Optimal-stopping-Wikipedia #

A sequence of ‘reward’ functions which depend on the observed values of the random variables in 1.: Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/c347dcdade06ab59406dacf0d929e03855856ee8 Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/4f93e8400461ac4d620680baee52030fa89911db Given those objects, the problem is as follows:

• You are observing the sequence of random variables, and at each step , you can choose to either stop observing or continue Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/add78d8608ad86e54951b8c8bd6c8d8416533d20
• If you stop observing at step , you will receive reward Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/67d30d30b6c2dbe4d6f150d699de040937ecc95f
• You want to choose a stopping rule to maximize your expected reward (or equivalently, minimize your expected loss)

Continuous time case[edit] #

Consider a gain processes Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/3145251bd7dcd62f06889457914d47d54447646a defined on a filtered probability space Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/5e3f1b6d200f2bc4fd12f17fcd4b9547da96ce09 and assume that Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/f5f3c8921a3b352de45446a6789b104458c9f90b is adapted to the filtration. The optimal stopping problem is to find the stopping time Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/0b4ac981f3c6efc49fbcb3ecd24f7bf152dad0a7 which maximizes the expected gain Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/c4da65227df8165056ee82f640793d8e4b37908f where Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/57a433d75842b2d6a28cd5f8ca9cf7dba459084f is called the value function. We consider an adapted strong Markov process Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/478bcaa73ef8daeb8bd07701b59c6384b689f131 defined on a filtered probability space Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/becca0fa5b0e6527db1e25d78299511b5320edbb where Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/03c8fe9e48980d22020c362b11762a216f8bee58 denotes the probability measure where the stochastic process starts at Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/87f9e315fd7e2ba406057a97300593c4802b53e4 .

A jump diffusion result[edit] #

Let Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/95734a78eb8407939c3496cbfd92763ced1e41e1 be a Lévy diffusion in Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/1bcd8908c9fa46eb979ef7b67d1bb65eb3692cbb given by the SDE Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/264bc8d76ca788b3eff6e45fa24b76c3201aba60 where Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/47136aad860d145f75f3eed3022df827cee94d7a is an Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/0a07d98bb302f3856cbabc47b2b9016692e3f7bc -dimensional Brownian motion, Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/b49f9e15c90b97d6d95aaf6bd1a4f520d66c2bb7 is an Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/829091f745070b9eb97a80244129025440a1cfac -dimensional compensated Poisson random measure, Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/aae4bec0dfe664f70a1b9cda15fd319fa1e454eb , Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/165b2ac51764fbee3ed5db71d915b53420333832 , and Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/454e9f9964b0205f0e19d54a5e902038bc1e095f are given functions such that a unique solution Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/5446d2e710df1848b39d3474304fa84dbdc60a05 exists. The optimal stopping problem is: Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/96e90fc8d59f61857be4ba95aff689714bfc5761 It turns out that under some regularity conditions,[5] the following verification theorem holds: If a function Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/7d9dd8e4893e28a7f6eabb88b72d49efc8ddeb39 satisfies

• where the continuation region is , Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/8a238ecdc084108386647a9f4928c99d54af39a4 Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/9b3f5a2a4a0459b28eb40706f67ea48f83d35b78
• on , and Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/3a8073395c6ee55e9f384471412c9d453ced655f Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/2302a18e269dbecc43c57c0c2aced3bfae15278d
• on , where is the infinitesimal generator of Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/b4b891a2ffc1861ecef13412bb7c69dd7e794891 Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/bc33a22fb3a9e91084b653ef5e58815ff05aba06 Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/280ae03440942ab348c2ca9b8db6b56ffa9618f8 then Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/812e58cf6049240099f528ebf2c4b403f7a9ebc7 for all Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/45c9dab32dcebce045fc69264dc531a98b9bc6c9 . Moreover, if
• on Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/ab904cee10099523faefb28ada29590acb97c578 Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/f34a0c600395e5d4345287e21fb26efd386990e6 Then Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/76c271acf29d6893d8a17d35018cc7d8d840ac50 for all and Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/72be0ff341a0ca9b991ed0249f29a229b223903f is an optimal stopping time. The solution is known to be[7]
• (Perpetual call) where and Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/62363453bd2df75ecda55d5ef3dba9d954f679a5 Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/0b5c45a5b03cdca86ea8deb1ec6e2c10ed35d099 Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/fe2ccc20c0bf5cf1d671556648d75d76656fca3d
• (Perpetual put) where and Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/e77d9317d3a58cf30457e68bf232480b6afc4a4f Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/fb8e5648b16dcd2cd97fdd295d5ea25ee2224d52 Optimal%20stopping%20-%20Wikipedia%2036ca95b1506f4eae8314e9de4cd135fa/10134e66db0d440076b296492c842f996f485e14 On the other hand, when the expiry date is finite, the problem is associated with a 2-dimensional free-boundary problem with no known closed-form solution.