3704:
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
1175:
3705:
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.
3695:
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.
2129:
1507:
4320:
2290:
4861:
4542:
3929:
3725:
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
5056:
660:
1784:
884:
3676:
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
506:
4707:
1932:
995:
4939:
1713:
2920:
5061:
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
4608:
2180:
1943:
2362:
3075:
1227:
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
3152:
4037:
2846:
1856:
1657:
1316:
2956:
2488:
2188:
277:
4373:
4714:
3012:
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.
2455:
2779:
2633:
2675:
4402:
4156:
3820:
3577:
3293:
3007:
797:
423:
151:
3247:
3201:
2596:
200:
3445:
1305:
913:
4148:
4094:
2512:
2414:
4944:
2742:
3530:
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.
2390:
1583:
695:
568:
560:
3792:
3772:
3674:
3641:
3525:
2549:
1817:
739:
5218:
3608:
3478:
3390:
3323:
1272:
356:
967:
1718:
4393:
3812:
3748:
3363:
3343:
2695:
1603:
1550:
1530:
1218:
987:
941:
719:
526:
329:
306:
802:
5334:
1188:
There are generally two approaches to solving optimal stopping problems. When the underlying process (or the gain process) is described by its unconditional
431:
5329:
4613:
1170:{\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).}
1864:
5473:
2515:
4866:
1662:
2861:
1231:
can often be utilized and this approach is referred to as the Markov method. The solution is usually obtained by solving the associated
2124:{\displaystyle V(y)=\sup _{\tau \leq \tau _{\mathcal {S}}}J^{\tau }(y)=\sup _{\tau \leq \tau _{\mathcal {S}}}\mathbb {E} _{y}\left.}
4547:
2140:
2295:
3037:
1502:{\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}
3086:
5427:
5301:
5261:
3943:
2784:
1822:
1612:
2925:
2285:{\displaystyle \phi \in C({\bar {\mathcal {S}}})\cap C^{1}({\mathcal {S}})\cap C^{2}({\mathcal {S}}\setminus \partial D)}
2460:
207:
4856:{\displaystyle V(x)={\begin{cases}K-x&x\in (0,c]\\(K-c)(x/c)^{\tilde {\gamma }}&x\in (c,\infty )\end{cases}}}
4328:
1308:
5063:
2421:
1193:
1189:
2747:
2601:
5490:
5529:
2641:
4537:{\displaystyle V(x)={\begin{cases}(b-K)(x/b)^{\gamma }&x\in (0,b)\\x-K&x\in [b,\infty )\end{cases}}}
4315:{\displaystyle \max \left\{{\frac {1}{2}}\sigma ^{2}x^{2}V''(x)+(r-\delta )xV'(x)-rV(x),g(x)-V(x)\right\}=0}
3924:{\displaystyle S_{t}=S_{0}\exp \left\{\left(r-\delta -{\frac {\sigma ^{2}}{2}}\right)t+\sigma B_{t}\right\}}
3480:) is the sequence of offers for your house, and the sequence of reward functions is how much you will earn.
1180:
This is sometimes called the MLS (which stand for Mayer, Lagrange, and supremum, respectively) formulation.
5539:
17:
5534:
3536:
3260:
2974:
750:
376:
110:
3206:
3160:
2557:
426:
159:
59:
5051:{\displaystyle \nu =(r-\delta )/\sigma -\sigma /2,\quad c={\tilde {\gamma }}K/({\tilde {\gamma }}-1).}
3395:
1281:
889:
5471:
F. Thomas Bruss. "The art of a right decision: Why decision makers want to know the odds-algorithm."
5093:
31:
4738:
4426:
4099:
4045:
2493:
2395:
62:
an expected reward or minimise an expected cost. Optimal stopping problems can be found in areas of
3610:
is the chance you pick the best object if you stop intentionally rejecting objects at step i, then
2703:
655:{\displaystyle V_{t}^{T}=\mathbb {E} G_{\tau ^{*}}=\sup _{t\leq \tau \leq T}\mathbb {E} G_{\tau }}
5088:
5071:
3751:
2369:
5451:
1559:
3029:
2852:
1606:
668:
538:
38:
2134:
It turns out that under some regularity conditions, the following verification theorem holds:
3777:
3757:
3646:
3613:
3497:
2521:
1789:
1232:
724:
3586:
3456:
3368:
3301:
1779:{\displaystyle \gamma :\mathbb {R} ^{k}\times \mathbb {R} ^{k}\to \mathbb {R} ^{k\times l}}
1250:
1192:, the appropriate solution technique is the martingale approach, so called because it uses
334:
71:
58:
is concerned with the problem of choosing a time to take a particular action, in order to
8:
5371:
5108:
1221:
946:
916:
87:
5160:
879:{\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\geq 0},\mathbb {P} _{x})}
5507:
5447:
5325:
5307:
5206:
5181:
5156:
5103:
5098:
4378:
3797:
3733:
3348:
3328:
2680:
1588:
1535:
1515:
1203:
972:
926:
920:
704:
511:
314:
291:
5423:
5388:
5351:
5311:
5297:
5257:
5210:
5198:
3727:
3489:
79:
5277:
3028:≥ 1) forms a sequence of independent, identically distributed random variables with
501:{\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\geq 0},\mathbb {P} )}
5499:
5415:
5407:
5380:
5343:
5289:
5249:
5190:
5176:
5142:
3718:
3714:
83:
75:
5461:
5347:
5241:
5134:
5083:
1553:
529:
4702:{\displaystyle \nu =(r-\delta )/\sigma -\sigma /2,\quad b=\gamma K/(\gamma -1).}
1275:
5503:
3678:
1236:
1228:
1197:
745:
698:
364:
to maximize your expected reward (or equivalently, minimize your expected loss)
5293:
5281:
5253:
1927:{\displaystyle \tau _{\mathcal {S}}=\inf\{t>0:Y_{t}\notin {\mathcal {S}}\}}
5523:
5392:
5355:
5202:
3690:
3015:
You wish to maximise the amount you get paid by choosing a stopping rule. If
533:
361:
3722:
5384:
744:
A more specific formulation is as follows. We consider an adapted strong
47:
5482:
5194:
4934:{\displaystyle {\tilde {\gamma }}=-({\sqrt {\nu ^{2}+2r}}+\nu )/\sigma }
5511:
5369:
MacQueen, J.; Miller Jr., R.G. (1960). "Optimal persistence policies".
63:
3450:
You wish to maximise the amount you earn by choosing a stopping rule.
288:
You are observing the sequence of random variables, and at each step
67:
1708:{\displaystyle \sigma :\mathbb {R} ^{k}\to \mathbb {R} ^{k\times m}}
5419:
2915:{\displaystyle \max \left\{{\mathcal {A}}\phi +L,M-\phi \right\}=0}
3794:
be the dividend rate and volatility of the stock. The stock price
202:
which depend on the observed values of the random variables in 1:
82:. Optimal stopping problems can often be written in the form of a
2851:
These conditions can also be written is a more compact form (the
3298:
You have a house and wish to sell it. Each day you are offered
532:
to the filtration. The optimal stopping problem is to find the
37:"Dynkin game" redirects here. For the coupling card trick, see
5414:. Stochastic Modelling and Applied Probability. Vol. 39.
3937:
When the option is perpetual, the optimal stopping problem is
5067:
4603:{\displaystyle \gamma =({\sqrt {\nu ^{2}+2r}}-\nu )/\sigma }
4849:
4530:
2175:{\displaystyle \phi :{\bar {\mathcal {S}}}\to \mathbb {R} }
153:, whose joint distribution is something assumed to be known
3345:
to continue advertising it. If you sell your house on day
2357:{\displaystyle D=\{y\in {\mathcal {S}}:\phi (y)>M(y)\}}
3070:{\displaystyle {\text{Bern}}\left({\frac {1}{2}}\right),}
1937:
be the bankruptcy time. The optimal stopping problem is:
3583:
is some large number) are the ranks of the objects, and
3147:{\displaystyle y_{i}={\frac {1}{i}}\sum _{k=1}^{i}X_{k}}
103:
Stopping rule problems are associated with two objects:
5483:"Search-theoretic models of the labor market: a survey"
4032:{\displaystyle V(x)=\sup _{\tau }\mathbb {E} _{x}\left}
78:). A key example of an optimal stopping problem is the
4395:
is the exercise boundary. The solution is known to be
2841:{\displaystyle \tau ^{*}=\inf\{t>0:Y_{t}\notin D\}}
1851:{\displaystyle {\mathcal {S}}\subset \mathbb {R} ^{k}}
1652:{\displaystyle b:\mathbb {R} ^{k}\to \mathbb {R} ^{k}}
4947:
4869:
4717:
4616:
4550:
4405:
4381:
4331:
4159:
4102:
4048:
3946:
3823:
3800:
3780:
3760:
3736:
3649:
3616:
3589:
3539:
3500:
3459:
3398:
3371:
3351:
3331:
3304:
3263:
3209:
3163:
3089:
3040:
2977:
2928:
2864:
2787:
2750:
2706:
2683:
2644:
2604:
2560:
2524:
2496:
2463:
2424:
2398:
2372:
2298:
2191:
2143:
1946:
1867:
1825:
1792:
1721:
1665:
1615:
1591:
1562:
1538:
1518:
1319:
1284:
1253:
1206:
1200:. In the discrete time case, if the planning horizon
998:
975:
949:
929:
892:
805:
753:
727:
707:
671:
571:
541:
514:
434:
379:
337:
317:
308:, you can choose to either stop observing or continue
294:
210:
162:
113:
5480:
1220:
is finite, the problem can also be easily solved by
5132:
2951:{\displaystyle {\mathcal {S}}\setminus \partial D.}
5368:
5139:Great Expectations: The Theory of Optimal Stopping
5050:
4933:
4855:
4701:
4602:
4536:
4387:
4367:
4314:
4142:
4088:
4031:
3923:
3806:
3786:
3766:
3742:
3668:
3635:
3602:
3571:
3519:
3472:
3439:
3384:
3357:
3337:
3317:
3287:
3241:
3195:
3146:
3069:
3001:
2950:
2914:
2840:
2773:
2736:
2689:
2669:
2627:
2590:
2543:
2506:
2483:{\displaystyle {\mathcal {S}}\setminus \partial D}
2482:
2449:
2408:
2384:
2356:
2284:
2174:
2123:
1926:
1850:
1811:
1778:
1707:
1651:
1597:
1577:
1544:
1524:
1501:
1299:
1266:
1212:
1169:
981:
961:
935:
907:
878:
791:
733:
713:
689:
654:
554:
520:
500:
417:
350:
323:
300:
271:
194:
145:
5362:
5521:
5332:(2010). "House-hunting without second moments".
4160:
4150:for a put option. The variational inequality is
3963:
2865:
2801:
2010:
1963:
1883:
1786:are given functions such that a unique solution
1119:
1015:
616:
284:Given those objects, the problem is as follows:
272:{\displaystyle y_{i}=y_{i}(x_{1},\ldots ,x_{i})}
5474:Newsletter of the European Mathematical Society
5074:, calculation of the optimal time to exercise.
5066:for various valuation methods here, as well as
4368:{\displaystyle x\in (0,\infty )\setminus \{b\}}
5405:
5239:
3249:are the objects associated with this problem.
5481:Rogerson, R.; Shimer, R.; Wright, R. (2005).
5324:
5286:Applied Stochastic Control of Jump Diffusions
5276:
5126:
1196:theory, the most important concept being the
4362:
4356:
2835:
2804:
2351:
2305:
1921:
1886:
5399:
5270:
5246:Optimal Stopping and Free-Boundary Problems
5149:
2450:{\displaystyle {\mathcal {A}}\phi +L\leq 0}
2774:{\displaystyle y\in {\bar {\mathcal {S}}}}
2628:{\displaystyle y\in {\bar {\mathcal {S}}}}
1858:be an open set (the solvency region) and
1242:
5318:
5235:
5233:
3974:
3265:
2979:
2168:
2036:
1838:
1760:
1745:
1730:
1689:
1674:
1639:
1624:
1406:
1287:
1038:
895:
863:
638:
591:
491:
5155:
799:defined on a filtered probability space
5248:. Lectures in Mathematics. ETH Zürich.
5169:
3681:of optimal stopping (Bruss algorithm).
2670:{\displaystyle {\mathcal {A}}\phi +L=0}
368:
86:, and are therefore often solved using
14:
5522:
5230:
98:
5175:
5064:Black–Scholes model#American options
3483:
3572:{\displaystyle R_{1},\ldots ,R_{n}}
3288:{\displaystyle \mathbb {E} (y_{i})}
3002:{\displaystyle \mathbb {E} (y_{i})}
1183:
989:, the optimal stopping problem is
792:{\displaystyle X=(X_{t})_{t\geq 0}}
418:{\displaystyle G=(G_{t})_{t\geq 0}}
146:{\displaystyle X_{1},X_{2},\ldots }
24:
5464:. "Sum the odds to one and stop."
4840:
4521:
4347:
3814:follows geometric Brownian motion
3699:
2939:
2931:
2875:
2761:
2647:
2615:
2499:
2474:
2466:
2427:
2401:
2316:
2273:
2265:
2239:
2208:
2154:
2026:
1979:
1916:
1874:
1828:
831:
817:
809:
728:
562:which maximizes the expected gain
460:
446:
438:
25:
5551:
5452:Who solved the secretary problem?
5162:Optimal Stopping and Applications
4353:
3708:
3242:{\displaystyle (y_{i})_{i\geq 1}}
3196:{\displaystyle (X_{i})_{i\geq 1}}
2936:
2591:{\displaystyle \phi (y)\geq V(y)}
2471:
2292:where the continuation region is
2270:
195:{\displaystyle (y_{i})_{i\geq 1}}
156:A sequence of 'reward' functions
5179:(2009). "Knowing When to Stop".
3934:under the risk-neutral measure.
3684:
3440:{\displaystyle y_{n}=(X_{n}-nk)}
3252:
1300:{\displaystyle \mathbb {R} ^{k}}
1190:finite-dimensional distributions
908:{\displaystyle \mathbb {P} _{x}}
5412:Methods of Mathematical Finance
4994:
4663:
3453:In this example, the sequence (
3295:does not necessarily converge)
2966:
1482:
107:A sequence of random variables
5491:Journal of Economic Literature
5042:
5030:
5021:
5007:
4966:
4954:
4920:
4888:
4876:
4843:
4831:
4816:
4807:
4792:
4789:
4777:
4770:
4758:
4727:
4721:
4693:
4681:
4635:
4623:
4589:
4557:
4524:
4512:
4488:
4476:
4459:
4444:
4441:
4429:
4415:
4409:
4350:
4338:
4298:
4292:
4283:
4277:
4268:
4262:
4250:
4244:
4230:
4218:
4212:
4206:
4143:{\displaystyle g(x)=(K-x)^{+}}
4131:
4118:
4112:
4106:
4089:{\displaystyle g(x)=(x-K)^{+}}
4077:
4064:
4058:
4052:
4021:
4008:
3956:
3950:
3663:
3650:
3630:
3617:
3514:
3501:
3434:
3412:
3282:
3269:
3224:
3210:
3178:
3164:
2996:
2983:
2853:integro-variational inequality
2765:
2731:
2725:
2716:
2710:
2619:
2585:
2579:
2570:
2564:
2538:
2525:
2507:{\displaystyle {\mathcal {A}}}
2409:{\displaystyle {\mathcal {S}}}
2348:
2342:
2333:
2327:
2279:
2260:
2244:
2234:
2218:
2212:
2201:
2164:
2158:
2104:
2091:
2067:
2054:
2003:
1997:
1956:
1950:
1806:
1793:
1755:
1684:
1634:
1569:
1476:
1458:
1452:
1443:
1421:
1380:
1367:
1352:
1339:
1156:
1143:
1106:
1093:
1069:
1056:
1008:
1002:
873:
843:
825:
806:
774:
760:
495:
472:
454:
435:
400:
386:
311:If you stop observing at step
266:
234:
177:
163:
27:Class of mathematical problems
13:
1:
5217:(For French translation, see
5114:
4042:where the payoff function is
2848:is an optimal stopping time.
2737:{\displaystyle \phi (y)=V(y)}
943:. Given continuous functions
93:
5348:10.1080/07474946.2010.487423
5119:
7:
5468:, Vol. 28, 1384–1391,(2000)
5077:
2961:
2385:{\displaystyle \phi \geq M}
74:(related to the pricing of
10:
5556:
5504:10.1257/002205105775362014
5440:
3688:
3487:
1578:{\displaystyle {\bar {N}}}
427:filtered probability space
331:, you will receive reward
36:
29:
5477:, Issue 62, 14–20, (2006)
5458:, Vol. 4.,282–296, (1989)
5294:10.1007/978-3-540-69826-5
5254:10.1007/978-3-7643-7390-0
5094:Optional stopping theorem
1605:-dimensional compensated
690:{\displaystyle V_{t}^{T}}
555:{\displaystyle \tau ^{*}}
32:Optional stopping theorem
3325:for your house, and pay
373:Consider a gain process
30:Not to be confused with
5137:; Siegmund, D. (1971).
5089:Markov decision process
3787:{\displaystyle \sigma }
3767:{\displaystyle \delta }
3752:risk-free interest rate
3669:{\displaystyle (y_{i})}
3636:{\displaystyle (R_{i})}
3520:{\displaystyle (X_{i})}
2544:{\displaystyle (Y_{t})}
2516:infinitesimal generator
1812:{\displaystyle (Y_{t})}
1243:A jump diffusion result
734:{\displaystyle \infty }
5052:
4935:
4857:
4703:
4604:
4538:
4389:
4369:
4316:
4144:
4096:for a call option and
4090:
4033:
3925:
3808:
3788:
3768:
3744:
3670:
3637:
3604:
3573:
3527:is a finite sequence)
3521:
3474:
3441:
3386:
3359:
3339:
3319:
3289:
3243:
3197:
3148:
3133:
3071:
3030:Bernoulli distribution
3003:
2952:
2916:
2842:
2775:
2738:
2691:
2671:
2629:
2592:
2545:
2508:
2484:
2451:
2410:
2386:
2358:
2286:
2176:
2125:
1928:
1852:
1813:
1780:
1709:
1653:
1607:Poisson random measure
1599:
1579:
1546:
1526:
1503:
1301:
1268:
1233:free-boundary problems
1214:
1171:
983:
963:
937:
909:
880:
793:
735:
715:
691:
656:
556:
522:
502:
419:
352:
325:
302:
273:
196:
147:
5466:Annals of Probability
5221:in the July issue of
5053:
4936:
4858:
4704:
4605:
4539:
4390:
4370:
4317:
4145:
4091:
4034:
3926:
3809:
3789:
3769:
3745:
3671:
3638:
3605:
3603:{\displaystyle y_{i}}
3574:
3522:
3475:
3473:{\displaystyle X_{i}}
3442:
3387:
3385:{\displaystyle y_{n}}
3360:
3340:
3320:
3318:{\displaystyle X_{n}}
3290:
3244:
3198:
3149:
3113:
3072:
3004:
2953:
2917:
2843:
2776:
2739:
2692:
2672:
2630:
2593:
2546:
2509:
2485:
2452:
2411:
2387:
2359:
2287:
2177:
2126:
1929:
1853:
1814:
1781:
1710:
1654:
1600:
1580:
1547:
1527:
1504:
1302:
1269:
1267:{\displaystyle Y_{t}}
1215:
1172:
984:
964:
938:
910:
881:
794:
736:
716:
692:
657:
557:
523:
503:
420:
360:You want to choose a
353:
351:{\displaystyle y_{i}}
326:
303:
274:
197:
148:
5530:Mathematical finance
5385:10.1287/opre.8.3.362
4945:
4867:
4715:
4614:
4548:
4403:
4379:
4329:
4157:
4100:
4046:
3944:
3821:
3798:
3778:
3758:
3734:
3647:
3614:
3587:
3537:
3498:
3457:
3396:
3369:
3349:
3329:
3302:
3261:
3207:
3161:
3087:
3038:
2975:
2926:
2862:
2785:
2748:
2704:
2681:
2642:
2602:
2558:
2522:
2494:
2461:
2422:
2396:
2370:
2296:
2189:
2141:
1944:
1865:
1823:
1790:
1719:
1663:
1613:
1589:
1560:
1536:
1516:
1317:
1282:
1251:
1204:
996:
973:
947:
927:
890:
803:
751:
725:
705:
669:
569:
539:
512:
432:
377:
369:Continuous time case
335:
315:
292:
208:
160:
111:
72:mathematical finance
5540:Dynamic programming
5456:Statistical Science
5406:Karatzas, Ioannis;
5372:Operations Research
5335:Sequential Analysis
5326:Ferguson, Thomas S.
5195:10.1511/2009.77.126
5157:Ferguson, Thomas S.
5109:Sequential analysis
3721:, the holder of an
3157:then the sequences
2087:
1222:dynamic programming
1089:
962:{\displaystyle M,L}
917:probability measure
686:
586:
88:dynamic programming
39:Dynkin's card trick
5535:Sequential methods
5448:Thomas S. Ferguson
5182:American Scientist
5104:Stochastic control
5099:Prophet inequality
5048:
4931:
4853:
4848:
4699:
4600:
4534:
4529:
4385:
4365:
4312:
4140:
4086:
4029:
3971:
3921:
3804:
3784:
3764:
3740:
3713:In the trading of
3666:
3633:
3600:
3569:
3517:
3470:
3437:
3382:
3355:
3335:
3315:
3285:
3239:
3193:
3144:
3067:
2999:
2948:
2912:
2838:
2771:
2734:
2687:
2667:
2625:
2588:
2541:
2504:
2480:
2447:
2406:
2382:
2354:
2282:
2172:
2121:
2073:
2033:
1986:
1924:
1848:
1809:
1776:
1705:
1649:
1595:
1575:
1542:
1522:
1499:
1297:
1264:
1210:
1167:
1139:
1075:
1035:
979:
959:
933:
921:stochastic process
905:
876:
789:
731:
711:
687:
672:
652:
636:
572:
552:
518:
498:
415:
348:
321:
298:
269:
192:
143:
99:Discrete time case
5429:978-0-387-94839-3
5408:Shreve, Steven E.
5330:Klass, Michael J.
5303:978-3-540-69825-8
5263:978-3-7643-2419-3
5177:Hill, Theodore P.
5033:
5010:
4912:
4879:
4819:
4581:
4399:(Perpetual call)
4388:{\displaystyle b}
4176:
3962:
3890:
3807:{\displaystyle S}
3743:{\displaystyle r}
3719:financial markets
3490:Secretary problem
3484:Secretary problem
3358:{\displaystyle n}
3338:{\displaystyle k}
3111:
3058:
3044:
2768:
2690:{\displaystyle D}
2622:
2215:
2161:
2009:
1962:
1598:{\displaystyle l}
1572:
1545:{\displaystyle m}
1525:{\displaystyle B}
1455:
1213:{\displaystyle T}
1118:
1014:
982:{\displaystyle K}
936:{\displaystyle x}
714:{\displaystyle T}
615:
521:{\displaystyle G}
324:{\displaystyle i}
301:{\displaystyle i}
80:secretary problem
16:(Redirected from
5547:
5515:
5487:
5434:
5433:
5403:
5397:
5396:
5366:
5360:
5359:
5322:
5316:
5315:
5274:
5268:
5267:
5242:Shiryaev, Albert
5237:
5228:
5214:
5173:
5167:
5166:
5153:
5147:
5146:
5143:Houghton Mifflin
5130:
5070:for a discrete,
5057:
5055:
5054:
5049:
5035:
5034:
5026:
5020:
5012:
5011:
5003:
4987:
4973:
4940:
4938:
4937:
4932:
4927:
4913:
4902:
4901:
4892:
4881:
4880:
4872:
4862:
4860:
4859:
4854:
4852:
4851:
4822:
4821:
4820:
4812:
4802:
4711:(Perpetual put)
4708:
4706:
4705:
4700:
4680:
4656:
4642:
4609:
4607:
4606:
4601:
4596:
4582:
4571:
4570:
4561:
4543:
4541:
4540:
4535:
4533:
4532:
4467:
4466:
4454:
4394:
4392:
4391:
4386:
4374:
4372:
4371:
4366:
4321:
4319:
4318:
4313:
4305:
4301:
4243:
4205:
4197:
4196:
4187:
4186:
4177:
4169:
4149:
4147:
4146:
4141:
4139:
4138:
4095:
4093:
4092:
4087:
4085:
4084:
4038:
4036:
4035:
4030:
4028:
4024:
4020:
4019:
4004:
4003:
3983:
3982:
3977:
3970:
3930:
3928:
3927:
3922:
3920:
3916:
3915:
3914:
3896:
3892:
3891:
3886:
3885:
3876:
3846:
3845:
3833:
3832:
3813:
3811:
3810:
3805:
3793:
3791:
3790:
3785:
3773:
3771:
3770:
3765:
3749:
3747:
3746:
3741:
3675:
3673:
3672:
3667:
3662:
3661:
3642:
3640:
3639:
3634:
3629:
3628:
3609:
3607:
3606:
3601:
3599:
3598:
3578:
3576:
3575:
3570:
3568:
3567:
3549:
3548:
3526:
3524:
3523:
3518:
3513:
3512:
3479:
3477:
3476:
3471:
3469:
3468:
3446:
3444:
3443:
3438:
3424:
3423:
3408:
3407:
3391:
3389:
3388:
3383:
3381:
3380:
3365:, you will earn
3364:
3362:
3361:
3356:
3344:
3342:
3341:
3336:
3324:
3322:
3321:
3316:
3314:
3313:
3294:
3292:
3291:
3286:
3281:
3280:
3268:
3248:
3246:
3245:
3240:
3238:
3237:
3222:
3221:
3202:
3200:
3199:
3194:
3192:
3191:
3176:
3175:
3153:
3151:
3150:
3145:
3143:
3142:
3132:
3127:
3112:
3104:
3099:
3098:
3076:
3074:
3073:
3068:
3063:
3059:
3051:
3045:
3042:
3008:
3006:
3005:
3000:
2995:
2994:
2982:
2957:
2955:
2954:
2949:
2935:
2934:
2921:
2919:
2918:
2913:
2905:
2901:
2879:
2878:
2847:
2845:
2844:
2839:
2828:
2827:
2797:
2796:
2780:
2778:
2777:
2772:
2770:
2769:
2764:
2759:
2743:
2741:
2740:
2735:
2696:
2694:
2693:
2688:
2676:
2674:
2673:
2668:
2651:
2650:
2634:
2632:
2631:
2626:
2624:
2623:
2618:
2613:
2597:
2595:
2594:
2589:
2550:
2548:
2547:
2542:
2537:
2536:
2513:
2511:
2510:
2505:
2503:
2502:
2489:
2487:
2486:
2481:
2470:
2469:
2456:
2454:
2453:
2448:
2431:
2430:
2415:
2413:
2412:
2407:
2405:
2404:
2391:
2389:
2388:
2383:
2363:
2361:
2360:
2355:
2320:
2319:
2291:
2289:
2288:
2283:
2269:
2268:
2259:
2258:
2243:
2242:
2233:
2232:
2217:
2216:
2211:
2206:
2181:
2179:
2178:
2173:
2171:
2163:
2162:
2157:
2152:
2130:
2128:
2127:
2122:
2117:
2113:
2103:
2102:
2086:
2081:
2066:
2065:
2045:
2044:
2039:
2032:
2031:
2030:
2029:
1996:
1995:
1985:
1984:
1983:
1982:
1933:
1931:
1930:
1925:
1920:
1919:
1910:
1909:
1879:
1878:
1877:
1857:
1855:
1854:
1849:
1847:
1846:
1841:
1832:
1831:
1818:
1816:
1815:
1810:
1805:
1804:
1785:
1783:
1782:
1777:
1775:
1774:
1763:
1754:
1753:
1748:
1739:
1738:
1733:
1714:
1712:
1711:
1706:
1704:
1703:
1692:
1683:
1682:
1677:
1658:
1656:
1655:
1650:
1648:
1647:
1642:
1633:
1632:
1627:
1604:
1602:
1601:
1596:
1584:
1582:
1581:
1576:
1574:
1573:
1565:
1551:
1549:
1548:
1543:
1531:
1529:
1528:
1523:
1508:
1506:
1505:
1500:
1492:
1491:
1457:
1456:
1448:
1436:
1435:
1417:
1416:
1415:
1414:
1409:
1395:
1394:
1379:
1378:
1351:
1350:
1332:
1331:
1306:
1304:
1303:
1298:
1296:
1295:
1290:
1273:
1271:
1270:
1265:
1263:
1262:
1229:Markov processes
1219:
1217:
1216:
1211:
1184:Solution methods
1176:
1174:
1173:
1168:
1163:
1159:
1155:
1154:
1138:
1105:
1104:
1088:
1083:
1068:
1067:
1047:
1046:
1041:
1034:
988:
986:
985:
980:
968:
966:
965:
960:
942:
940:
939:
934:
914:
912:
911:
906:
904:
903:
898:
885:
883:
882:
877:
872:
871:
866:
857:
856:
841:
840:
835:
834:
821:
820:
798:
796:
795:
790:
788:
787:
772:
771:
740:
738:
737:
732:
720:
718:
717:
712:
696:
694:
693:
688:
685:
680:
661:
659:
658:
653:
651:
650:
641:
635:
611:
610:
609:
608:
594:
585:
580:
561:
559:
558:
553:
551:
550:
527:
525:
524:
519:
508:and assume that
507:
505:
504:
499:
494:
486:
485:
470:
469:
464:
463:
450:
449:
424:
422:
421:
416:
414:
413:
398:
397:
357:
355:
354:
349:
347:
346:
330:
328:
327:
322:
307:
305:
304:
299:
278:
276:
275:
270:
265:
264:
246:
245:
233:
232:
220:
219:
201:
199:
198:
193:
191:
190:
175:
174:
152:
150:
149:
144:
136:
135:
123:
122:
84:Bellman equation
76:American options
52:optimal stopping
50:, the theory of
21:
5555:
5554:
5550:
5549:
5548:
5546:
5545:
5544:
5520:
5519:
5518:
5485:
5462:F. Thomas Bruss
5443:
5438:
5437:
5430:
5404:
5400:
5367:
5363:
5323:
5319:
5304:
5275:
5271:
5264:
5240:Peskir, Goran;
5238:
5231:
5223:Pour la Science
5174:
5170:
5154:
5150:
5131:
5127:
5122:
5117:
5084:Halting problem
5080:
5025:
5024:
5016:
5002:
5001:
4983:
4969:
4946:
4943:
4942:
4923:
4897:
4893:
4891:
4871:
4870:
4868:
4865:
4864:
4847:
4846:
4823:
4811:
4810:
4806:
4798:
4774:
4773:
4750:
4734:
4733:
4716:
4713:
4712:
4676:
4652:
4638:
4615:
4612:
4611:
4592:
4566:
4562:
4560:
4549:
4546:
4545:
4528:
4527:
4504:
4492:
4491:
4468:
4462:
4458:
4450:
4422:
4421:
4404:
4401:
4400:
4380:
4377:
4376:
4330:
4327:
4326:
4236:
4198:
4192:
4188:
4182:
4178:
4168:
4167:
4163:
4158:
4155:
4154:
4134:
4130:
4101:
4098:
4097:
4080:
4076:
4047:
4044:
4043:
4015:
4011:
3993:
3989:
3988:
3984:
3978:
3973:
3972:
3966:
3945:
3942:
3941:
3910:
3906:
3881:
3877:
3875:
3862:
3858:
3857:
3853:
3841:
3837:
3828:
3824:
3822:
3819:
3818:
3799:
3796:
3795:
3779:
3776:
3775:
3759:
3756:
3755:
3735:
3732:
3731:
3730:set-up and let
3723:American option
3711:
3702:
3700:Parking problem
3693:
3687:
3657:
3653:
3648:
3645:
3644:
3624:
3620:
3615:
3612:
3611:
3594:
3590:
3588:
3585:
3584:
3563:
3559:
3544:
3540:
3538:
3535:
3534:
3508:
3504:
3499:
3496:
3495:
3494:(Example where
3492:
3486:
3464:
3460:
3458:
3455:
3454:
3419:
3415:
3403:
3399:
3397:
3394:
3393:
3376:
3372:
3370:
3367:
3366:
3350:
3347:
3346:
3330:
3327:
3326:
3309:
3305:
3303:
3300:
3299:
3276:
3272:
3264:
3262:
3259:
3258:
3257:(Example where
3255:
3227:
3223:
3217:
3213:
3208:
3205:
3204:
3181:
3177:
3171:
3167:
3162:
3159:
3158:
3138:
3134:
3128:
3117:
3103:
3094:
3090:
3088:
3085:
3084:
3050:
3046:
3041:
3039:
3036:
3035:
3023:
2990:
2986:
2978:
2976:
2973:
2972:
2971:(Example where
2969:
2964:
2930:
2929:
2927:
2924:
2923:
2874:
2873:
2872:
2868:
2863:
2860:
2859:
2823:
2819:
2792:
2788:
2786:
2783:
2782:
2760:
2758:
2757:
2749:
2746:
2745:
2705:
2702:
2701:
2682:
2679:
2678:
2646:
2645:
2643:
2640:
2639:
2635:. Moreover, if
2614:
2612:
2611:
2603:
2600:
2599:
2559:
2556:
2555:
2532:
2528:
2523:
2520:
2519:
2498:
2497:
2495:
2492:
2491:
2465:
2464:
2462:
2459:
2458:
2426:
2425:
2423:
2420:
2419:
2400:
2399:
2397:
2394:
2393:
2371:
2368:
2367:
2315:
2314:
2297:
2294:
2293:
2264:
2263:
2254:
2250:
2238:
2237:
2228:
2224:
2207:
2205:
2204:
2190:
2187:
2186:
2167:
2153:
2151:
2150:
2142:
2139:
2138:
2098:
2094:
2082:
2077:
2061:
2057:
2050:
2046:
2040:
2035:
2034:
2025:
2024:
2020:
2013:
1991:
1987:
1978:
1977:
1973:
1966:
1945:
1942:
1941:
1915:
1914:
1905:
1901:
1873:
1872:
1868:
1866:
1863:
1862:
1842:
1837:
1836:
1827:
1826:
1824:
1821:
1820:
1800:
1796:
1791:
1788:
1787:
1764:
1759:
1758:
1749:
1744:
1743:
1734:
1729:
1728:
1720:
1717:
1716:
1693:
1688:
1687:
1678:
1673:
1672:
1664:
1661:
1660:
1643:
1638:
1637:
1628:
1623:
1622:
1614:
1611:
1610:
1590:
1587:
1586:
1564:
1563:
1561:
1558:
1557:
1554:Brownian motion
1537:
1534:
1533:
1517:
1514:
1513:
1487:
1483:
1447:
1446:
1428:
1424:
1410:
1405:
1404:
1403:
1399:
1390:
1386:
1374:
1370:
1346:
1342:
1327:
1323:
1318:
1315:
1314:
1291:
1286:
1285:
1283:
1280:
1279:
1258:
1254:
1252:
1249:
1248:
1245:
1237:Stefan problems
1205:
1202:
1201:
1186:
1150:
1146:
1122:
1100:
1096:
1084:
1079:
1063:
1059:
1052:
1048:
1042:
1037:
1036:
1018:
997:
994:
993:
974:
971:
970:
948:
945:
944:
928:
925:
924:
899:
894:
893:
891:
888:
887:
867:
862:
861:
846:
842:
836:
830:
829:
828:
816:
815:
804:
801:
800:
777:
773:
767:
763:
752:
749:
748:
726:
723:
722:
721:can take value
706:
703:
702:
681:
676:
670:
667:
666:
646:
642:
637:
619:
604:
600:
599:
595:
590:
581:
576:
570:
567:
566:
546:
542:
540:
537:
536:
513:
510:
509:
490:
475:
471:
465:
459:
458:
457:
445:
444:
433:
430:
429:
403:
399:
393:
389:
378:
375:
374:
371:
342:
338:
336:
333:
332:
316:
313:
312:
293:
290:
289:
260:
256:
241:
237:
228:
224:
215:
211:
209:
206:
205:
180:
176:
170:
166:
161:
158:
157:
131:
127:
118:
114:
112:
109:
108:
101:
96:
42:
35:
28:
23:
22:
15:
12:
11:
5:
5553:
5543:
5542:
5537:
5532:
5517:
5516:
5478:
5469:
5459:
5444:
5442:
5439:
5436:
5435:
5428:
5420:10.1007/b98840
5398:
5379:(3): 362–380.
5361:
5342:(3): 236–244.
5317:
5302:
5269:
5262:
5229:
5227:
5226:
5189:(2): 126–133.
5168:
5148:
5124:
5123:
5121:
5118:
5116:
5113:
5112:
5111:
5106:
5101:
5096:
5091:
5086:
5079:
5076:
5059:
5058:
5047:
5044:
5041:
5038:
5032:
5029:
5023:
5019:
5015:
5009:
5006:
5000:
4997:
4993:
4990:
4986:
4982:
4979:
4976:
4972:
4968:
4965:
4962:
4959:
4956:
4953:
4950:
4930:
4926:
4922:
4919:
4916:
4911:
4908:
4905:
4900:
4896:
4890:
4887:
4884:
4878:
4875:
4850:
4845:
4842:
4839:
4836:
4833:
4830:
4827:
4824:
4818:
4815:
4809:
4805:
4801:
4797:
4794:
4791:
4788:
4785:
4782:
4779:
4776:
4775:
4772:
4769:
4766:
4763:
4760:
4757:
4754:
4751:
4749:
4746:
4743:
4740:
4739:
4737:
4732:
4729:
4726:
4723:
4720:
4709:
4698:
4695:
4692:
4689:
4686:
4683:
4679:
4675:
4672:
4669:
4666:
4662:
4659:
4655:
4651:
4648:
4645:
4641:
4637:
4634:
4631:
4628:
4625:
4622:
4619:
4599:
4595:
4591:
4588:
4585:
4580:
4577:
4574:
4569:
4565:
4559:
4556:
4553:
4531:
4526:
4523:
4520:
4517:
4514:
4511:
4508:
4505:
4503:
4500:
4497:
4494:
4493:
4490:
4487:
4484:
4481:
4478:
4475:
4472:
4469:
4465:
4461:
4457:
4453:
4449:
4446:
4443:
4440:
4437:
4434:
4431:
4428:
4427:
4425:
4420:
4417:
4414:
4411:
4408:
4384:
4364:
4361:
4358:
4355:
4352:
4349:
4346:
4343:
4340:
4337:
4334:
4323:
4322:
4311:
4308:
4304:
4300:
4297:
4294:
4291:
4288:
4285:
4282:
4279:
4276:
4273:
4270:
4267:
4264:
4261:
4258:
4255:
4252:
4249:
4246:
4242:
4239:
4235:
4232:
4229:
4226:
4223:
4220:
4217:
4214:
4211:
4208:
4204:
4201:
4195:
4191:
4185:
4181:
4175:
4172:
4166:
4162:
4137:
4133:
4129:
4126:
4123:
4120:
4117:
4114:
4111:
4108:
4105:
4083:
4079:
4075:
4072:
4069:
4066:
4063:
4060:
4057:
4054:
4051:
4040:
4039:
4027:
4023:
4018:
4014:
4010:
4007:
4002:
3999:
3996:
3992:
3987:
3981:
3976:
3969:
3965:
3961:
3958:
3955:
3952:
3949:
3932:
3931:
3919:
3913:
3909:
3905:
3902:
3899:
3895:
3889:
3884:
3880:
3874:
3871:
3868:
3865:
3861:
3856:
3852:
3849:
3844:
3840:
3836:
3831:
3827:
3803:
3783:
3763:
3739:
3710:
3709:Option trading
3707:
3701:
3698:
3689:Main article:
3686:
3683:
3679:odds algorithm
3665:
3660:
3656:
3652:
3632:
3627:
3623:
3619:
3597:
3593:
3566:
3562:
3558:
3555:
3552:
3547:
3543:
3516:
3511:
3507:
3503:
3488:Main article:
3485:
3482:
3467:
3463:
3436:
3433:
3430:
3427:
3422:
3418:
3414:
3411:
3406:
3402:
3379:
3375:
3354:
3334:
3312:
3308:
3284:
3279:
3275:
3271:
3267:
3254:
3251:
3236:
3233:
3230:
3226:
3220:
3216:
3212:
3190:
3187:
3184:
3180:
3174:
3170:
3166:
3155:
3154:
3141:
3137:
3131:
3126:
3123:
3120:
3116:
3110:
3107:
3102:
3097:
3093:
3078:
3077:
3066:
3062:
3057:
3054:
3049:
3019:
2998:
2993:
2989:
2985:
2981:
2968:
2965:
2963:
2960:
2959:
2958:
2947:
2944:
2941:
2938:
2933:
2911:
2908:
2904:
2900:
2897:
2894:
2891:
2888:
2885:
2882:
2877:
2871:
2867:
2837:
2834:
2831:
2826:
2822:
2818:
2815:
2812:
2809:
2806:
2803:
2800:
2795:
2791:
2767:
2763:
2756:
2753:
2733:
2730:
2727:
2724:
2721:
2718:
2715:
2712:
2709:
2698:
2697:
2686:
2666:
2663:
2660:
2657:
2654:
2649:
2621:
2617:
2610:
2607:
2587:
2584:
2581:
2578:
2575:
2572:
2569:
2566:
2563:
2552:
2551:
2540:
2535:
2531:
2527:
2501:
2479:
2476:
2473:
2468:
2446:
2443:
2440:
2437:
2434:
2429:
2417:
2403:
2381:
2378:
2375:
2365:
2353:
2350:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2318:
2313:
2310:
2307:
2304:
2301:
2281:
2278:
2275:
2272:
2267:
2262:
2257:
2253:
2249:
2246:
2241:
2236:
2231:
2227:
2223:
2220:
2214:
2210:
2203:
2200:
2197:
2194:
2170:
2166:
2160:
2156:
2149:
2146:
2137:If a function
2132:
2131:
2120:
2116:
2112:
2109:
2106:
2101:
2097:
2093:
2090:
2085:
2080:
2076:
2072:
2069:
2064:
2060:
2056:
2053:
2049:
2043:
2038:
2028:
2023:
2019:
2016:
2012:
2008:
2005:
2002:
1999:
1994:
1990:
1981:
1976:
1972:
1969:
1965:
1961:
1958:
1955:
1952:
1949:
1935:
1934:
1923:
1918:
1913:
1908:
1904:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1876:
1871:
1845:
1840:
1835:
1830:
1808:
1803:
1799:
1795:
1773:
1770:
1767:
1762:
1757:
1752:
1747:
1742:
1737:
1732:
1727:
1724:
1702:
1699:
1696:
1691:
1686:
1681:
1676:
1671:
1668:
1646:
1641:
1636:
1631:
1626:
1621:
1618:
1594:
1571:
1568:
1541:
1521:
1510:
1509:
1498:
1495:
1490:
1486:
1481:
1478:
1475:
1472:
1469:
1466:
1463:
1460:
1454:
1451:
1445:
1442:
1439:
1434:
1431:
1427:
1423:
1420:
1413:
1408:
1402:
1398:
1393:
1389:
1385:
1382:
1377:
1373:
1369:
1366:
1363:
1360:
1357:
1354:
1349:
1345:
1341:
1338:
1335:
1330:
1326:
1322:
1294:
1289:
1261:
1257:
1244:
1241:
1209:
1198:Snell envelope
1185:
1182:
1178:
1177:
1166:
1162:
1158:
1153:
1149:
1145:
1142:
1137:
1134:
1131:
1128:
1125:
1121:
1117:
1114:
1111:
1108:
1103:
1099:
1095:
1092:
1087:
1082:
1078:
1074:
1071:
1066:
1062:
1058:
1055:
1051:
1045:
1040:
1033:
1030:
1027:
1024:
1021:
1017:
1013:
1010:
1007:
1004:
1001:
978:
958:
955:
952:
932:
902:
897:
875:
870:
865:
860:
855:
852:
849:
845:
839:
833:
827:
824:
819:
814:
811:
808:
786:
783:
780:
776:
770:
766:
762:
759:
756:
746:Markov process
730:
710:
699:value function
697:is called the
684:
679:
675:
663:
662:
649:
645:
640:
634:
631:
628:
625:
622:
618:
614:
607:
603:
598:
593:
589:
584:
579:
575:
549:
545:
517:
497:
493:
489:
484:
481:
478:
474:
468:
462:
456:
453:
448:
443:
440:
437:
412:
409:
406:
402:
396:
392:
388:
385:
382:
370:
367:
366:
365:
358:
345:
341:
320:
309:
297:
282:
281:
280:
279:
268:
263:
259:
255:
252:
249:
244:
240:
236:
231:
227:
223:
218:
214:
189:
186:
183:
179:
173:
169:
165:
154:
142:
139:
134:
130:
126:
121:
117:
100:
97:
95:
92:
56:early stopping
26:
9:
6:
4:
3:
2:
5552:
5541:
5538:
5536:
5533:
5531:
5528:
5527:
5525:
5513:
5509:
5505:
5501:
5498:(4): 959–88.
5497:
5493:
5492:
5484:
5479:
5476:
5475:
5470:
5467:
5463:
5460:
5457:
5453:
5449:
5446:
5445:
5431:
5425:
5421:
5417:
5413:
5409:
5402:
5394:
5390:
5386:
5382:
5378:
5374:
5373:
5365:
5357:
5353:
5349:
5345:
5341:
5337:
5336:
5331:
5327:
5321:
5313:
5309:
5305:
5299:
5295:
5291:
5287:
5283:
5279:
5273:
5265:
5259:
5255:
5251:
5247:
5243:
5236:
5234:
5224:
5220:
5216:
5215:
5212:
5208:
5204:
5200:
5196:
5192:
5188:
5184:
5183:
5178:
5172:
5164:
5163:
5158:
5152:
5144:
5140:
5136:
5129:
5125:
5110:
5107:
5105:
5102:
5100:
5097:
5095:
5092:
5090:
5087:
5085:
5082:
5081:
5075:
5073:
5069:
5065:
5045:
5039:
5036:
5027:
5017:
5013:
5004:
4998:
4995:
4991:
4988:
4984:
4980:
4977:
4974:
4970:
4963:
4960:
4957:
4951:
4948:
4928:
4924:
4917:
4914:
4909:
4906:
4903:
4898:
4894:
4885:
4882:
4873:
4837:
4834:
4828:
4825:
4813:
4803:
4799:
4795:
4786:
4783:
4780:
4767:
4764:
4761:
4755:
4752:
4747:
4744:
4741:
4735:
4730:
4724:
4718:
4710:
4696:
4690:
4687:
4684:
4677:
4673:
4670:
4667:
4664:
4660:
4657:
4653:
4649:
4646:
4643:
4639:
4632:
4629:
4626:
4620:
4617:
4597:
4593:
4586:
4583:
4578:
4575:
4572:
4567:
4563:
4554:
4551:
4518:
4515:
4509:
4506:
4501:
4498:
4495:
4485:
4482:
4479:
4473:
4470:
4463:
4455:
4451:
4447:
4438:
4435:
4432:
4423:
4418:
4412:
4406:
4398:
4397:
4396:
4382:
4359:
4344:
4341:
4335:
4332:
4309:
4306:
4302:
4295:
4289:
4286:
4280:
4274:
4271:
4265:
4259:
4256:
4253:
4247:
4240:
4237:
4233:
4227:
4224:
4221:
4215:
4209:
4202:
4199:
4193:
4189:
4183:
4179:
4173:
4170:
4164:
4153:
4152:
4151:
4135:
4127:
4124:
4121:
4115:
4109:
4103:
4081:
4073:
4070:
4067:
4061:
4055:
4049:
4025:
4016:
4012:
4005:
4000:
3997:
3994:
3990:
3985:
3979:
3967:
3959:
3953:
3947:
3940:
3939:
3938:
3935:
3917:
3911:
3907:
3903:
3900:
3897:
3893:
3887:
3882:
3878:
3872:
3869:
3866:
3863:
3859:
3854:
3850:
3847:
3842:
3838:
3834:
3829:
3825:
3817:
3816:
3815:
3801:
3781:
3761:
3753:
3737:
3729:
3728:Black–Scholes
3724:
3720:
3716:
3706:
3697:
3692:
3691:Search theory
3685:Search theory
3682:
3680:
3658:
3654:
3625:
3621:
3595:
3591:
3582:
3564:
3560:
3556:
3553:
3550:
3545:
3541:
3531:
3528:
3509:
3505:
3491:
3481:
3465:
3461:
3451:
3448:
3431:
3428:
3425:
3420:
3416:
3409:
3404:
3400:
3377:
3373:
3352:
3332:
3310:
3306:
3296:
3277:
3273:
3253:House selling
3250:
3234:
3231:
3228:
3218:
3214:
3188:
3185:
3182:
3172:
3168:
3139:
3135:
3129:
3124:
3121:
3118:
3114:
3108:
3105:
3100:
3095:
3091:
3083:
3082:
3081:
3064:
3060:
3055:
3052:
3047:
3034:
3033:
3032:
3031:
3027:
3022:
3018:
3013:
3010:
2991:
2987:
2945:
2942:
2909:
2906:
2902:
2898:
2895:
2892:
2889:
2886:
2883:
2880:
2869:
2858:
2857:
2856:
2854:
2849:
2832:
2829:
2824:
2820:
2816:
2813:
2810:
2807:
2798:
2793:
2789:
2754:
2751:
2728:
2722:
2719:
2713:
2707:
2684:
2664:
2661:
2658:
2655:
2652:
2638:
2637:
2636:
2608:
2605:
2582:
2576:
2573:
2567:
2561:
2533:
2529:
2517:
2477:
2444:
2441:
2438:
2435:
2432:
2418:
2379:
2376:
2373:
2366:
2345:
2339:
2336:
2330:
2324:
2321:
2311:
2308:
2302:
2299:
2276:
2255:
2251:
2247:
2229:
2225:
2221:
2198:
2195:
2192:
2185:
2184:
2183:
2147:
2144:
2135:
2118:
2114:
2110:
2107:
2099:
2095:
2088:
2083:
2078:
2074:
2070:
2062:
2058:
2051:
2047:
2041:
2021:
2017:
2014:
2006:
2000:
1992:
1988:
1974:
1970:
1967:
1959:
1953:
1947:
1940:
1939:
1938:
1911:
1906:
1902:
1898:
1895:
1892:
1889:
1880:
1869:
1861:
1860:
1859:
1843:
1833:
1801:
1797:
1771:
1768:
1765:
1750:
1740:
1735:
1725:
1722:
1700:
1697:
1694:
1679:
1669:
1666:
1644:
1629:
1619:
1616:
1608:
1592:
1566:
1555:
1552:-dimensional
1539:
1519:
1496:
1493:
1488:
1484:
1479:
1473:
1470:
1467:
1464:
1461:
1449:
1440:
1437:
1432:
1429:
1425:
1418:
1411:
1400:
1396:
1391:
1387:
1383:
1375:
1371:
1364:
1361:
1358:
1355:
1347:
1343:
1336:
1333:
1328:
1324:
1320:
1313:
1312:
1311:
1310:
1307:given by the
1292:
1278:diffusion in
1277:
1259:
1255:
1240:
1238:
1234:
1230:
1225:
1223:
1207:
1199:
1195:
1191:
1181:
1164:
1160:
1151:
1147:
1140:
1135:
1132:
1129:
1126:
1123:
1115:
1112:
1109:
1101:
1097:
1090:
1085:
1080:
1076:
1072:
1064:
1060:
1053:
1049:
1043:
1031:
1028:
1025:
1022:
1019:
1011:
1005:
999:
992:
991:
990:
976:
956:
953:
950:
930:
922:
918:
900:
868:
858:
853:
850:
847:
837:
822:
812:
784:
781:
778:
768:
764:
757:
754:
747:
742:
708:
700:
682:
677:
673:
647:
643:
632:
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534:stopping time
531:
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425:defined on a
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359:
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5133:Chow, Y.S.;
5128:
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5219:cover story
5135:Robbins, H.
3009:converges)
48:mathematics
5524:Categories
5141:. Boston:
5115:References
5072:tree based
2182:satisfies
1194:martingale
923:starts at
919:where the
94:Definition
64:statistics
5393:0030-364X
5356:0747-4946
5312:123531718
5282:Sulem, A.
5211:124798270
5203:1545-2786
5120:Citations
5037:−
5031:~
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5008:~
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5410:(1998).
5284:(2007).
5244:(2006).
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5159:(2007).
5078:See also
4325:for all
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3392:, where
2962:Examples
2744:for all
2598:for all
2490:, where
60:maximise
5512:4129380
5441:Sources
5165:. UCLA.
3750:be the
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3080:and if
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701:. Here
530:adapted
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4863:where
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4375:where
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1585:is an
1512:where
969:, and
886:where
665:where
70:, and
5508:JSTOR
5486:(PDF)
5308:S2CID
5207:S2CID
5068:Fugit
3024:(for
2700:Then
2554:then
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1274:be a
5424:ISBN
5389:ISSN
5352:ISSN
5298:ISBN
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