Introduction the doob meyer decomposition of square of a square integrable martingale was a landmark result that led to the development of stochastic integration with respect to a martingale as outlined by doob. Let mt,ft be a continuous, square integrable martingale. Its existence follows from the doob meyer decomposition theorem and, for continuous local martingales, it is the same as the quadratic variation. A setindexed submartingale can be decomposed into the sum of a weak martingale.
Nicolaus copernicus university abstract both the doobmeyer and the graversenrao decomposition theorems can be proved following an approach based on predictable compensators of discretizations and weakl1 technique, which was developed by k. A nonnegative rightcontinuous submartingale is of class d. In mathematics, quadratic variation is used in the analysis of stochastic processes such as brownian motion and other martingales. We provide a short and elementary prove of the doobmeyer decomposition. The quadratic variation of continuous time stochastic. The quadratic variation x,x of a semimartingale xis one of the main tools of stochastic. Stat331 some key results for counting process martingales. Doobmeyer decomposition, an important theorem about the existence of compensator processes. The condition of the lemma is invariant with respect to an equivalent change of measure p. This is often called the predictable quadratic variation of. Natural decomposition of processes and weak dirichlet processes. In 1953, doob published the doob decomposition theorem which gives a unique decomposition for certain discrete time martingales. At the end of the chapter we discuss the quadratic variation process of a local martingale, a key concept in martingale theory based stochastic analysis.
Towards a general doobmeyer decomposition theorem adam jakubowski. Doob meyer decomposition and quadratic variation of processes in m. Its existence follows from the doobmeyer decomposition theorem and, for continuous local martingales, it is the same as the quadratic variation. Finally, we state the result that links the optional decomposition with the predictable decomposition of doob and meyer. By the convexity condition discussed earlier, is a submartingale provided it is integrable, ie is squareintegrable, and so the process in its doobmeyer decomposition is. The quadratic variation process of xt, often denoted hxit or hx,xit, exists and is equal to at, the increasing part of the doobmeyer decomposition that is, x2 t. Quadratic variation is just one kind of variation of a process. Such doobmeyer decomposition will be called the natural decomposition of x. He conjectured a continuous time version of the theorem and in two publications in 1962 and 1963 paulandre meyer proved such a theorem, which became known as the doobmeyer decomposition. The essential point is that in the limit which converges in the mean square sense under our hypotheses the quadratic covariation receives.
Stochastic integrals and quadratic variation springerlink. Clt for approximate quadratic variation 3 random positive numbers, going to 0. Quantum stochatic integrals and doobmeyer decomposition. Such a decomposition of x is a doobmeyer type decomposition. Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets d. To start our fourth chapter we introduce random measures, point processes, and poisson processes. The quadratic variation of continuous time stochastic processes in vector lattices. Advanced computational methodslecture 2 1 brief introduction.
This book presents a concise and rigorous treatment of stochastic calculus. Theorem 1 doobmeyer let m t, a t bbe a right continuous, square integrable martingale with left hand limits. Conditional expectation and conditional probability. Last time doob decomposition discrete time doobmeyer decomposition continuous time quadratic variation of stochastic process quadratic variation of brownian motion martingale characterization of brownian motion todays lecture. Oct 20, 2004 setindexed strong martingales and a form of predictability for setindexed processes are defined. Following that we introduce the doob meyer decomposition. This predictable quadratic variation is sometimes denoted.
A particularly relevant example is the square of a martingale, that is, where is a martingale. Can at be interpreted as a quadratic variation along a. Following that we introduce the doobmeyer decomposition. The theorem was proved by and is named for joseph l. By the convexity condition discussed earlier, is a submartingale provided it is integrable, ie is squareintegrable, and so the process in its doob meyer decomposition is increasing. Optional decomposition of supermartingales and hedging. Doobs maximal inequality doobmeyer decomposition continuous squareintegrable martingales, quadratic variation brownian motion brownian motion, construction of brownian motion, markov property re ection principle, distribution of rst passage time distribution of brownian motion and its. Therefore, the approximate quadratic variation is simply x. The existence of the quadratic variation process was originally deduced from the doobmeyer decomposition.
By the convexity condition discussed earlier, is a submartingale provided it is integrable, ie is squareintegrable, and so the process in its doobmeyer decomposition is increasing. Under a natural integrability condition, we show that any setindexed strong submartingale can be decomposed in the doobmeyer sense. The doobmeyer decomposition theorem for continuous semimartingales is stated but the proof is omitted. Pdf quantum stochatic integrals and doobmeyer decomposition. Doobs maximal inequality doobmeyer decomposition continuous squareintegrable martingales, quadratic variation brownian motion brownian motion, construction of brownian motion, markov property re ection principle, distribution of rst passage time distribution of brownian motion and its running maximum.
Feb 12, 2017 by the convexity condition discussed earlier, is a submartingale provided it is integrable, ie is squareintegrable, and so the process in its doob meyer decomposition is increasing. Introduction stochastic processes and stochastic integrals the martingale m n a the doob. But recall the quadratic variation property of the brownian motion. 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. In finance, the stochastic calculus is applied to pricing options by no arbitrage. Central limit theorem for approximate quadratic variations. In this paper we consider continuoustime stochastic processes in the measurefree setting of vector lattices. The counting process and martingale framework 2005 wiley.
Both the doobmeyer and the graversenrao theorems can be proved following an approach based on predictable compensators of discretizations and weakl1 tech. Quantum stochatic integrals and doobmeyer decomposition core. Theorem 1 doob meyer let m t, a t bbe a right continuous, square integrable martingale with left hand limits. The present construction was inspired by rogers and williams 2000b. And, even in this case one needs assumptions on the function t xt, such as it is absolutely continuous. Massachusetts institute of technology ito integral. Setindexed strong martingales and a form of predictability for setindexed processes are defined. Before we state and prove the doobmeyer decomp osition theorem we. We consider various continuity notions of such processes and in the case that x t is a submartingale we investigate how its continuity influences that of the natural increasing process appearing in its doobmeyer decomposition. The admissible function of a submartingale is defined and some class d conditions are given which allow the extension of the function to a. In biology, it is applied to populations models, and in engineering. Natural decomposition of processes and weak dirichlet. Central limit theorem for approximate quadratic variations of. The existence of the quadratic variation process was originally deduced from the doob meyer decomposition.
It also gives its main applications in finance, biology and engineering. For the quadratic variation, we also make the greatly simplifying assumption that tn,i i. Setindexed martingales and submartingales are defined and studied. Doob inequalities and doobmeyer decomposition eventually. Doob meyer decomposition, an important theorem about the existence of compensator processes. One of them is the monograph 1 by jacod and shiryaev. Fisk 1966 showed how the quadratic variation can also be obtained directly from the process, as in proposition 17. A setindexed submartingale can be decomposed into the sum of a weak martingale and an. Lastly in chapter three, we discuss two di erent ways to extend our main result to higher dimensions. Doobmeyer decomposition for setindexed submartingales.
Central limit theorem for approximate quadratic variations of pure jump ito. By the convexity condition discussed earlier, is a submartingale provided it is integrable, ie is squareintegrable, and so the process in its doob meyer decomposition is. Introduction in the existing theories of quantum stochastic integration we face a. Introduction to stochastic calculus with applications. The doobmeyer decomposition theorem is a theorem in stochastic calculus stating the conditions under which a submartingale may be decomposed in a unique way as the sum of a martingale and an increasing predictable process. In fact, m2 is right continuous submartingale, so it is of class dl.
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