3693:
A special example of an application of search theory is the task of optimal selection of parking space by a driver going to the opera (theater, shopping, etc.). Approaching the destination, the driver goes down the street along which there are parking spaces – usually, only some places in the parking
1164:
3694:
lot are free. The goal is clearly visible, so the distance from the target is easily assessed. The driver's task is to choose a free parking space as close to the destination as possible without turning around so that the distance from this place to the destination is the shortest.
3684:
Economists have studied a number of optimal stopping problems similar to the 'secretary problem', and typically call this type of analysis 'search theory'. Search theory has especially focused on a worker's search for a high-wage job, or a consumer's search for a low-priced good.
2118:
1496:
4309:
2279:
4850:
4531:
3918:
3714:
is allowed to exercise the right to buy (or sell) the underlying asset at a predetermined price at any time before or at the expiry date. Therefore, the valuation of
American options is essentially an optimal stopping problem. Consider a classical
5045:
649:
1773:
873:
3665:
are the sequences associated with this problem. This problem was solved in the early 1960s by several people. An elegant solution to the secretary problem and several modifications of this problem is provided by the more recent
495:
4696:
1921:
984:
4928:
1702:
2909:
5050:
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. Various numerical methods can, however, be used. See
4597:
2169:
1932:
2351:
3064:
1216:
When the underlying process is determined by a family of (conditional) transition functions leading to a Markov family of transition probabilities, powerful analytical tools provided by the theory of
3141:
4026:
2835:
1845:
1646:
1305:
2945:
2477:
2177:
266:
4362:
4703:
3001:
You have a fair coin and are repeatedly tossing it. Each time, before it is tossed, you can choose to stop tossing it and get paid (in dollars, say) the average number of heads observed.
2444:
2768:
2622:
2664:
4391:
4145:
3809:
3566:
3282:
2996:
786:
412:
140:
3236:
3190:
2585:
189:
3434:
1294:
902:
4137:
4083:
2501:
2403:
4933:
2731:
3519:
You are observing a sequence of objects which can be ranked from best to worst. You wish to choose a stopping rule which maximises your chance of picking the best object.
2379:
1572:
684:
557:
549:
3781:
3761:
3663:
3630:
3514:
2538:
1806:
728:
5462:
5207:
3597:
3467:
3379:
3312:
1261:
345:
2504:
956:
1707:
4382:
3801:
3737:
3352:
3332:
2684:
1592:
1539:
1519:
1207:
976:
930:
708:
515:
318:
295:
791:
1177:
There are generally two approaches to solving optimal stopping problems. When the underlying process (or the gain process) is described by its unconditional
420:
4602:
1159:{\displaystyle V(x)=\sup _{0\leq \tau \leq T}\mathbb {E} _{x}\left(M(X_{\tau })+\int _{0}^{\tau }L(X_{t})dt+\sup _{0\leq t\leq \tau }K(X_{t})\right).}
1853:
4855:
5416:
5290:
5250:
1651:
2850:
1220:
can often be utilized and this approach is referred to as the Markov method. The solution is usually obtained by solving the associated
2113:{\displaystyle V(y)=\sup _{\tau \leq \tau _{\mathcal {S}}}J^{\tau }(y)=\sup _{\tau \leq \tau _{\mathcal {S}}}\mathbb {E} _{y}\left.}
4536:
2129:
1297:
2284:
3026:
1491:{\displaystyle dY_{t}=b(Y_{t})dt+\sigma (Y_{t})dB_{t}+\int _{\mathbb {R} ^{k}}\gamma (Y_{t-},z){\bar {N}}(dt,dz),\quad Y_{0}=y}
1182:
1178:
3075:
5479:
3932:
2773:
1811:
1601:
5518:
2914:
2274:{\displaystyle \phi \in C({\bar {\mathcal {S}}})\cap C^{1}({\mathcal {S}})\cap C^{2}({\mathcal {S}}\setminus \partial D)}
5528:
2449:
5523:
196:
4845:{\displaystyle V(x)={\begin{cases}K-x&x\in (0,c]\\(K-c)(x/c)^{\tilde {\gamma }}&x\in (c,\infty )\end{cases}}}
4317:
415:
48:
5082:
2410:
20:
2736:
2590:
5052:
2630:
5077:
5060:
4526:{\displaystyle V(x)={\begin{cases}(b-K)(x/b)^{\gamma }&x\in (0,b)\\x-K&x\in [b,\infty )\end{cases}}}
4304:{\displaystyle \max \left\{{\frac {1}{2}}\sigma ^{2}x^{2}V''(x)+(r-\delta )xV'(x)-rV(x),g(x)-V(x)\right\}=0}
3913:{\displaystyle S_{t}=S_{0}\exp \left\{\left(r-\delta -{\frac {\sigma ^{2}}{2}}\right)t+\sigma B_{t}\right\}}
3740:
3469:) is the sequence of offers for your house, and the sequence of reward functions is how much you will earn.
1169:
This is sometimes called the MLS (which stand for Mayer, Lagrange, and supremum, respectively) formulation.
3018:
2841:
1595:
27:
5323:
3525:
3249:
2963:
739:
365:
99:
1221:
3195:
3149:
2546:
148:
5318:
5040:{\displaystyle \nu =(r-\delta )/\sigma -\sigma /2,\quad c={\tilde {\gamma }}K/({\tilde {\gamma }}-1).}
3384:
1270:
878:
5460:
F. Thomas Bruss. "The art of a right decision: Why decision makers want to know the odds-algorithm."
60:
4727:
4415:
4088:
4034:
2482:
2384:
51:
an expected reward or minimise an expected cost. Optimal stopping problems can be found in areas of
5360:
5097:
3599:
is the chance you pick the best object if you stop intentionally rejecting objects at step i, then
2692:
1210:
905:
76:
644:{\displaystyle V_{t}^{T}=\mathbb {E} G_{\tau ^{*}}=\sup _{t\leq \tau \leq T}\mathbb {E} G_{\tau }}
5496:
5436:
5314:
5296:
5195:
5170:
5145:
5092:
5087:
2358:
909:
5440:
3716:
1548:
657:
527:
5412:
5377:
5340:
5286:
5266:
5246:
5187:
3478:
2123:
It turns out that under some regularity conditions, the following verification theorem holds:
68:
3766:
3746:
3635:
3602:
3486:
2510:
1778:
713:
5488:
5404:
5396:
5369:
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5238:
5179:
5165:
5131:
3707:
3703:
72:
64:
3575:
3445:
3357:
3290:
1768:{\displaystyle \gamma :\mathbb {R} ^{k}\times \mathbb {R} ^{k}\to \mathbb {R} ^{k\times l}}
1239:
1181:, the appropriate solution technique is the martingale approach, so called because it uses
323:
5450:
5230:
5123:
5072:
1542:
518:
47:
is concerned with the problem of choosing a time to take a particular action, in order to
1264:
935:
5149:
868:{\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\geq 0},\mathbb {P} _{x})}
5270:
4367:
3786:
3722:
3667:
3337:
3317:
2669:
1577:
1524:
1504:
1225:
1217:
1192:
1186:
961:
915:
734:
693:
687:
500:
303:
280:
5512:
5300:
5199:
3679:
522:
350:
3017:≥ 1) forms a sequence of independent, identically distributed random variables with
490:{\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\geq 0},\mathbb {P} )}
3711:
5336:
4691:{\displaystyle \nu =(r-\delta )/\sigma -\sigma /2,\quad b=\gamma K/(\gamma -1).}
36:
5492:
353:
to maximize your expected reward (or equivalently, minimize your expected loss)
5282:
5242:
1916:{\displaystyle \tau _{\mathcal {S}}=\inf\{t>0:Y_{t}\notin {\mathcal {S}}\}}
52:
5381:
5344:
5191:
3004:
You wish to maximise the amount you get paid by choosing a stopping rule. If
56:
5373:
733:
A more specific formulation is as follows. We consider an adapted strong
5471:
5183:
4923:{\displaystyle {\tilde {\gamma }}=-({\sqrt {\nu ^{2}+2r}}+\nu )/\sigma }
5500:
5358:
MacQueen, J.; Miller Jr., R.G. (1960). "Optimal persistence policies".
3439:
You wish to maximise the amount you earn by choosing a stopping rule.
277:
You are observing the sequence of random variables, and at each step
1697:{\displaystyle \sigma :\mathbb {R} ^{k}\to \mathbb {R} ^{k\times m}}
5408:
2904:{\displaystyle \max \left\{{\mathcal {A}}\phi +L,M-\phi \right\}=0}
5056:
3783:
be the dividend rate and volatility of the stock. The stock price
191:
which depend on the observed values of the random variables in 1:
71:. Optimal stopping problems can often be written in the form of a
2840:
These conditions can also be written is a more compact form (the
3287:
You have a house and wish to sell it. Each day you are offered
521:
to the filtration. The optimal stopping problem is to find the
26:"Dynkin game" redirects here. For the coupling card trick, see
5403:. Stochastic Modelling and Applied Probability. Vol. 39.
3926:
When the option is perpetual, the optimal stopping problem is
4592:{\displaystyle \gamma =({\sqrt {\nu ^{2}+2r}}-\nu )/\sigma }
2920:
2864:
2750:
2636:
2604:
2488:
2455:
2416:
2390:
2305:
2254:
2228:
2197:
2143:
2015:
1968:
1905:
1863:
1817:
820:
806:
449:
435:
4838:
4519:
2164:{\displaystyle \phi :{\bar {\mathcal {S}}}\to \mathbb {R} }
142:, whose joint distribution is something assumed to be known
3334:
to continue advertising it. If you sell your house on day
2346:{\displaystyle D=\{y\in {\mathcal {S}}:\phi (y)>M(y)\}}
3059:{\displaystyle {\text{Bern}}\left({\frac {1}{2}}\right),}
1926:
be the bankruptcy time. The optimal stopping problem is:
3572:
is some large number) are the ranks of the objects, and
3136:{\displaystyle y_{i}={\frac {1}{i}}\sum _{k=1}^{i}X_{k}}
92:
Stopping rule problems are associated with two objects:
5472:"Search-theoretic models of the labor market: a survey"
4021:{\displaystyle V(x)=\sup _{\tau }\mathbb {E} _{x}\left}
67:). A key example of an optimal stopping problem is the
4384:
is the exercise boundary. The solution is known to be
2830:{\displaystyle \tau ^{*}=\inf\{t>0:Y_{t}\notin D\}}
1840:{\displaystyle {\mathcal {S}}\subset \mathbb {R} ^{k}}
1641:{\displaystyle b:\mathbb {R} ^{k}\to \mathbb {R} ^{k}}
4936:
4858:
4706:
4605:
4539:
4394:
4370:
4320:
4148:
4091:
4037:
3935:
3812:
3789:
3769:
3749:
3725:
3638:
3605:
3578:
3528:
3489:
3448:
3387:
3360:
3340:
3320:
3293:
3252:
3198:
3152:
3078:
3029:
2966:
2917:
2853:
2776:
2739:
2695:
2672:
2633:
2593:
2549:
2513:
2485:
2452:
2413:
2387:
2361:
2287:
2180:
2132:
1935:
1856:
1814:
1781:
1710:
1654:
1604:
1580:
1551:
1527:
1507:
1308:
1273:
1242:
1195:
1189:. In the discrete time case, if the planning horizon
987:
964:
938:
918:
881:
794:
742:
716:
696:
660:
560:
530:
503:
423:
368:
326:
306:
297:, you can choose to either stop observing or continue
283:
199:
151:
102:
1209:
is finite, the problem can also be easily solved by
2940:{\displaystyle {\mathcal {S}}\setminus \partial D.}
5128:Great Expectations: The Theory of Optimal Stopping
5039:
4922:
4844:
4690:
4591:
4525:
4376:
4356:
4303:
4131:
4077:
4020:
3912:
3795:
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3624:
3591:
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3508:
3461:
3428:
3373:
3346:
3326:
3306:
3276:
3230:
3184:
3135:
3058:
2990:
2939:
2903:
2829:
2762:
2725:
2678:
2658:
2616:
2579:
2532:
2495:
2472:{\displaystyle {\mathcal {S}}\setminus \partial D}
2471:
2438:
2397:
2373:
2345:
2273:
2163:
2112:
1915:
1839:
1800:
1767:
1696:
1640:
1586:
1566:
1533:
1513:
1490:
1288:
1255:
1201:
1158:
970:
950:
924:
896:
867:
780:
722:
702:
678:
643:
543:
509:
489:
406:
339:
312:
289:
260:
183:
134:
5321:(2010). "House-hunting without second moments".
4149:
4139:for a put option. The variational inequality is
3952:
2854:
2790:
1999:
1952:
1872:
1775:are given functions such that a unique solution
1108:
1004:
605:
273:Given those objects, the problem is as follows:
261:{\displaystyle y_{i}=y_{i}(x_{1},\ldots ,x_{i})}
5463:Newsletter of the European Mathematical Society
5063:, calculation of the optimal time to exercise.
5055:for various valuation methods here, as well as
4357:{\displaystyle x\in (0,\infty )\setminus \{b\}}
3238:are the objects associated with this problem.
5470:Rogerson, R.; Shimer, R.; Wright, R. (2005).
5275:Applied Stochastic Control of Jump Diffusions
1185:theory, the most important concept being the
8:
4351:
4345:
2824:
2793:
2340:
2294:
1910:
1875:
5235:Optimal Stopping and Free-Boundary Problems
2439:{\displaystyle {\mathcal {A}}\phi +L\leq 0}
2763:{\displaystyle y\in {\bar {\mathcal {S}}}}
2617:{\displaystyle y\in {\bar {\mathcal {S}}}}
1847:be an open set (the solvency region) and
5224:
5222:
5014:
5013:
5005:
4991:
4990:
4972:
4958:
4935:
4912:
4886:
4880:
4860:
4859:
4857:
4800:
4799:
4787:
4722:
4705:
4665:
4641:
4627:
4604:
4581:
4555:
4549:
4538:
4451:
4439:
4410:
4393:
4369:
4319:
4181:
4171:
4157:
4147:
4123:
4090:
4069:
4036:
4004:
3982:
3967:
3963:
3962:
3955:
3934:
3899:
3870:
3864:
3830:
3817:
3811:
3788:
3768:
3748:
3724:
3646:
3637:
3613:
3604:
3583:
3577:
3552:
3533:
3527:
3497:
3488:
3453:
3447:
3408:
3392:
3386:
3365:
3359:
3339:
3319:
3298:
3292:
3265:
3254:
3253:
3251:
3216:
3206:
3197:
3170:
3160:
3151:
3127:
3117:
3106:
3092:
3083:
3077:
3039:
3030:
3028:
2979:
2968:
2967:
2965:
2919:
2918:
2916:
2863:
2862:
2852:
2812:
2781:
2775:
2749:
2747:
2746:
2738:
2694:
2671:
2635:
2634:
2632:
2603:
2601:
2600:
2592:
2548:
2521:
2512:
2487:
2486:
2484:
2454:
2453:
2451:
2415:
2414:
2412:
2389:
2388:
2386:
2360:
2304:
2303:
2286:
2253:
2252:
2243:
2227:
2226:
2217:
2196:
2194:
2193:
2179:
2157:
2156:
2142:
2140:
2139:
2131:
2087:
2071:
2066:
2050:
2029:
2025:
2024:
2014:
2013:
2002:
1980:
1967:
1966:
1955:
1934:
1904:
1903:
1894:
1862:
1861:
1855:
1831:
1827:
1826:
1816:
1815:
1813:
1789:
1780:
1753:
1749:
1748:
1738:
1734:
1733:
1723:
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1718:
1709:
1682:
1678:
1677:
1667:
1663:
1662:
1653:
1632:
1628:
1627:
1617:
1613:
1612:
1603:
1579:
1553:
1552:
1550:
1526:
1506:
1476:
1436:
1435:
1417:
1399:
1395:
1394:
1392:
1379:
1363:
1335:
1316:
1307:
1280:
1276:
1275:
1272:
1247:
1241:
1194:
1139:
1111:
1089:
1073:
1068:
1052:
1031:
1027:
1026:
1007:
986:
963:
937:
917:
888:
884:
883:
880:
856:
852:
851:
835:
825:
819:
818:
805:
804:
793:
766:
756:
741:
715:
695:
670:
665:
659:
635:
627:
626:
608:
593:
588:
580:
579:
570:
565:
559:
535:
529:
502:
480:
479:
464:
454:
448:
447:
434:
433:
422:
392:
382:
367:
331:
325:
305:
282:
249:
230:
217:
204:
198:
169:
159:
150:
120:
107:
101:
788:defined on a filtered probability space
5237:. Lectures in Mathematics. ETH Zürich.
5114:
4342:
3670:of optimal stopping (Bruss algorithm).
2925:
2659:{\displaystyle {\mathcal {A}}\phi +L=0}
2460:
2259:
75:, and are therefore often solved using
7:
5053:Black–Scholes model#American options
3561:{\displaystyle R_{1},\ldots ,R_{n}}
3277:{\displaystyle \mathbb {E} (y_{i})}
2991:{\displaystyle \mathbb {E} (y_{i})}
978:, the optimal stopping problem is
781:{\displaystyle X=(X_{t})_{t\geq 0}}
407:{\displaystyle G=(G_{t})_{t\geq 0}}
135:{\displaystyle X_{1},X_{2},\ldots }
5453:. "Sum the odds to one and stop."
4829:
4510:
4336:
3803:follows geometric Brownian motion
2928:
2463:
2262:
798:
717:
551:which maximizes the expected gain
427:
14:
5441:Who solved the secretary problem?
5151:Optimal Stopping and Applications
3231:{\displaystyle (y_{i})_{i\geq 1}}
3185:{\displaystyle (X_{i})_{i\geq 1}}
2580:{\displaystyle \phi (y)\geq V(y)}
2281:where the continuation region is
184:{\displaystyle (y_{i})_{i\geq 1}}
145:A sequence of 'reward' functions
5168:(2009). "Knowing When to Stop".
3923:under the risk-neutral measure.
3429:{\displaystyle y_{n}=(X_{n}-nk)}
1289:{\displaystyle \mathbb {R} ^{k}}
1179:finite-dimensional distributions
897:{\displaystyle \mathbb {P} _{x}}
5401:Methods of Mathematical Finance
4983:
4652:
3442:In this example, the sequence (
3284:does not necessarily converge)
1471:
96:A sequence of random variables
5480:Journal of Economic Literature
5031:
5019:
5010:
4996:
4955:
4943:
4909:
4877:
4865:
4832:
4820:
4805:
4796:
4781:
4778:
4766:
4759:
4747:
4716:
4710:
4682:
4670:
4624:
4612:
4578:
4546:
4513:
4501:
4477:
4465:
4448:
4433:
4430:
4418:
4404:
4398:
4339:
4327:
4287:
4281:
4272:
4266:
4257:
4251:
4239:
4233:
4219:
4207:
4201:
4195:
4132:{\displaystyle g(x)=(K-x)^{+}}
4120:
4107:
4101:
4095:
4078:{\displaystyle g(x)=(x-K)^{+}}
4066:
4053:
4047:
4041:
4010:
3997:
3945:
3939:
3652:
3639:
3619:
3606:
3503:
3490:
3423:
3401:
3271:
3258:
3213:
3199:
3167:
3153:
2985:
2972:
2842:integro-variational inequality
2754:
2720:
2714:
2705:
2699:
2608:
2574:
2568:
2559:
2553:
2527:
2514:
2496:{\displaystyle {\mathcal {A}}}
2398:{\displaystyle {\mathcal {S}}}
2337:
2331:
2322:
2316:
2268:
2249:
2233:
2223:
2207:
2201:
2190:
2153:
2147:
2093:
2080:
2056:
2043:
1992:
1986:
1945:
1939:
1795:
1782:
1744:
1673:
1623:
1558:
1465:
1447:
1441:
1432:
1410:
1369:
1356:
1341:
1328:
1145:
1132:
1095:
1082:
1058:
1045:
997:
991:
862:
832:
814:
795:
763:
749:
484:
461:
443:
424:
389:
375:
300:If you stop observing at step
255:
223:
166:
152:
16:Class of mathematical problems
1:
5206:(For French translation, see
4031:where the payoff function is
2837:is an optimal stopping time.
2726:{\displaystyle \phi (y)=V(y)}
932:. Given continuous functions
5337:10.1080/07474946.2010.487423
5457:, Vol. 28, 1384–1391,(2000)
2374:{\displaystyle \phi \geq M}
63:(related to the pricing of
5547:
5493:10.1257/002205105775362014
3677:
3476:
1567:{\displaystyle {\bar {N}}}
416:filtered probability space
320:, you will receive reward
25:
18:
5466:, Issue 62, 14–20, (2006)
5447:, Vol. 4.,282–296, (1989)
5283:10.1007/978-3-540-69826-5
5243:10.1007/978-3-7643-7390-0
5083:Optional stopping theorem
1594:-dimensional compensated
679:{\displaystyle V_{t}^{T}}
544:{\displaystyle \tau ^{*}}
21:Optional stopping theorem
3314:for your house, and pay
362:Consider a gain process
19:Not to be confused with
5126:; Siegmund, D. (1971).
5078:Markov decision process
3776:{\displaystyle \sigma }
3756:{\displaystyle \delta }
3741:risk-free interest rate
3658:{\displaystyle (y_{i})}
3625:{\displaystyle (R_{i})}
3509:{\displaystyle (X_{i})}
2533:{\displaystyle (Y_{t})}
2505:infinitesimal generator
1801:{\displaystyle (Y_{t})}
1232:A jump diffusion result
723:{\displaystyle \infty }
5041:
4924:
4846:
4692:
4593:
4527:
4378:
4358:
4305:
4133:
4085:for a call option and
4079:
4022:
3914:
3797:
3777:
3757:
3733:
3659:
3626:
3593:
3562:
3516:is a finite sequence)
3510:
3463:
3430:
3375:
3348:
3328:
3308:
3278:
3232:
3186:
3137:
3122:
3060:
3019:Bernoulli distribution
2992:
2941:
2905:
2831:
2764:
2727:
2680:
2660:
2618:
2581:
2534:
2497:
2473:
2440:
2399:
2375:
2347:
2275:
2165:
2114:
1917:
1841:
1802:
1769:
1698:
1642:
1596:Poisson random measure
1588:
1568:
1535:
1515:
1492:
1290:
1257:
1222:free-boundary problems
1203:
1160:
972:
952:
926:
898:
869:
782:
724:
704:
680:
645:
545:
511:
491:
408:
341:
314:
291:
262:
185:
136:
5455:Annals of Probability
5210:in the July issue of
5042:
4925:
4847:
4693:
4594:
4528:
4379:
4359:
4306:
4134:
4080:
4023:
3915:
3798:
3778:
3758:
3734:
3660:
3627:
3594:
3592:{\displaystyle y_{i}}
3563:
3511:
3464:
3462:{\displaystyle X_{i}}
3431:
3376:
3374:{\displaystyle y_{n}}
3349:
3329:
3309:
3307:{\displaystyle X_{n}}
3279:
3233:
3187:
3138:
3102:
3061:
2993:
2942:
2906:
2832:
2765:
2728:
2681:
2661:
2619:
2582:
2535:
2498:
2474:
2441:
2400:
2376:
2348:
2276:
2166:
2115:
1918:
1842:
1803:
1770:
1699:
1643:
1589:
1569:
1536:
1516:
1493:
1291:
1258:
1256:{\displaystyle Y_{t}}
1204:
1161:
973:
953:
927:
899:
870:
783:
725:
705:
681:
646:
546:
512:
492:
409:
349:You want to choose a
342:
340:{\displaystyle y_{i}}
315:
292:
263:
186:
137:
5519:Mathematical finance
5374:10.1287/opre.8.3.362
4934:
4856:
4704:
4603:
4537:
4392:
4368:
4318:
4146:
4089:
4035:
3933:
3810:
3787:
3767:
3747:
3723:
3636:
3603:
3576:
3526:
3487:
3446:
3385:
3358:
3338:
3318:
3291:
3250:
3196:
3150:
3076:
3027:
2964:
2915:
2851:
2774:
2737:
2693:
2670:
2631:
2591:
2547:
2511:
2483:
2450:
2411:
2385:
2359:
2285:
2178:
2130:
1933:
1854:
1812:
1779:
1708:
1652:
1602:
1578:
1549:
1525:
1505:
1306:
1271:
1240:
1193:
985:
962:
936:
916:
879:
792:
740:
714:
694:
658:
558:
528:
501:
421:
366:
358:Continuous time case
324:
304:
281:
197:
149:
100:
61:mathematical finance
5529:Dynamic programming
5445:Statistical Science
5395:Karatzas, Ioannis;
5361:Operations Research
5324:Sequential Analysis
5315:Ferguson, Thomas S.
5184:10.1511/2009.77.126
5146:Ferguson, Thomas S.
5098:Sequential analysis
3710:, the holder of an
3146:then the sequences
2076:
1211:dynamic programming
1078:
951:{\displaystyle M,L}
906:probability measure
675:
575:
77:dynamic programming
28:Dynkin's card trick
5524:Sequential methods
5437:Thomas S. Ferguson
5171:American Scientist
5093:Stochastic control
5088:Prophet inequality
5037:
4920:
4842:
4837:
4688:
4589:
4523:
4518:
4374:
4354:
4301:
4129:
4075:
4018:
3960:
3910:
3793:
3773:
3753:
3729:
3702:In the trading of
3655:
3622:
3589:
3558:
3506:
3459:
3426:
3371:
3344:
3324:
3304:
3274:
3228:
3182:
3133:
3056:
2988:
2937:
2901:
2827:
2760:
2723:
2676:
2656:
2614:
2577:
2530:
2493:
2469:
2436:
2395:
2371:
2343:
2271:
2161:
2110:
2062:
2022:
1975:
1913:
1837:
1798:
1765:
1694:
1638:
1584:
1564:
1531:
1511:
1488:
1286:
1253:
1199:
1156:
1128:
1064:
1024:
968:
948:
922:
910:stochastic process
894:
865:
778:
720:
700:
676:
661:
641:
625:
561:
541:
507:
487:
404:
337:
310:
287:
258:
181:
132:
88:Discrete time case
5418:978-0-387-94839-3
5397:Shreve, Steven E.
5319:Klass, Michael J.
5292:978-3-540-69825-8
5252:978-3-7643-2419-3
5166:Hill, Theodore P.
5022:
4999:
4901:
4868:
4808:
4570:
4388:(Perpetual call)
4377:{\displaystyle b}
4165:
3951:
3879:
3796:{\displaystyle S}
3732:{\displaystyle r}
3708:financial markets
3479:Secretary problem
3473:Secretary problem
3347:{\displaystyle n}
3327:{\displaystyle k}
3100:
3047:
3033:
2757:
2679:{\displaystyle D}
2611:
2204:
2150:
1998:
1951:
1587:{\displaystyle l}
1561:
1534:{\displaystyle m}
1514:{\displaystyle B}
1444:
1202:{\displaystyle T}
1107:
1003:
971:{\displaystyle K}
925:{\displaystyle x}
703:{\displaystyle T}
604:
510:{\displaystyle G}
313:{\displaystyle i}
290:{\displaystyle i}
69:secretary problem
5536:
5504:
5476:
5423:
5422:
5392:
5386:
5385:
5355:
5349:
5348:
5311:
5305:
5304:
5263:
5257:
5256:
5231:Shiryaev, Albert
5226:
5217:
5203:
5162:
5156:
5155:
5142:
5136:
5135:
5132:Houghton Mifflin
5119:
5059:for a discrete,
5046:
5044:
5043:
5038:
5024:
5023:
5015:
5009:
5001:
5000:
4992:
4976:
4962:
4929:
4927:
4926:
4921:
4916:
4902:
4891:
4890:
4881:
4870:
4869:
4861:
4851:
4849:
4848:
4843:
4841:
4840:
4811:
4810:
4809:
4801:
4791:
4700:(Perpetual put)
4697:
4695:
4694:
4689:
4669:
4645:
4631:
4598:
4596:
4595:
4590:
4585:
4571:
4560:
4559:
4550:
4532:
4530:
4529:
4524:
4522:
4521:
4456:
4455:
4443:
4383:
4381:
4380:
4375:
4363:
4361:
4360:
4355:
4310:
4308:
4307:
4302:
4294:
4290:
4232:
4194:
4186:
4185:
4176:
4175:
4166:
4158:
4138:
4136:
4135:
4130:
4128:
4127:
4084:
4082:
4081:
4076:
4074:
4073:
4027:
4025:
4024:
4019:
4017:
4013:
4009:
4008:
3993:
3992:
3972:
3971:
3966:
3959:
3919:
3917:
3916:
3911:
3909:
3905:
3904:
3903:
3885:
3881:
3880:
3875:
3874:
3865:
3835:
3834:
3822:
3821:
3802:
3800:
3799:
3794:
3782:
3780:
3779:
3774:
3762:
3760:
3759:
3754:
3738:
3736:
3735:
3730:
3664:
3662:
3661:
3656:
3651:
3650:
3631:
3629:
3628:
3623:
3618:
3617:
3598:
3596:
3595:
3590:
3588:
3587:
3567:
3565:
3564:
3559:
3557:
3556:
3538:
3537:
3515:
3513:
3512:
3507:
3502:
3501:
3468:
3466:
3465:
3460:
3458:
3457:
3435:
3433:
3432:
3427:
3413:
3412:
3397:
3396:
3380:
3378:
3377:
3372:
3370:
3369:
3354:, you will earn
3353:
3351:
3350:
3345:
3333:
3331:
3330:
3325:
3313:
3311:
3310:
3305:
3303:
3302:
3283:
3281:
3280:
3275:
3270:
3269:
3257:
3237:
3235:
3234:
3229:
3227:
3226:
3211:
3210:
3191:
3189:
3188:
3183:
3181:
3180:
3165:
3164:
3142:
3140:
3139:
3134:
3132:
3131:
3121:
3116:
3101:
3093:
3088:
3087:
3065:
3063:
3062:
3057:
3052:
3048:
3040:
3034:
3031:
2997:
2995:
2994:
2989:
2984:
2983:
2971:
2946:
2944:
2943:
2938:
2924:
2923:
2910:
2908:
2907:
2902:
2894:
2890:
2868:
2867:
2836:
2834:
2833:
2828:
2817:
2816:
2786:
2785:
2769:
2767:
2766:
2761:
2759:
2758:
2753:
2748:
2732:
2730:
2729:
2724:
2685:
2683:
2682:
2677:
2665:
2663:
2662:
2657:
2640:
2639:
2623:
2621:
2620:
2615:
2613:
2612:
2607:
2602:
2586:
2584:
2583:
2578:
2539:
2537:
2536:
2531:
2526:
2525:
2502:
2500:
2499:
2494:
2492:
2491:
2478:
2476:
2475:
2470:
2459:
2458:
2445:
2443:
2442:
2437:
2420:
2419:
2404:
2402:
2401:
2396:
2394:
2393:
2380:
2378:
2377:
2372:
2352:
2350:
2349:
2344:
2309:
2308:
2280:
2278:
2277:
2272:
2258:
2257:
2248:
2247:
2232:
2231:
2222:
2221:
2206:
2205:
2200:
2195:
2170:
2168:
2167:
2162:
2160:
2152:
2151:
2146:
2141:
2119:
2117:
2116:
2111:
2106:
2102:
2092:
2091:
2075:
2070:
2055:
2054:
2034:
2033:
2028:
2021:
2020:
2019:
2018:
1985:
1984:
1974:
1973:
1972:
1971:
1922:
1920:
1919:
1914:
1909:
1908:
1899:
1898:
1868:
1867:
1866:
1846:
1844:
1843:
1838:
1836:
1835:
1830:
1821:
1820:
1807:
1805:
1804:
1799:
1794:
1793:
1774:
1772:
1771:
1766:
1764:
1763:
1752:
1743:
1742:
1737:
1728:
1727:
1722:
1703:
1701:
1700:
1695:
1693:
1692:
1681:
1672:
1671:
1666:
1647:
1645:
1644:
1639:
1637:
1636:
1631:
1622:
1621:
1616:
1593:
1591:
1590:
1585:
1573:
1571:
1570:
1565:
1563:
1562:
1554:
1540:
1538:
1537:
1532:
1520:
1518:
1517:
1512:
1497:
1495:
1494:
1489:
1481:
1480:
1446:
1445:
1437:
1425:
1424:
1406:
1405:
1404:
1403:
1398:
1384:
1383:
1368:
1367:
1340:
1339:
1321:
1320:
1295:
1293:
1292:
1287:
1285:
1284:
1279:
1262:
1260:
1259:
1254:
1252:
1251:
1218:Markov processes
1208:
1206:
1205:
1200:
1173:Solution methods
1165:
1163:
1162:
1157:
1152:
1148:
1144:
1143:
1127:
1094:
1093:
1077:
1072:
1057:
1056:
1036:
1035:
1030:
1023:
977:
975:
974:
969:
957:
955:
954:
949:
931:
929:
928:
923:
903:
901:
900:
895:
893:
892:
887:
874:
872:
871:
866:
861:
860:
855:
846:
845:
830:
829:
824:
823:
810:
809:
787:
785:
784:
779:
777:
776:
761:
760:
729:
727:
726:
721:
709:
707:
706:
701:
685:
683:
682:
677:
674:
669:
650:
648:
647:
642:
640:
639:
630:
624:
600:
599:
598:
597:
583:
574:
569:
550:
548:
547:
542:
540:
539:
516:
514:
513:
508:
497:and assume that
496:
494:
493:
488:
483:
475:
474:
459:
458:
453:
452:
439:
438:
413:
411:
410:
405:
403:
402:
387:
386:
346:
344:
343:
338:
336:
335:
319:
317:
316:
311:
296:
294:
293:
288:
267:
265:
264:
259:
254:
253:
235:
234:
222:
221:
209:
208:
190:
188:
187:
182:
180:
179:
164:
163:
141:
139:
138:
133:
125:
124:
112:
111:
73:Bellman equation
65:American options
41:optimal stopping
39:, the theory of
5546:
5545:
5539:
5538:
5537:
5535:
5534:
5533:
5509:
5508:
5507:
5474:
5469:
5451:F. Thomas Bruss
5432:
5427:
5426:
5419:
5394:
5393:
5389:
5357:
5356:
5352:
5313:
5312:
5308:
5293:
5265:
5264:
5260:
5253:
5229:Peskir, Goran;
5228:
5227:
5220:
5212:Pour la Science
5164:
5163:
5159:
5144:
5143:
5139:
5121:
5120:
5116:
5111:
5106:
5073:Halting problem
5069:
4932:
4931:
4882:
4854:
4853:
4836:
4835:
4812:
4795:
4763:
4762:
4739:
4723:
4702:
4701:
4601:
4600:
4551:
4535:
4534:
4517:
4516:
4493:
4481:
4480:
4457:
4447:
4411:
4390:
4389:
4366:
4365:
4316:
4315:
4225:
4187:
4177:
4167:
4156:
4152:
4144:
4143:
4119:
4087:
4086:
4065:
4033:
4032:
4000:
3978:
3977:
3973:
3961:
3931:
3930:
3895:
3866:
3851:
3847:
3846:
3842:
3826:
3813:
3808:
3807:
3785:
3784:
3765:
3764:
3745:
3744:
3721:
3720:
3719:set-up and let
3712:American option
3700:
3691:
3689:Parking problem
3682:
3676:
3642:
3634:
3633:
3609:
3601:
3600:
3579:
3574:
3573:
3548:
3529:
3524:
3523:
3493:
3485:
3484:
3483:(Example where
3481:
3475:
3449:
3444:
3443:
3404:
3388:
3383:
3382:
3361:
3356:
3355:
3336:
3335:
3316:
3315:
3294:
3289:
3288:
3261:
3248:
3247:
3246:(Example where
3244:
3212:
3202:
3194:
3193:
3166:
3156:
3148:
3147:
3123:
3079:
3074:
3073:
3035:
3025:
3024:
3012:
2975:
2962:
2961:
2960:(Example where
2958:
2953:
2913:
2912:
2861:
2857:
2849:
2848:
2808:
2777:
2772:
2771:
2735:
2734:
2691:
2690:
2668:
2667:
2629:
2628:
2624:. Moreover, if
2589:
2588:
2545:
2544:
2517:
2509:
2508:
2481:
2480:
2448:
2447:
2409:
2408:
2383:
2382:
2357:
2356:
2283:
2282:
2239:
2213:
2176:
2175:
2128:
2127:
2083:
2046:
2039:
2035:
2023:
2009:
1976:
1962:
1931:
1930:
1890:
1857:
1852:
1851:
1825:
1810:
1809:
1785:
1777:
1776:
1747:
1732:
1717:
1706:
1705:
1676:
1661:
1650:
1649:
1626:
1611:
1600:
1599:
1576:
1575:
1547:
1546:
1543:Brownian motion
1523:
1522:
1503:
1502:
1472:
1413:
1393:
1388:
1375:
1359:
1331:
1312:
1304:
1303:
1274:
1269:
1268:
1243:
1238:
1237:
1234:
1226:Stefan problems
1191:
1190:
1175:
1135:
1085:
1048:
1041:
1037:
1025:
983:
982:
960:
959:
934:
933:
914:
913:
882:
877:
876:
850:
831:
817:
790:
789:
762:
752:
738:
737:
712:
711:
710:can take value
692:
691:
656:
655:
631:
589:
584:
556:
555:
531:
526:
525:
499:
498:
460:
446:
419:
418:
388:
378:
364:
363:
360:
327:
322:
321:
302:
301:
279:
278:
245:
226:
213:
200:
195:
194:
165:
155:
147:
146:
116:
103:
98:
97:
90:
85:
31:
24:
17:
12:
11:
5:
5544:
5543:
5540:
5532:
5531:
5526:
5521:
5511:
5510:
5506:
5505:
5467:
5458:
5448:
5433:
5431:
5428:
5425:
5424:
5417:
5409:10.1007/b98840
5387:
5368:(3): 362–380.
5350:
5331:(3): 236–244.
5306:
5291:
5258:
5251:
5218:
5216:
5215:
5178:(2): 126–133.
5157:
5137:
5113:
5112:
5110:
5107:
5105:
5102:
5101:
5100:
5095:
5090:
5085:
5080:
5075:
5068:
5065:
5048:
5047:
5036:
5033:
5030:
5027:
5021:
5018:
5012:
5008:
5004:
4998:
4995:
4989:
4986:
4982:
4979:
4975:
4971:
4968:
4965:
4961:
4957:
4954:
4951:
4948:
4945:
4942:
4939:
4919:
4915:
4911:
4908:
4905:
4900:
4897:
4894:
4889:
4885:
4879:
4876:
4873:
4867:
4864:
4839:
4834:
4831:
4828:
4825:
4822:
4819:
4816:
4813:
4807:
4804:
4798:
4794:
4790:
4786:
4783:
4780:
4777:
4774:
4771:
4768:
4765:
4764:
4761:
4758:
4755:
4752:
4749:
4746:
4743:
4740:
4738:
4735:
4732:
4729:
4728:
4726:
4721:
4718:
4715:
4712:
4709:
4698:
4687:
4684:
4681:
4678:
4675:
4672:
4668:
4664:
4661:
4658:
4655:
4651:
4648:
4644:
4640:
4637:
4634:
4630:
4626:
4623:
4620:
4617:
4614:
4611:
4608:
4588:
4584:
4580:
4577:
4574:
4569:
4566:
4563:
4558:
4554:
4548:
4545:
4542:
4520:
4515:
4512:
4509:
4506:
4503:
4500:
4497:
4494:
4492:
4489:
4486:
4483:
4482:
4479:
4476:
4473:
4470:
4467:
4464:
4461:
4458:
4454:
4450:
4446:
4442:
4438:
4435:
4432:
4429:
4426:
4423:
4420:
4417:
4416:
4414:
4409:
4406:
4403:
4400:
4397:
4373:
4353:
4350:
4347:
4344:
4341:
4338:
4335:
4332:
4329:
4326:
4323:
4312:
4311:
4300:
4297:
4293:
4289:
4286:
4283:
4280:
4277:
4274:
4271:
4268:
4265:
4262:
4259:
4256:
4253:
4250:
4247:
4244:
4241:
4238:
4235:
4231:
4228:
4224:
4221:
4218:
4215:
4212:
4209:
4206:
4203:
4200:
4197:
4193:
4190:
4184:
4180:
4174:
4170:
4164:
4161:
4155:
4151:
4126:
4122:
4118:
4115:
4112:
4109:
4106:
4103:
4100:
4097:
4094:
4072:
4068:
4064:
4061:
4058:
4055:
4052:
4049:
4046:
4043:
4040:
4029:
4028:
4016:
4012:
4007:
4003:
3999:
3996:
3991:
3988:
3985:
3981:
3976:
3970:
3965:
3958:
3954:
3950:
3947:
3944:
3941:
3938:
3921:
3920:
3908:
3902:
3898:
3894:
3891:
3888:
3884:
3878:
3873:
3869:
3863:
3860:
3857:
3854:
3850:
3845:
3841:
3838:
3833:
3829:
3825:
3820:
3816:
3792:
3772:
3752:
3728:
3699:
3698:Option trading
3696:
3690:
3687:
3678:Main article:
3675:
3672:
3668:odds algorithm
3654:
3649:
3645:
3641:
3621:
3616:
3612:
3608:
3586:
3582:
3555:
3551:
3547:
3544:
3541:
3536:
3532:
3505:
3500:
3496:
3492:
3477:Main article:
3474:
3471:
3456:
3452:
3425:
3422:
3419:
3416:
3411:
3407:
3403:
3400:
3395:
3391:
3368:
3364:
3343:
3323:
3301:
3297:
3273:
3268:
3264:
3260:
3256:
3243:
3240:
3225:
3222:
3219:
3215:
3209:
3205:
3201:
3179:
3176:
3173:
3169:
3163:
3159:
3155:
3144:
3143:
3130:
3126:
3120:
3115:
3112:
3109:
3105:
3099:
3096:
3091:
3086:
3082:
3067:
3066:
3055:
3051:
3046:
3043:
3038:
3008:
2987:
2982:
2978:
2974:
2970:
2957:
2954:
2952:
2949:
2948:
2947:
2936:
2933:
2930:
2927:
2922:
2900:
2897:
2893:
2889:
2886:
2883:
2880:
2877:
2874:
2871:
2866:
2860:
2856:
2826:
2823:
2820:
2815:
2811:
2807:
2804:
2801:
2798:
2795:
2792:
2789:
2784:
2780:
2756:
2752:
2745:
2742:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2701:
2698:
2687:
2686:
2675:
2655:
2652:
2649:
2646:
2643:
2638:
2610:
2606:
2599:
2596:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2555:
2552:
2541:
2540:
2529:
2524:
2520:
2516:
2490:
2468:
2465:
2462:
2457:
2435:
2432:
2429:
2426:
2423:
2418:
2406:
2392:
2370:
2367:
2364:
2354:
2342:
2339:
2336:
2333:
2330:
2327:
2324:
2321:
2318:
2315:
2312:
2307:
2302:
2299:
2296:
2293:
2290:
2270:
2267:
2264:
2261:
2256:
2251:
2246:
2242:
2238:
2235:
2230:
2225:
2220:
2216:
2212:
2209:
2203:
2199:
2192:
2189:
2186:
2183:
2159:
2155:
2149:
2145:
2138:
2135:
2126:If a function
2121:
2120:
2109:
2105:
2101:
2098:
2095:
2090:
2086:
2082:
2079:
2074:
2069:
2065:
2061:
2058:
2053:
2049:
2045:
2042:
2038:
2032:
2027:
2017:
2012:
2008:
2005:
2001:
1997:
1994:
1991:
1988:
1983:
1979:
1970:
1965:
1961:
1958:
1954:
1950:
1947:
1944:
1941:
1938:
1924:
1923:
1912:
1907:
1902:
1897:
1893:
1889:
1886:
1883:
1880:
1877:
1874:
1871:
1865:
1860:
1834:
1829:
1824:
1819:
1797:
1792:
1788:
1784:
1762:
1759:
1756:
1751:
1746:
1741:
1736:
1731:
1726:
1721:
1716:
1713:
1691:
1688:
1685:
1680:
1675:
1670:
1665:
1660:
1657:
1635:
1630:
1625:
1620:
1615:
1610:
1607:
1583:
1560:
1557:
1530:
1510:
1499:
1498:
1487:
1484:
1479:
1475:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1443:
1440:
1434:
1431:
1428:
1423:
1420:
1416:
1412:
1409:
1402:
1397:
1391:
1387:
1382:
1378:
1374:
1371:
1366:
1362:
1358:
1355:
1352:
1349:
1346:
1343:
1338:
1334:
1330:
1327:
1324:
1319:
1315:
1311:
1283:
1278:
1250:
1246:
1233:
1230:
1198:
1187:Snell envelope
1174:
1171:
1167:
1166:
1155:
1151:
1147:
1142:
1138:
1134:
1131:
1126:
1123:
1120:
1117:
1114:
1110:
1106:
1103:
1100:
1097:
1092:
1088:
1084:
1081:
1076:
1071:
1067:
1063:
1060:
1055:
1051:
1047:
1044:
1040:
1034:
1029:
1022:
1019:
1016:
1013:
1010:
1006:
1002:
999:
996:
993:
990:
967:
947:
944:
941:
921:
891:
886:
864:
859:
854:
849:
844:
841:
838:
834:
828:
822:
816:
813:
808:
803:
800:
797:
775:
772:
769:
765:
759:
755:
751:
748:
745:
735:Markov process
719:
699:
688:value function
686:is called the
673:
668:
664:
652:
651:
638:
634:
629:
623:
620:
617:
614:
611:
607:
603:
596:
592:
587:
582:
578:
573:
568:
564:
538:
534:
506:
486:
482:
478:
473:
470:
467:
463:
457:
451:
445:
442:
437:
432:
429:
426:
401:
398:
395:
391:
385:
381:
377:
374:
371:
359:
356:
355:
354:
347:
334:
330:
309:
298:
286:
271:
270:
269:
268:
257:
252:
248:
244:
241:
238:
233:
229:
225:
220:
216:
212:
207:
203:
178:
175:
172:
168:
162:
158:
154:
143:
131:
128:
123:
119:
115:
110:
106:
89:
86:
84:
81:
45:early stopping
15:
13:
10:
9:
6:
4:
3:
2:
5542:
5541:
5530:
5527:
5525:
5522:
5520:
5517:
5516:
5514:
5502:
5498:
5494:
5490:
5487:(4): 959–88.
5486:
5482:
5481:
5473:
5468:
5465:
5464:
5459:
5456:
5452:
5449:
5446:
5442:
5438:
5435:
5434:
5429:
5420:
5414:
5410:
5406:
5402:
5398:
5391:
5388:
5383:
5379:
5375:
5371:
5367:
5363:
5362:
5354:
5351:
5346:
5342:
5338:
5334:
5330:
5326:
5325:
5320:
5316:
5310:
5307:
5302:
5298:
5294:
5288:
5284:
5280:
5276:
5272:
5268:
5262:
5259:
5254:
5248:
5244:
5240:
5236:
5232:
5225:
5223:
5219:
5213:
5209:
5205:
5204:
5201:
5197:
5193:
5189:
5185:
5181:
5177:
5173:
5172:
5167:
5161:
5158:
5153:
5152:
5147:
5141:
5138:
5133:
5129:
5125:
5118:
5115:
5108:
5103:
5099:
5096:
5094:
5091:
5089:
5086:
5084:
5081:
5079:
5076:
5074:
5071:
5070:
5066:
5064:
5062:
5058:
5054:
5034:
5028:
5025:
5016:
5006:
5002:
4993:
4987:
4984:
4980:
4977:
4973:
4969:
4966:
4963:
4959:
4952:
4949:
4946:
4940:
4937:
4917:
4913:
4906:
4903:
4898:
4895:
4892:
4887:
4883:
4874:
4871:
4862:
4826:
4823:
4817:
4814:
4802:
4792:
4788:
4784:
4775:
4772:
4769:
4756:
4753:
4750:
4744:
4741:
4736:
4733:
4730:
4724:
4719:
4713:
4707:
4699:
4685:
4679:
4676:
4673:
4666:
4662:
4659:
4656:
4653:
4649:
4646:
4642:
4638:
4635:
4632:
4628:
4621:
4618:
4615:
4609:
4606:
4586:
4582:
4575:
4572:
4567:
4564:
4561:
4556:
4552:
4543:
4540:
4507:
4504:
4498:
4495:
4490:
4487:
4484:
4474:
4471:
4468:
4462:
4459:
4452:
4444:
4440:
4436:
4427:
4424:
4421:
4412:
4407:
4401:
4395:
4387:
4386:
4385:
4371:
4348:
4333:
4330:
4324:
4321:
4298:
4295:
4291:
4284:
4278:
4275:
4269:
4263:
4260:
4254:
4248:
4245:
4242:
4236:
4229:
4226:
4222:
4216:
4213:
4210:
4204:
4198:
4191:
4188:
4182:
4178:
4172:
4168:
4162:
4159:
4153:
4142:
4141:
4140:
4124:
4116:
4113:
4110:
4104:
4098:
4092:
4070:
4062:
4059:
4056:
4050:
4044:
4038:
4014:
4005:
4001:
3994:
3989:
3986:
3983:
3979:
3974:
3968:
3956:
3948:
3942:
3936:
3929:
3928:
3927:
3924:
3906:
3900:
3896:
3892:
3889:
3886:
3882:
3876:
3871:
3867:
3861:
3858:
3855:
3852:
3848:
3843:
3839:
3836:
3831:
3827:
3823:
3818:
3814:
3806:
3805:
3804:
3790:
3770:
3750:
3742:
3726:
3718:
3717:Black–Scholes
3713:
3709:
3705:
3697:
3695:
3688:
3686:
3681:
3680:Search theory
3674:Search theory
3673:
3671:
3669:
3647:
3643:
3614:
3610:
3584:
3580:
3571:
3553:
3549:
3545:
3542:
3539:
3534:
3530:
3520:
3517:
3498:
3494:
3480:
3472:
3470:
3454:
3450:
3440:
3437:
3420:
3417:
3414:
3409:
3405:
3398:
3393:
3389:
3366:
3362:
3341:
3321:
3299:
3295:
3285:
3266:
3262:
3242:House selling
3241:
3239:
3223:
3220:
3217:
3207:
3203:
3177:
3174:
3171:
3161:
3157:
3128:
3124:
3118:
3113:
3110:
3107:
3103:
3097:
3094:
3089:
3084:
3080:
3072:
3071:
3070:
3053:
3049:
3044:
3041:
3036:
3023:
3022:
3021:
3020:
3016:
3011:
3007:
3002:
2999:
2980:
2976:
2955:
2950:
2934:
2931:
2898:
2895:
2891:
2887:
2884:
2881:
2878:
2875:
2872:
2869:
2858:
2847:
2846:
2845:
2843:
2838:
2821:
2818:
2813:
2809:
2805:
2802:
2799:
2796:
2787:
2782:
2778:
2743:
2740:
2717:
2711:
2708:
2702:
2696:
2673:
2653:
2650:
2647:
2644:
2641:
2627:
2626:
2625:
2597:
2594:
2571:
2565:
2562:
2556:
2550:
2522:
2518:
2506:
2466:
2433:
2430:
2427:
2424:
2421:
2407:
2368:
2365:
2362:
2355:
2334:
2328:
2325:
2319:
2313:
2310:
2300:
2297:
2291:
2288:
2265:
2244:
2240:
2236:
2218:
2214:
2210:
2187:
2184:
2181:
2174:
2173:
2172:
2136:
2133:
2124:
2107:
2103:
2099:
2096:
2088:
2084:
2077:
2072:
2067:
2063:
2059:
2051:
2047:
2040:
2036:
2030:
2010:
2006:
2003:
1995:
1989:
1981:
1977:
1963:
1959:
1956:
1948:
1942:
1936:
1929:
1928:
1927:
1900:
1895:
1891:
1887:
1884:
1881:
1878:
1869:
1858:
1850:
1849:
1848:
1832:
1822:
1790:
1786:
1760:
1757:
1754:
1739:
1729:
1724:
1714:
1711:
1689:
1686:
1683:
1668:
1658:
1655:
1633:
1618:
1608:
1605:
1597:
1581:
1555:
1544:
1541:-dimensional
1528:
1508:
1485:
1482:
1477:
1473:
1468:
1462:
1459:
1456:
1453:
1450:
1438:
1429:
1426:
1421:
1418:
1414:
1407:
1400:
1389:
1385:
1380:
1376:
1372:
1364:
1360:
1353:
1350:
1347:
1344:
1336:
1332:
1325:
1322:
1317:
1313:
1309:
1302:
1301:
1300:
1299:
1296:given by the
1281:
1267:diffusion in
1266:
1248:
1244:
1231:
1229:
1227:
1223:
1219:
1214:
1212:
1196:
1188:
1184:
1180:
1172:
1170:
1153:
1149:
1140:
1136:
1129:
1124:
1121:
1118:
1115:
1112:
1104:
1101:
1098:
1090:
1086:
1079:
1074:
1069:
1065:
1061:
1053:
1049:
1042:
1038:
1032:
1020:
1017:
1014:
1011:
1008:
1000:
994:
988:
981:
980:
979:
965:
945:
942:
939:
919:
911:
907:
889:
857:
847:
842:
839:
836:
826:
811:
801:
773:
770:
767:
757:
753:
746:
743:
736:
731:
697:
689:
671:
666:
662:
636:
632:
621:
618:
615:
612:
609:
601:
594:
590:
585:
576:
571:
566:
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554:
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536:
532:
524:
523:stopping time
520:
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465:
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440:
430:
417:
414:defined on a
399:
396:
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352:
351:stopping rule
348:
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275:
274:
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246:
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5267:Øksendal, B.
5261:
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5211:
5175:
5169:
5160:
5150:
5140:
5127:
5122:Chow, Y.S.;
5117:
5049:
4313:
4030:
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3701:
3692:
3683:
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3245:
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3009:
3005:
3003:
3000:
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2956:Coin tossing
2839:
2688:
2542:
2125:
2122:
1925:
1808:exists. Let
1500:
1235:
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1168:
904:denotes the
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272:
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5208:cover story
5124:Robbins, H.
2998:converges)
37:mathematics
5513:Categories
5130:. Boston:
5104:References
5061:tree based
2171:satisfies
1183:martingale
912:starts at
908:where the
83:Definition
53:statistics
5382:0030-364X
5345:0747-4946
5301:123531718
5271:Sulem, A.
5200:124798270
5192:1545-2786
5109:Citations
5026:−
5020:~
5017:γ
4997:~
4994:γ
4970:σ
4967:−
4964:σ
4953:δ
4950:−
4938:ν
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4884:ν
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4866:~
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4806:~
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2007:≤
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1982:τ
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1960:≤
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613:≤
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469:≥
428:Ω
397:≥
240:…
174:≥
130:…
57:economics
5399:(1998).
5273:(2007).
5233:(2006).
5214:(2009).)
5148:(2007).
5067:See also
4314:for all
4230:′
4192:″
3381:, where
2951:Examples
2733:for all
2587:for all
2479:, where
49:maximise
5501:4129380
5430:Sources
5154:. UCLA.
3739:be the
3704:options
3069:and if
2503:is the
1521:is an
690:. Here
519:adapted
5499:
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4852:where
4533:where
4364:where
3192:, and
1704:, and
1574:is an
1501:where
958:, and
875:where
654:where
59:, and
5497:JSTOR
5475:(PDF)
5297:S2CID
5196:S2CID
5057:Fugit
3013:(for
2689:Then
2543:then
2405:, and
1263:be a
5413:ISBN
5378:ISSN
5341:ISSN
5287:ISBN
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