmeyerdecompositionforg supermartingale systems, which does not require any rightcontinuity on the system, nor thatthe. 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. Supermartingale decomposition with a general index set. Yuliya mishura, olena ragulina, in ruin probabilities, 2016. Supermartingaledecompositiontheoremunder gexpectation. The new hexagon patterns to download the pdf, click on the project name. Part of the lecture notes in mathematics book series lnm, volume 1686.
The interested reader is referred to the book by baccelli, cohen, olsder. In particular, gerber used the martingale approach to find the famous exponential bound, which is known as the lundberg inequality, for. The decomposition book line by michael roger is a new spin on an old concept. Lectures on mathematical finance and related topics. Supermartingales as radonnikodym densities 5 f t t 0. Pdf optional decomposition and lagrange multipliers. The theorem was proved by and is named for joseph l. 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. Regularity and decomposition of twoparameter supermartingales. Supermartingale with constant expectation is a martingale.
Characterisation of honest times and optional semimartingales of. Let v be a bounded below q supermartingale for all q e mx. 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. Supermartingale definition of supermartingale by the.
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. Some illustrative examples based on onedimensional diffusion processes are then provided. G 0,t, the representation of the solution to gbsde and the approximation method via penalization. M i, it follows that there exist a d dimensional predictable process a, and a predictable process ctaking values in the set of nonnegativede. By doob s theorem, a supermartingale indexed by the natural numbers decomposes uniquely into the difference of a martingale and an increasing, predictable process. As an application, we show how the maxplus supermartingale decomposition allows, in particular, to. Optional decomposition of supermartingales and hedging. Rigorous mathematical finance relies strongly on two additional fields.
Another interesting application concerns the portfolio insurance. Mathematical finance extended from lectures of fall 2012. Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. 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.
Each book features creative imagery and a distinct aesthetic. 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. 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. Submartingale article about submartingale by the free.
Honest times, azemas supermartingale, additive decomposition, multiplica. The realization of its potential and the fundamental development of the subject, however, are due to j. If a gambler repeatedly plays a game with positive expectation, his payoff over time is a submartingale. 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. Martingale probability theory project gutenberg self. 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. If you have trouble downloading to your device, you can download on your personal computer. Kotani 2006 has characterised the martingale property of a onedimensional diffusion in natural scale in terms of the classification of its boundaries.
It is a new nonlinear doobmeyer decomposition theorem for the supermartingale with continuous trajectory 1. 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. Optional decomposition for continuous semimartingales. A less general decomposition is based on a further uniform integrability condition for processes. To solve our problem, we used the supermartingale approach, which generalizes the martingale approach introduced by gerber ger 73.
By the doob meyer decomposition, any supermartingale is locally the. The reader can also refer to the recent book of aksamit and jeanblanc 3 for a modern english. 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. Supermartingale decompositions, maxplus algebra, running supremum process, american options, optimal stopping, l.
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. Research article nonlinear decomposition of doobmeyer s. The wellknown doobmeyer decomposition of a supermartingale as the difference of a martingale and an increasing process is extended in several ways for. Informally, the martingale convergence theorem typically refers to the result that any supermartingale satisfying a certain boundedness. July 2015 abstract weprovideageneraldoobmeyerdecompositionforgsupermartingale systems, which does not require any rightcontinuity on the system, nor thatthe. The main ideas are to apply the uniformly continuous property of s. The introduction of the potentials of increasing processes suggests writing the doob decomposition of a finite positive supermartingale in a slightly different way. Kramkov, optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Strong supermartingales and limits of nonnegative martingales. Decomposition of supermartingale into ui martingale and supermartingale. When new information decreases that ignorance, it changes our probabilities. We then provide all the relevant characteristics of a default time i. Discretetime martingale march 20, 2002 the notion of martingale is fundamental in stochastic analysis. Martingale theory illustrates the history of mathematical probability.
Generalised submartingale property proved through use of doob decomposition theorem. Optional decomposition for continuous semimartingales under arbitrary. 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. Maxplus decomposition of supermartingales and convex. For supermartingales, the same decomposition works, but now z t is nonincreasing. If a gambler repeatedly plays a game with negative expectation, his payoff over time is a supermartingale. 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. On the submartingale supermartingale property of di. Tis equivalent to another stochastic process y t,t. The strap of a horses harness that connects the girth to the noseband and is designed to prevent the horse from throwing back its. 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. Some remarks on the optional decomposition theorem springerlink. In particular, a martingale is a sequence of random variables i.