Optional stopping theorem
The next four lectures will be devoted to the foundational theorems of the theory of continuous time martingales. All of these theorems are due to Joseph Doob. The following first theorem shows that martingales behave in a very optional stopping theorem way with respect to stopping times.
The following properties are equivalent: Let us assume that is a martingale with respect to the filtrationwhose paths are right continuous and locally bounded. Let now be a stopping time of the filtration that is almost surely bounded by.
Let us first assume that takes its values in a finite set: Thanks to the martingale optional stopping theorem, we have. The theorem is therefore proved if takes its values in a finite set. If takes an infinite number of values, we approximate by the following sequence of stopping times:. The stopping time takes its values in a finite set and optional stopping theorem. To conclude the proof of the first part of the proposition, it is therefore enough to show that.
For this, we are going to prove that the family is uniformly integrable. Conversely, let us now assume that for any, almost surely bounded stopping time of the filtration such thatwe have. By using the stopping time. The hypothesis that the paths of be right continuous and locally bounded is actually not strictly necessary, however the hypothesis that the stopping time be almost surely bounded is essential, as it is proved in the following exercise. Let be a filtration defined on a probability space and let be a continuous martingale that is a martingale with continuous paths with respect to the filtration such that almost surely.
Forwe denote. Show that is a stopping time of the filtration. Prove that is not almost surely bounded. Let be a filtration defined on a probability space and let be a continuous martingale with respect to the filtration. Deduce that the stochastic process is a martingale with optional stopping theorem to the filtration.
Let be a filtration defined on a probability space and let be a submartingale with respect to the filtration whose paths are continuous. You are commenting using your WordPress. You are commenting using your Twitter account. You are commenting using your Facebook account. Notify me of new comments via email. Research and Lecture notes. Home About this blog Stochastic Calculus lectures Rough paths theory Curvature dimension inequalities Mathematicians Hypoelliptic operators Diffusions on foliated manifolds Diffusions on manifolds.
If takes an infinite number of values, we approximate by the following sequence of stopping times: To conclude the proof of the first part of the proposition, it is therefore enough to optional stopping theorem that For optional stopping theorem, we are going to prove that the family is uniformly integrable. By using the stopping time we are led to which implies the martingale property for.
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In probability theorythe optional stopping theorem or Doob 's optional sampling theorem says that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial expected value. Since martingales can be used to model the wealth of a gambler participating in a fair game, the optional optional stopping theorem theorem says that, on average, nothing can be gained by stopping play based on the information obtainable so far i.
Certain conditions are necessary for this result to hold true. In particular, the theorem applies to doubling strategies. The optional stopping theorem is an important tool optional stopping theorem mathematical finance in the context of the fundamental theorem of asset pricing. Assume that one of the following three conditions holds:. Similarly, if the stochastic process X is a submartingale or a supermartingale and one of the above conditions holds, then.
Otherwise, writing the stopped process as. By the monotone convergence theorem. Therefore, under any one of the three conditions in the theorem, the stopped process is dominated by an integrable random variable M.
Similarly, if X is a submartingale or supermartingale, respectively, change the equality in the last two formulas to the appropriate inequality. From Wikipedia, the free encyclopedia.
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