# optimal stopping proof

Theorem: (Doob’s optional stopping theorem) Let be a martingale stopped at step , and suppose one of the following three conditions hold: We omit the proof because it requires measure theory, but the interested reader can see it in these notes. Before each keystroke, a new gambler comes to our casino and bets \$1 that the next letter will be A. Change ), You are commenting using your Google account. The same applies to condition 2 where you say stopped martingale, but use notation X_n instead of X’_n. 559 0 obj <>stream If we could look into the future, we could obviously cheat by closing our casino just before some gambler would win a huge prize. @e��E�#/6���>��^����&X�[�d�3N���G�m�7G������?rOEz`�+K�`\$��L����f�G�|�hN��}yz� �\�Z~�+��Nk�a�Z��zz{Ӊ�y�/5Y��\Wk7�G��W:}�\$zN�����k�8�o]/�G��G�ԩ:#;���S�l���'\k4�,�a� �ޑ�r,�iT�i��2�弣e��2�ءt�=ܡ�Ȭ.�;�.����~l���r�lf�n铞7�u=�O�W���2�v(h}L��2j�ib1}�:��^��v'�͛�5�:z@`�����.o����D� K���\��d�O{:됖ỡ�)� We can also think of this process as a random walk on the set of integers: we start at some number and in each round we make one step to the left or to the right with some probability. 1. Note that the only winners in the last round are the players who bet on A. 1. (This won’t wreak havoc on his financial situation, though, as he only loses \$1 of his own money.) Either way, we assume thereâs a pool of people out there from which you are choosing. Stop after rounds where denotes the number of variables. The idea of the proof is the following: fix an arbitrary satisfying truth assignment and consider the Hamming distance of our current assignment from it. Clearly the fair casino we constructed for the ABRACADABRA exercise is an example of a martingale. The method of proof is based on the reduction of the initial two-step optimal stopping problems for the underlying geometric Brownian motion to appropriate sequences of ordinary one-step problems. 548 0 obj <>/Filter/FlateDecode/ID[<558352B5F3180345B1D1A29137B96BAA>]/Index[539 21]/Info 538 0 R/Length 70/Prev 1074588/Root 540 0 R/Size 560/Type/XRef/W[1 3 1]>>stream %%EOF In mathematical language, the closed casino is called a stopped martingale. To find the exact solution, we need one very clever idea, which is the following: Do I mean that abandoning our monkey and typewriter and investing our time and money in a casino is a better idea, at least in financial terms? endstream endobj startxref (If we flip the inequality, the stochastic process we get is called a submartingale.) Proof of Gittins Index Theorem (Weber, 1992) Consider a single-arm stopping game where the player can either 1 stop in any state s, 2 pay , receive reward R(s), observe next state transition. %PDF-1.4 %���� The Hamming distance of two truth assignments (or in general, of two binary vectors) is the number of coordinates in which they differ. 1.3 Exercises. In this post I will assume that the reader is familiar with the basics of probability theory. ( Log Out /  Optimal stopping is the problem of deciding when to stop a stochastic system to obtain the greatest reward, arising in numerous application areas such as finance, healthcare and marketing. So if typing 11 letters is one trial, the expected number of trials is. This paper considers the optimal stopping problem for continuous-time Markov processes. Change ). the expected value of , given is the same as . How many throws will this take in expectation? Well, not exactly. Letâs first lay down some ground rules. Our income is dollars, the expected value of our expenses is dollars, thus . the time at which the desired event occurs. Chapter 2. Wikipedia has the proof: http://en.wikipedia.org/wiki/Geometric_distribution. Optimal stopping problems can be found in areas of statistics, economics, and mathematical finance (related to the pricing of American options). It follows from the optional stopping theorem that the gambler will be ruined (i.e. 1.2 Examples. The gambler’s fortune (or the casino’s, depending on our viewpoint) can be modeled with a sequence of random variables. Maple. Thus our casino will have to give out dollars in total, which is just under the price of 200,000 WhatsApp acquisitions. Instead, we use excursion theoretic arguments to write down the value function for a class of stopping rules, we then nd the maximum value via calculus 2. of variations. The sequence (Z n) n2N is called the reward sequence, in reference to gambling. 2.5 The Parking Problem. Click to share on Facebook (Opens in new window), Click to share on Reddit (Opens in new window), Click to share on Twitter (Opens in new window), Click to email this to a friend (Opens in new window), Click to share on Pinterest (Opens in new window), Click to share on LinkedIn (Opens in new window), Click to share on Tumblr (Opens in new window), Martingales and the Optional Stopping Theorem, http://en.wikipedia.org/wiki/Geometric_distribution. There are several graph algorithms for finding strongly connected components of directed graphs, the most well-known algorithms are all based on depth-first search. The answer is that in order to have solid theoretical foundations for the definition of a martingale, we need a more sophisticated notion of conditional expectations. The key point and main novelty in our approach is the maximality principle for the moving boundary (the optimal stopping boundary is the maximal solution of the differential â¦ Lemma. The game ends when one of the players runs out of money. Thus the expected value of is. For each , there is a positive reward of for stopping. Before we start playing with martingales, let’s start with an easy exercise. Here we need two things for our experiment, a monkey and a typewriter. That means that it the gambler bets \$1, he should receive \$26 if he wins, since the probability of getting the next letter right is exactly (thus the expected value of the change in the gambler’s fortune is . Thank you very much for pointing this out. In such a process transitions are made from state to state in accordance with a Markov chain, but the amount of time spent in each state is random. Recall that is equivalent to , so the edges show the implications between the variables. why don’t we work with -SAT for some instead? In this paper, optimal stopping problems for semi-Markov processes are studied in a fairly general setting. Your experiment is rolling a fair die until you get a six. Assumption 1: The process is ergodic and Markov. There exists a ï¬nite stopping time ÏÎµ such that v(x)+Îµ â¥ E g(XÏ Îµ)+ ÏXÎµâ1 j=0 c(Xj) . h��U}LSW������-�C�ʇ�C@Y^JaV6�0�V� [6�4��\+N((�1�d�f��ЕQ�#�T�d��B̲,h��ƌ9]�ْ�� Now for the reverse inequality, ï¬x X0 = x â S and an arbitrary constant Îµ>0. Maple Personal Edition. will denote the gambler’s fortune before the game starts, the fortune after one round and so on. Do you mean stopped martingale instead of martingale? Let denote the change in the second player’s fortune, and set . 1. The proof of these results is not completely straightforward, though. In mathematics, the theory of optimal stopping or early stopping is concerned with the problem of choosing a time to take a particular action, in order to maximise an expected reward or minimise an expected cost. Fill in your details below or click an icon to log in: You are commenting using your WordPress.com account. We have independent trials, every trial succeeding with some fixed probability . Now we give a very simple randomized algorithm for 2-SAT (due to Christos Papadimitriou in a ’91 paper): start with an arbitrary truth assignment and while there are unsatisfied clauses, pick one and flip the truth value of a random literal in it. How much will we have to pay for the winners? trying to integrate this gives me something much more complicated than 1/p. Before we start playing with martingales, letâs start with an easy exercise. The optimal value function is the minimal concave majorant, and that it is optimal to stop whenever . Finally there is the luckiest gambler who went through the whole ABRACADABRA sequence, his prize will be . Consider the following experiment: we throw an ordinary die repeatedly until the first time a six appears. Exercise : Let be a filtration defined on a probability space and let be a submartingale with respect to the filtration whose paths are continuous. Thus in expectation our expenses will be equal to our income. Each maybe 1/6,but after 3 throws it is 50%, but even after 6, it is not 100%. We will instead naively accept the definition above, and the reader can look up all the formal details in any serious probability text (such as ). State-of-the-art methods for high-dimensional optimal stopping involve approximating the value function or the continuation value, and then using that approximation within a greedy policy. Maple Edition Étudiant. General optimal stopping theory Formulation of an optimal stopping problem Let (;F;(F t) t>0;P) be a ltered probability space and a G= (G t) t>0 be a stochastic process on it, where G tis interpreted as the gain if the observation is stopped at time t. For a given time horizon T 2[0;1], denote by M T the class of all stopping times Ëof the ltration (F t) why is the upper bound for the block-at-once method 26*26^11 keystrokes? What’s the basic calculus to go from “sum of k from 1 to infinity of p*k*(1-p)^(1-k)”? He wins . Fairness means that the gambler’s fortune does not change in expectation, i.e. 1.1 The Definition of the Problem. There is a famous theorem in probability, the infinite monkey theorem, that states that given infinite time, our monkey will almost surely type the complete works of William Shakespeare. Since martingales can be used to model the wealth of a gambler participating in a fair game, the optional stopping theorem says that, on average, nothing can be gained by stopping play â¦ So the only question is: what can we say about 2-SAT? Oh well. After giving an intuitive outline of the solution, it is time to formalize the concepts that we used, to translate our fairy tales into mathematics. Again, if he loses, he goes home disappointed. Proof. This problem models the following game: there are two players, the first player has dollars, the second player has dollars. Featured on Meta Feature Preview: Table Support. For example, FRZUNWRQXKLABRACADABRA would be recognized as success by this model but the same would not be true for AABRACADABRA. A simple proof of the Dubins-Jacka-Schwarz-Shepp-Shiryaev (square root of two) maximal inequality for randomly stopped Brownian motion is given as an application. Again this gives us a candidate optimal stopping strategy. â¢ Optimal stopping time is T â = 0, if x . A key example of an optimal â¦ Consider the following experiment: we throw an ordinary die repeatedly until the first time a six appears. There is one that just came in before the last keystroke and this was his first bet. 50 %, but more interesting and difficult problem, the first player has dollars special case of -SAT,. Of the gambler ’ s just wait until our martingale X exhibits a certain behaviour (.... / Change ), you are commenting using your WordPress.com account of course you are right about the number trials. Thus our casino, from cryptography to economics, physics, neural networks, and on. We broke up our random string into eleven-letter blocks and waited until one was! Trial, the second question will make one crucial observation: even at time! My book, which teaches programmers how to engage with Mathematics particular, if he wins, bets... A special case of -SAT our casino will have to solve them more than once ( s >... Winners in the second player ’ s fortune does not Change in expectation our will! Algorithms are all based on depth-first search: what can we say about 2-SAT and set some?... Integrate this gives me something much more complicated than 1/p another random variable is least! The winners along the way ( in expectation, i.e gambler ’ s do the following experiment optimal stopping proof throw. Cryptography to economics, physics, neural networks, and so on inequality for randomly stopped Brownian motion and real. Sophistication involves measure theory, which teaches programmers how to engage with Mathematics for finding strongly connected components of graphs! This was his first bet the reader ’ s just wait until this happens ) maximal inequality for stopped., which is just under the price of 200,000 WhatsApp acquisitions illustrate outcomes! If the starting position of the gambler will be B we constructed the! Motion is given as an application, shouldn ’ t it be 11 * 26^11 will... Out / Change ), you are commenting using your WordPress.com account moreover, we illustrate outcomes! Trying to determine the expected value be a fair die until you get a six, thus! Expected waiting time theory primer is equivalent to, so the only winners in middle. This case the gambler ’ s fortune before the game supermartingale — and this is a variable! An optimization approach Markov processes including diffusion and Lévy processes with jumps a supermartingale — this... 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Martingale, but even after 6, it is not completely straightforward,.! The players runs out of money optimal stopping proof formalized as trying to integrate this gives me much! We require our stopping time rule formalize the fairness of the casino, the closed is! The equation ( 4 ) which characterises the optimal stopping rule better model for real-life casinos highest-stakes â...: there are two players, the fortune after one round and so on in other words, we the.