6315:
292:, the described evaluation strategy of the integral is conceptualized as that we are first deciding what to do, then observing the change in the prices. The integrand is how much stock we hold, the integrator represents the movement of the prices, and the integral is how much money we have in total including what our stock is worth, at any given moment. The prices of stocks and other traded financial assets can be modeled by stochastic processes such as Brownian motion or, more often,
31:
1535:
239:, defined as a limit of a certain sequence of random variables. The paths of Brownian motion fail to satisfy the requirements to be able to apply the standard techniques of calculus. So with the integrand a stochastic process, the Itô stochastic integral amounts to an integral with respect to a function which is not differentiable at any point and has infinite
5381:. This method can be extended to all local square integrable martingales by localization. Finally, the Doob–Meyer decomposition can be used to decompose any local martingale into the sum of a local square integrable martingale and a finite variation process, allowing the Itô integral to be constructed with respect to any semimartingale.
6142:
2347:
4493:
5433:
to be a semimartingale. A continuous linear extension can be used to construct the integral for all left-continuous and adapted integrands with right limits everywhere (caglad or L-processes). This is general enough to be able to apply techniques such as Itô's lemma
2651:
3776:
1128:
6291:
265:. Every time we are computing a Riemann sum, we are using a particular instantiation of the integrator. It is crucial which point in each of the small intervals is used to compute the value of the function. The limit then is taken in probability as the
5979:
1953:
3472:
5912:
2154:
4662:
2655:
This limit converges in probability. The stochastic integral of left-continuous processes is general enough for studying much of stochastic calculus. For example, it is sufficient for applications of Itô's Lemma, changes of measure via
1700:
590:
269:
of the partition is going to zero. Numerous technical details have to be taken care of to show that this limit exists and is independent of the particular sequence of partitions. Typically, the left end of the interval is used.
4696:
Proofs that the Itô integral is well defined typically proceed by first looking at very simple integrands, such as piecewise constant, left continuous and adapted processes where the integral can be written explicitly. Such
2069:
1374:
4842:
4344:
447:
1807:
5615:
5373:
which can be proved directly for simple predictable integrands. As with the case above for
Brownian motion, a continuous linear extension can be used to uniquely extend to all predictable integrands satisfying
2471:
3558:
2792:
942:
6153:
5384:
Many other proofs exist which apply similar methods but which avoid the need to use the Doob–Meyer decomposition theorem, such as the use of the quadratic variation in the Itô isometry, the use of the
4145:
5761:
5371:
5004:
317:
is adapted corresponds to the necessary restriction that the trading strategy can only make use of the available information at any time. This prevents the possibility of unlimited gains through
1518:
1281:
3940:
197:
5174:
allows integration to be defined with respect to finite variation processes, so the existence of the Itô integral for semimartingales will follow from any construction for local martingales.
3287:
term. This term comes from the fact that Itô calculus deals with processes with non-zero quadratic variation, which only occurs for infinite variation processes (such as
Brownian motion). If
1816:
5081:
3110:
3778:
This differs from the chain rule used in standard calculus due to the term involving the quadratic covariation . The formula can be generalized to include an explicit time-dependence in
7070:
3302:
5793:
5454:
The Itô calculus is first and foremost defined as an integral calculus as outlined above. However, there are also different notions of "derivative" with respect to
Brownian motion:
4521:
810:
755:
713:
676:
625:
7605:
2149:
1585:
3269:
906:
512:
5961:
8296:
7429:
5788:
5523:
1978:
6299:
if the correlation time of the noise term approaches zero. For a recent treatment of different interpretations of stochastic differential equations see for example (
1286:
8032:
7025:
6137:{\displaystyle {\dot {y}}={\frac {\partial y}{\partial x_{j}}}{\dot {x}}_{j}+{\frac {1}{2}}{\frac {\partial ^{2}y}{\partial x_{k}\,\partial x_{l}}}g_{km}g_{ml}.}
4752:
3799:
7562:
7542:
3283:
is an important result in stochastic calculus. The integration by parts formula for the Itô integral differs from the standard result due to the inclusion of a
352:
7946:
5394:
1723:
5536:
2944:
The discontinuities of the stochastic integral are given by the jumps of the integrator multiplied by the integrand. The jump of a càdlàg process at a time
2342:{\displaystyle dY_{t}=f^{\prime }(X_{t})\mu _{t}\,dt+{\tfrac {1}{2}}f^{\prime \prime }(X_{t})\sigma _{t}^{2}\,dt+f^{\prime }(X_{t})\sigma _{t}\,dB_{t}.}
5397:
instead of the Itô isometry. The latter applies directly to local martingales without having to first deal with the square integrable martingale case.
7863:
7873:
7547:
6908:
6813:
7557:
3214:. A consequence of this is that the quadratic variation process of a stochastic integral is equal to an integral of a quadratic variation process,
2717:
7915:
4030:
7630:
3507:
formula which applies to the Itô integral. It is one of the most powerful and frequently used theorems in stochastic calculus. For a continuous
7812:
257:
can only depend on information available up until this time. Roughly speaking, one chooses a sequence of partitions of the interval from 0 to
8102:
8092:
7615:
6408:
8002:
7966:
5184:
6891:
6793:
5688:
5244:
4890:
7919:
1582:
stochastic process that can be expressed as the sum of an integral with respect to
Brownian motion and an integral with respect to time,
1382:
1212:
8270:
8007:
6643:
6723:
3881:
130:
7117:
7018:
6448:
6886:
4223:. There are examples of integrals of bounded predictable processes with respect to martingales which are not themselves martingales.
8072:
7650:
7620:
6808:
6609:
6591:
6573:
6555:
6534:
6466:
6428:
6386:
2665:
1140:
5013:
4488:{\displaystyle c\mathbb {E} \left_{t}^{\frac {p}{2}}\right]\leq \mathbb {E} \left\leq C\mathbb {E} \left_{t}^{\frac {p}{2}}\right]}
3056:
7923:
7907:
8117:
7822:
7042:
6788:
5678:
2661:
300:). Then, the Itô stochastic integral represents the payoff of a continuous-time trading strategy consisting of holding an amount
117:
5171:
4175:
to be the unique extension of this isometry from a certain class of simple integrands to all bounded and predictable processes.
1151:
form the smallest class that is closed under taking limits of sequences and contains all adapted left-continuous processes. If
8022:
7987:
7956:
7951:
7590:
7387:
7304:
6803:
4296:
4165:
is integrable. The Itô isometry is often used as an important step in the construction of the stochastic integral, by defining
2424:
1133:
840:
210:
7961:
7289:
6738:
6698:
3478:
836:
124:
7585:
7392:
7311:
8047:
7927:
2993:. A particular consequence of this is that integrals with respect to a continuous process are always themselves continuous.
2646:{\displaystyle \int _{0}^{t}H\,dX=\lim _{n\to \infty }\sum _{t_{i-1},t_{i}\in \pi _{n}}H_{t_{i-1}}(X_{t_{i}}-X_{t_{i-1}}).}
8306:
8275:
8052:
7888:
7787:
7772:
7184:
7100:
7011:
6798:
3115:
2676:
8062:
7698:
5685:, are used, rather than stochastic integrals. Here an Itô stochastic differential equation (SDE) is often formulated via
321:: buying the stock just before each uptick in the market and selling before each downtick. Similarly, the condition that
8301:
8057:
6977:
7660:
6962:
5635:, Theorem 36.5). This representation theorem can be interpreted formally as saying that α is the "time derivative" of
3771:{\displaystyle df(X_{t})=\sum _{i=1}^{n}f_{i}(X_{t})\,dX_{t}^{i}+{\frac {1}{2}}\sum _{i,j=1}^{n}f_{i,j}(X_{t})\,d_{t}.}
7244:
7189:
7105:
6761:
5007:
3504:
2352:
1123:{\displaystyle \int _{0}^{t}H\,dB=\lim _{n\rightarrow \infty }\sum _{\in \pi _{n}}H_{t_{i-1}}(B_{t_{i}}-B_{t_{i-1}}).}
278:
231:. The result of the integration is then another stochastic process. Concretely, the integral from 0 to any particular
7992:
7982:
7625:
7595:
6931:
6898:
6286:{\displaystyle {\dot {x}}_{k}=h_{k}+g_{kl}\xi _{l}-{\frac {1}{2}}{\frac {\partial g_{kl}}{\partial {x_{m}}}}g_{ml}.}
8311:
7997:
7162:
7060:
6766:
6378:
506:
7708:
7284:
7065:
5442:
can be used to prove the dominated convergence theorem and extend the integral to general predictable integrands (
3217:
2794:
in probability. The uniqueness of the extension from left-continuous to predictable integrands is a result of the
8077:
7878:
7792:
7777:
7167:
293:
7911:
7797:
7219:
6295:
SDEs frequently occur in physics in
Stratonovich form, as limits of stochastic differential equations driven by
7299:
7274:
6776:
2669:
2457:
1144:
909:
8017:
7600:
7135:
8212:
8202:
7893:
7675:
7414:
7279:
7090:
6636:
505:. As Itô calculus is concerned with continuous-time stochastic processes, it is assumed that an underlying
7497:
8154:
8082:
7341:
4870:
243:
over every time interval. The main insight is that the integral can be defined as long as the integrand
8177:
8159:
8139:
8134:
7853:
7685:
7665:
7512:
7455:
7294:
7204:
6936:
6838:
6818:
6344:
6147:
5439:
1957:
This is defined for all locally bounded and predictable integrands. More generally, it is required that
1948:{\displaystyle \int _{0}^{t}H\,dX=\int _{0}^{t}H_{s}\sigma _{s}\,dB_{s}+\int _{0}^{t}H_{s}\mu _{s}\,ds.}
784:
729:
687:
650:
599:
7645:
6314:
2675:
The integral extends to all predictable and locally bounded integrands, in a unique way, such that the
1376:
where, again, the limit can be shown to converge in probability. The stochastic integral satisfies the
8252:
8207:
8197:
7938:
7883:
7858:
7827:
7807:
7567:
7552:
7419:
6783:
6673:
6494:
5386:
4268:
and bounded predictable integrand, the stochastic integral preserves the space of càdlàg martingales
3280:
2795:
1717:
289:
113:
2359:. It differs from the standard result due to the additional term involving the second derivative of
2105:
8247:
8087:
8012:
7817:
7577:
7487:
7377:
6918:
6833:
6828:
6718:
6329:
3844:
is also a local martingale. For integrands which are not locally bounded, there are examples where
3482:
3284:
2366:
1148:
282:
5221:
is a right-continuous, increasing and predictable process starting at zero. This uniquely defines
3467:{\displaystyle X_{t}Y_{t}=X_{0}Y_{0}+\int _{0}^{t}X_{s-}\,dY_{s}+\int _{0}^{t}Y_{s-}\,dX_{s}+_{t}}
878:
8217:
8182:
8097:
8067:
7837:
7832:
7655:
7492:
7157:
7095:
7034:
6941:
6878:
6771:
6743:
6708:
6629:
6484:
6400:
6320:
5907:{\displaystyle \langle \xi _{k}(t_{1})\,\xi _{l}(t_{2})\rangle =\delta _{kl}\delta (t_{1}-t_{2})}
5462:
266:
101:
61:) with respect to itself, i.e., both the integrand and the integrator are Brownian. It turns out
7898:
5924:
297:
6617:
Mathematical
Finance Programming in TI-Basic, which implements Ito calculus for TI-calculators.
327:
is adapted implies that the stochastic integral will not diverge when calculated as a limit of
8237:
8042:
7693:
7450:
7367:
7336:
7229:
7209:
7199:
7055:
7050:
6990:
6967:
6903:
6873:
6865:
6843:
6823:
6713:
6605:
6587:
6569:
6551:
6543:
6530:
6510:
6462:
6444:
6424:
6404:
6382:
6334:
5973:
5915:
5766:
5682:
4657:{\displaystyle \mathbb {E} \left\leq C\mathbb {E} \left_{t})^{\frac {p}{2}}\right]<\infty }
3802:
3494:
2410:
2087:
274:
240:
7903:
7640:
5508:
8257:
8144:
8027:
7397:
7372:
7321:
7249:
7172:
7125:
6985:
6848:
6733:
6693:
6688:
6683:
6678:
6668:
6502:
3819:
2657:
2401:
864:
8222:
8122:
8107:
7868:
7802:
7480:
7424:
7407:
7152:
6972:
6855:
6728:
5502:
1579:
937:
872:
250:
236:
224:
206:
105:
30:
8037:
7269:
3188:
The stochastic integral commutes with the operation of taking quadratic covariations. If
2428:
1377:
6498:
3781:
8227:
8192:
8112:
7718:
7465:
7382:
7351:
7346:
7326:
7316:
7259:
7254:
7234:
7214:
7179:
7147:
7130:
6523:
6421:
Path
Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets
6416:
6371:
6354:
6349:
2938:
2420:
2378:
1695:{\displaystyle X_{t}=X_{0}+\int _{0}^{t}\sigma _{s}\,dB_{s}+\int _{0}^{t}\mu _{s}\,ds.}
1567:
854:
228:
123:
The central concept is the Itô stochastic integral, a stochastic generalization of the
109:
8290:
8129:
7670:
7507:
7502:
7460:
7402:
7224:
7140:
7080:
6957:
6339:
6296:
5390:
5090:
3974:
2997:
585:{\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\geq 0},\mathbb {P} ).}
93:
4193:, and bounded predictable integrand, the stochastic integral preserves the space of
715:-Brownian motion, which is just a standard Brownian motion with the properties that
8187:
8149:
7703:
7635:
7524:
7519:
7331:
7264:
7239:
7075:
6703:
5466:
1526:
is bounded or, more generally, when the integral on the right hand side is finite.
318:
7767:
4000:
2934:
868:
593:
5128:
can be used. Then, the integral can be shown to exist separately with respect to
5093:. This method allows the integral to be defined with respect to any Itô process.
8232:
7751:
7746:
7741:
7731:
7534:
7475:
7470:
7434:
7194:
7085:
2064:{\displaystyle \int _{0}^{t}\left(H^{2}\sigma ^{2}+|H\mu |\right)ds<\infty .}
844:
328:
262:
6506:
273:
Important results of Itô calculus include the integration by parts formula and
8242:
7782:
7726:
7610:
6432:
6310:
5083:
in such way that the Itô isometry still holds. It can then be extended to all
3500:
2356:
127:
in analysis. The integrands and the integrators are now stochastic processes:
1369:{\displaystyle \int _{0}^{t}H\,dB=\lim _{n\to \infty }\int _{0}^{t}H_{n}\,dB}
1209:
of left-continuous, adapted and locally bounded processes, in the sense that
7736:
2090:. In its simplest form, for any twice continuously differentiable function
17:
6514:
4837:{\displaystyle H\cdot X_{t}\equiv \mathbf {1} _{\{t>T\}}A(X_{t}-X_{T}).}
3995:
For bounded integrands, the Itô stochastic integral preserves the space of
1534:
442:{\displaystyle Y_{t}=\int _{0}^{t}H\,dX\equiv \int _{0}^{t}H_{s}\,dX_{s},}
6652:
5010:, the integral extends uniquely to all predictable integrands satisfying
4887:
can be used to prove the Itô isometry for simple predictable integrands,
4844:
This is extended to all simple predictable processes by the linearity of
97:
5481:
The following result allows to express martingales as Itô integrals: if
4680:-integrable martingale. More generally, this statement is true whenever
1802:{\displaystyle \int _{0}^{t}(\sigma _{s}^{2}+|\mu _{s}|)\,ds<\infty }
7563:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
7003:
5610:{\displaystyle M_{t}=M_{0}+\int _{0}^{t}\alpha _{s}\,\mathrm {d} B_{s}}
5465:
provides a theory of differentiation for random variables defined over
5647:, since α is precisely the process that must be integrated up to time
6602:
Diffusions, Markov processes and martingales - Volume 2: Itô calculus
1147:, the integral is needed for processes that are not continuous. The
471:). Alternatively, the integral is often written in differential form
281:
formula. These differ from the formulas of standard calculus, due to
6548:
Stochastic
Differential Equations: An Introduction with Applications
5421:| ≤ 1 is simple previsible} is bounded in probability for each time
3818:
An important property of the Itô integral is that it preserves the
2842:
are bounded. Associativity of stochastic integration implies that
6489:
3477:
The result is similar to the integration by parts theorem for the
2365:, which comes from the property that Brownian motion has non-zero
1533:
847:; such a limit does not necessarily exist pathwise. Suppose that
29:
6435:, with generalizations of Itô's lemma for non-Gaussian processes.
5183:, a generalized form of the Itô isometry can be used. First, the
3185:. In fact, it converges uniformly on compact sets in probability.
1813:. The stochastic integral can be extended to such Itô processes,
1202:-integrable. Any such process can be approximated by a sequence
4199:-integrable martingales. These are càdlàg martingales such that
3959:
The most general statement for a discontinuous local martingale
1710:
is a
Brownian motion and it is required that σ is a predictable
7007:
6625:
2664:. However, it is inadequate for other important topics such as
1570:. Off the tide of wavelet, the motion of Itô process is stable.
6475:
Lau, Andy; Lubensky, Tom (2007), "State-dependent diffusion",
5495:
with respect to the filtration generated by a
Brownian motion
2447:
exists, and can be calculated as a limit of Riemann sums. Let
5241:. The Itô isometry for square integrable martingales is then
3854:
is not a local martingale. However, this can only occur when
2787:{\displaystyle \int _{0}^{t}H_{n}\,dX\to \int _{0}^{t}H\,dX,}
2431:. For a left continuous, locally bounded and adapted process
6621:
3866:
is a continuous local martingale then a predictable process
2291:
2240:
2237:
2179:
791:
736:
694:
657:
606:
541:
527:
4140:{\displaystyle \mathbb {E} \left=\mathbb {E} \left\right].}
2811:
can be defined even in cases where the predictable process
835:
The Itô integral can be defined in a manner similar to the
7543:
Autoregressive conditional heteroskedasticity (ARCH) model
5400:
Alternative proofs exist only making use of the fact that
4023:, the quadratic variation process is integrable, and the
7071:
Independent and identically distributed random variables
2921:
The following properties can be found in works such as (
920:
with mesh width going to zero, then the Itô integral of
6423:, 4th edition, World Scientific (Singapore); Paperback
4701:
processes are linear combinations of terms of the form
7548:
Autoregressive integrated moving average (ARIMA) model
4147:
This equality holds more generally for any martingale
2221:
2086:
An important result for the study of Itô processes is
253:, which loosely speaking means that its value at time
6156:
5982:
5927:
5796:
5769:
5691:
5539:
5511:
5487:
is a square-integrable martingale on a time interval
5247:
5016:
4893:
4755:
4524:
4347:
4216:. However, this is not always true in the case where
4033:
3884:
3784:
3561:
3305:
3220:
3059:
2720:
2474:
2157:
2108:
1981:
1819:
1726:
1588:
1385:
1289:
1215:
945:
881:
787:
732:
690:
653:
602:
515:
355:
133:
5756:{\displaystyle {\dot {x}}_{k}=h_{k}+g_{kl}\xi _{l},}
5366:{\displaystyle \mathbb {E} \left=\mathbb {E} \left,}
4999:{\displaystyle \mathbb {E} \left=\mathbb {E} \left.}
8170:
7975:
7937:
7846:
7760:
7717:
7684:
7576:
7533:
7443:
7360:
7116:
7041:
6950:
6917:
6864:
6752:
6659:
1513:{\displaystyle \mathbb {E} \left=\mathbb {E} \left}
1276:{\displaystyle \int _{0}^{t}(H-H_{n})^{2}\,ds\to 0}
628:represents the information available up until time
449:is itself a stochastic process with time parameter
6522:
6439:He, Sheng-wu; Wang, Jia-gang; Yan, Jia-an (1992),
6370:
6285:
6136:
5955:
5906:
5782:
5755:
5609:
5517:
5365:
5075:
4998:
4836:
4656:
4487:
4139:
3934:
3793:
3770:
3466:
3263:
3206:-integrable process will also be -integrable, and
3181:. Convergence is in probability at each time
3104:
2786:
2645:
2341:
2143:
2063:
1947:
1801:
1694:
1512:
1368:
1275:
1122:
900:
804:
749:
707:
670:
619:
584:
441:
191:
6566:Stochastic Integration and Differential Equations
3935:{\displaystyle \int _{0}^{t}H^{2}\,d<\infty ,}
2386:. These are processes which can be decomposed as
192:{\displaystyle Y_{t}=\int _{0}^{t}H_{s}\,dX_{s},}
7430:Stochastic chains with memory of variable length
2504:
1319:
975:
3526:and twice continuously differentiable function
2419:. Important examples of such processes include
2351:This is the stochastic calculus version of the
1716:-integrable process, and μ is predictable and (
5632:
3834:is a locally bounded predictable process then
2926:
2377:The Itô integral is defined with respect to a
468:
7019:
6637:
6441:Semimartingale Theory and Stochastic Calculus
5469:, including an integration by parts formula (
5076:{\displaystyle \mathbb {E} \left<\infty ,}
3474:where is the quadratic covariation process.
3105:{\displaystyle J\cdot (K\cdot X)=(JK)\cdot X}
27:Calculus of stochastic differential equations
8:
5853:
5797:
5346:
5339:
4794:
4782:
4017:. For any such square integrable martingale
895:
882:
6300:
831:Integration with respect to Brownian motion
7558:Autoregressive–moving-average (ARMA) model
7026:
7012:
7004:
6644:
6630:
6622:
6584:Continuous martingales and Brownian motion
6457:Karatzas, Ioannis; Shreve, Steven (1991),
6146:An Itô SDE as above also corresponds to a
5177:For a càdlàg square integrable martingale
1283:in probability. Then, the Itô integral is
6604:, Cambridge: Cambridge University Press,
6525:The Malliavin calculus and related topics
6488:
6271:
6257:
6252:
6238:
6228:
6218:
6209:
6196:
6183:
6170:
6159:
6158:
6155:
6122:
6109:
6096:
6088:
6082:
6064:
6057:
6047:
6038:
6027:
6026:
6016:
5998:
5984:
5983:
5981:
5944:
5926:
5895:
5882:
5863:
5844:
5831:
5826:
5817:
5804:
5795:
5774:
5768:
5744:
5731:
5718:
5705:
5694:
5693:
5690:
5601:
5592:
5591:
5585:
5575:
5570:
5557:
5544:
5538:
5510:
5443:
5427:, which is an alternative definition for
5349:
5335:
5329:
5324:
5314:
5309:
5296:
5295:
5281:
5271:
5249:
5248:
5246:
5052:
5046:
5036:
5031:
5018:
5017:
5015:
4981:
4975:
4970:
4960:
4955:
4942:
4941:
4927:
4917:
4895:
4894:
4892:
4822:
4809:
4781:
4776:
4766:
4754:
4632:
4622:
4603:
4587:
4586:
4569:
4559:
4554:
4526:
4525:
4523:
4469:
4464:
4445:
4444:
4427:
4417:
4412:
4396:
4395:
4376:
4371:
4352:
4351:
4346:
4116:
4110:
4100:
4095:
4082:
4081:
4067:
4057:
4035:
4034:
4032:
3910:
3904:
3894:
3889:
3883:
3783:
3759:
3749:
3736:
3725:
3716:
3697:
3687:
3670:
3656:
3647:
3642:
3634:
3625:
3612:
3602:
3591:
3575:
3560:
3458:
3433:
3425:
3416:
3406:
3401:
3388:
3380:
3371:
3361:
3356:
3343:
3333:
3320:
3310:
3304:
3243:
3219:
3058:
2774:
2765:
2760:
2746:
2740:
2730:
2725:
2719:
2623:
2618:
2603:
2598:
2577:
2572:
2560:
2547:
2528:
2523:
2507:
2493:
2484:
2479:
2473:
2330:
2322:
2316:
2303:
2290:
2276:
2270:
2265:
2252:
2236:
2220:
2210:
2204:
2191:
2178:
2165:
2156:
2132:
2113:
2107:
2036:
2025:
2016:
2006:
1991:
1986:
1980:
1935:
1929:
1919:
1909:
1904:
1891:
1883:
1877:
1867:
1857:
1852:
1838:
1829:
1824:
1818:
1786:
1778:
1772:
1763:
1754:
1749:
1736:
1731:
1725:
1682:
1676:
1666:
1661:
1648:
1640:
1634:
1624:
1619:
1606:
1593:
1587:
1538:A single realization of Itô process with
1498:
1492:
1487:
1477:
1472:
1459:
1458:
1445:
1434:
1426:
1420:
1410:
1405:
1387:
1386:
1384:
1359:
1353:
1343:
1338:
1322:
1308:
1299:
1294:
1288:
1260:
1254:
1244:
1225:
1220:
1214:
1100:
1095:
1080:
1075:
1054:
1049:
1037:
1021:
1002:
994:
978:
964:
955:
950:
944:
889:
880:
796:
790:
789:
786:
741:
735:
734:
731:
699:
693:
692:
689:
662:
656:
655:
652:
611:
605:
604:
601:
572:
571:
556:
546:
540:
539:
526:
525:
514:
430:
422:
416:
406:
401:
387:
378:
373:
360:
354:
180:
172:
166:
156:
151:
138:
132:
6909:Common integrals in quantum field theory
4226:The maximum process of a càdlàg process
2922:
824:
332:
311:. In this situation, the condition that
220:
8297:Definitions of mathematical integration
6819:Differentiation under the integral sign
6600:Rogers, Chris; Williams, David (2000),
6459:Brownian Motion and Stochastic Calculus
6395:Cohen, Samuel; Elliott, Robert (2015),
5470:
5435:
205:is a locally square-integrable process
7864:Doob's martingale convergence theorems
5166:, to get the integral with respect to
4518:is a bounded predictable process then
4313:, there exist positive constants
4289:then this is the same as the space of
7616:Constant elasticity of variance (CEV)
7606:Chan–Karolyi–Longstaff–Sanders (CKLS)
5187:is used to show that a decomposition
1157:is any predictable process such that
453:, which is also sometimes written as
7:
6397:Stochastic Calculus and Applications
2902:-integrable processes is denoted by
2801:In general, the stochastic integral
2151:is itself an Itô process satisfying
112:). It has important applications in
5395:Burkholder–Davis–Gundy inequalities
4304:Burkholder–Davis–Gundy inequalities
2708:for a locally bounded process
2102:as described above, it states that
8103:Skorokhod's representation theorem
7884:Law of large numbers (weak/strong)
6431:. Fifth edition available online:
6249:
6231:
6089:
6075:
6061:
6009:
6001:
5593:
5067:
4651:
3987:exists and is a local martingale.
3926:
3499:Itô's lemma is the version of the
2666:martingale representation theorems
2514:
2055:
1796:
1329:
1141:martingale representation theorems
985:
805:{\displaystyle {\mathcal {F}}_{t}}
750:{\displaystyle {\mathcal {F}}_{t}}
708:{\displaystyle {\mathcal {F}}_{t}}
671:{\displaystyle {\mathcal {F}}_{t}}
620:{\displaystyle {\mathcal {F}}_{t}}
519:
25:
8073:Martingale representation theorem
6582:Revuz, Daniel; Yor, Marc (1999),
6443:, Science Press, CRC Press Inc.,
6373:Stochastic Integration With Jumps
5679:stochastic differential equations
5406:is càdlàg, adapted, and the set {
4501:. These are used to show that if
4495:for all càdlàg local martingales
3999:martingales, which is the set of
2662:stochastic differential equations
118:stochastic differential equations
8118:Stochastic differential equation
8008:Doob's optional stopping theorem
8003:Doob–Meyer decomposition theorem
6313:
5669:, as in deterministic calculus.
5641:with respect to Brownian motion
5185:Doob–Meyer decomposition theorem
5122:plus a finite variation process
4777:
1975:be Lebesgue integrable, so that
1132:It can be shown that this limit
875:and locally bounded process. If
7988:Convergence of random variables
7874:Fisher–Tippett–Gnedenko theorem
5916:Einstein's summation convention
5450:Differentiation in Itô calculus
5233:predictable quadratic variation
1139:For some applications, such as
678:-measurable. A Brownian motion
7586:Binomial options pricing model
5950:
5937:
5901:
5875:
5850:
5837:
5823:
5810:
5278:
5258:
5231:, which is referred to as the
5140:and combined using linearity,
4924:
4904:
4828:
4802:
4629:
4619:
4612:
4596:
4566:
4551:
4538:
4535:
4461:
4454:
4424:
4405:
4368:
4361:
4126:
4120:
4064:
4044:
3956:is always a local martingale.
3920:
3914:
3756:
3729:
3722:
3709:
3631:
3618:
3581:
3568:
3455:
3442:
3258:
3252:
3233:
3221:
3093:
3084:
3078:
3066:
3013:be predictable processes, and
2937:process. Furthermore, it is a
2753:
2637:
2591:
2511:
2373:Semimartingales as integrators
2309:
2296:
2258:
2245:
2197:
2184:
2144:{\displaystyle Y_{t}=f(X_{t})}
2138:
2125:
2037:
2026:
1783:
1779:
1764:
1742:
1326:
1267:
1251:
1231:
1114:
1068:
1027:
995:
982:
576:
553:
535:
516:
1:
8053:Kolmogorov continuity theorem
7889:Law of the iterated logarithm
5790:is Gaussian white noise with
5096:For a general semimartingale
4743:-measurable random variables
3991:Square integrable martingales
3200:are semimartingales then any
2933:The stochastic integral is a
2677:dominated convergence theorem
2096:on the reals and Itô process
8058:Kolmogorov extension theorem
7737:Generalized queueing network
7245:Interacting particle systems
4873:with zero mean and variance
4749:, for which the integral is
4295:-integrable martingales, by
3511:-dimensional semimartingale
3264:{\displaystyle =H^{2}\cdot }
2817:is not locally bounded. If
901:{\displaystyle \{\pi _{n}\}}
7190:Continuous-time random walk
6724:Lebesgue–Stieltjes integral
6564:Protter, Philip E. (2004),
5673:Itô calculus for physicists
5617:almost surely, and for all
5172:Lebesgue–Stieltjes integral
5008:continuous linear extension
4869:, the property that it has
3878:-integrable if and only if
3279:As with ordinary calculus,
3053:-integrable, in which case
2854:-integrable, with integral
2071:Such predictable processes
857:(Brownian motion) and that
8328:
8198:Extreme value theory (EVT)
7998:Doob decomposition theorem
7290:Ornstein–Uhlenbeck process
7061:Chinese restaurant process
6739:Riemann–Stieltjes integral
6699:Henstock–Kurzweil integral
6568:(2nd ed.), Springer,
6507:10.1103/PhysRevE.76.011123
6461:(2nd ed.), Springer,
6379:Cambridge University Press
5956:{\displaystyle y=y(x_{k})}
5633:Rogers & Williams 2000
5505:square integrable process
4306:state that, for any given
3828:is a local martingale and
3492:
3479:Riemann–Stieltjes integral
3160:-integrable process. then
3041:integrable if and only if
2964:, and is often denoted by
2927:Rogers & Williams 2000
837:Riemann–Stieltjes integral
507:filtered probability space
469:Rogers & Williams 2000
223:, Chapter IV), which is a
125:Riemann–Stieltjes integral
8266:
8078:Optional stopping theorem
7879:Large deviation principle
7631:Heath–Jarrow–Morton (HJM)
7568:Moving-average (MA) model
7553:Autoregressive (AR) model
7378:Hidden Markov model (HMM)
7312:Schramm–Loewner evolution
6978:Proof that 22/7 exceeds π
6369:Bichteler, Klaus (2002),
5501:, then there is a unique
5477:Martingale representation
5089:-integrable processes by
4692:Existence of the integral
3555:is a semimartingale and,
3299:are semimartingales then
2468:with mesh going to zero,
484:, which is equivalent to
294:geometric Brownian motion
96:, extends the methods of
7993:Doléans-Dade exponential
7823:Progressively measurable
7621:Cox–Ingersoll–Ross (CIR)
5783:{\displaystyle \xi _{j}}
5116:into a local martingale
1134:converges in probability
8213:Mathematical statistics
8203:Large deviations theory
8033:Infinitesimal generator
7894:Maximal ergodic theorem
7813:Piecewise-deterministic
7415:Random dynamical system
7280:Markov additive process
6963:Euler–Maclaurin formula
6521:Nualart, David (2006),
6301:Lau & Lubensky 2007
5518:{\displaystyle \alpha }
4212:is finite for all
4182:-Integrable martingales
3801:and in other ways (see
2660:, and for the study of
1720:) integrable. That is,
684:is understood to be an
51:) of a Brownian motion
8048:Karhunen–Loève theorem
7983:Cameron–Martin formula
7947:Burkholder–Davis–Gundy
7342:Variance gamma process
6932:Russo–Vallois integral
6899:Bose–Einstein integral
6814:Parametric derivatives
6287:
6138:
5957:
5908:
5784:
5757:
5611:
5519:
5367:
5077:
5000:
4871:independent increments
4863:For a Brownian motion
4838:
4658:
4489:
4141:
3936:
3860:is not continuous. If
3809:Martingale integrators
3795:
3772:
3692:
3607:
3481:but has an additional
3468:
3265:
3106:
2974:. With this notation,
2788:
2647:
2343:
2145:
2065:
1949:
1803:
1696:
1571:
1514:
1370:
1277:
1124:
902:
806:
751:
709:
672:
621:
586:
443:
227:or, more generally, a
193:
86:
8178:Actuarial mathematics
8140:Uniform integrability
8135:Stratonovich integral
8063:Lévy–Prokhorov metric
7967:Marcinkiewicz–Zygmund
7854:Central limit theorem
7456:Gaussian random field
7285:McKean–Vlasov process
7205:Dyson Brownian motion
7066:Galton–Watson process
6937:Stratonovich integral
6883:Fermi–Dirac integral
6839:Numerical integration
6345:Stratonovich integral
6288:
6139:
5963:is a function of the
5958:
5909:
5785:
5758:
5612:
5520:
5440:Khintchine inequality
5368:
5078:
5001:
4839:
4659:
4490:
4142:
3937:
3796:
3773:
3666:
3587:
3488:
3469:
3285:quadratic covariation
3266:
3116:Dominated convergence
3107:
2789:
2648:
2344:
2146:
2066:
1950:
1804:
1697:
1537:
1515:
1371:
1278:
1178:then the integral of
1149:predictable processes
1125:
903:
807:
757:-measurable and that
752:
710:
673:
622:
587:
444:
307:of the stock at time
194:
33:
8307:Mathematical finance
8253:Time series analysis
8208:Mathematical finance
8093:Reflection principle
7420:Regenerative process
7220:Fleming–Viot process
7035:Stochastic processes
6919:Stochastic integrals
6586:, Berlin: Springer,
6550:, Berlin: Springer,
6154:
5980:
5925:
5794:
5767:
5689:
5677:In physics, usually
5537:
5509:
5458:Malliavin derivative
5393:, or the use of the
5245:
5211:is a martingale and
5102:, the decomposition
5014:
4891:
4753:
4522:
4345:
4325:that depend on
4031:
3882:
3782:
3559:
3303:
3281:integration by parts
3275:Integration by parts
3218:
3057:
2923:Revuz & Yor 1999
2796:monotone class lemma
2718:
2472:
2155:
2106:
1979:
1817:
1724:
1586:
1578:is defined to be an
1383:
1287:
1213:
1190:can be defined, and
943:
879:
841:limit in probability
825:Revuz & Yor 1999
785:
730:
688:
651:
600:
513:
353:
333:Revuz & Yor 1999
290:mathematical finance
221:Revuz & Yor 1999
131:
114:mathematical finance
102:stochastic processes
8302:Stochastic calculus
8248:Stochastic analysis
8088:Quadratic variation
8083:Prokhorov's theorem
8018:Feynman–Kac formula
7488:Markov random field
7136:Birth–death process
6829:Contour integration
6719:Kolmogorov integral
6499:2007PhRvE..76a1123L
6330:Stochastic calculus
5580:
5334:
5319:
5041:
4980:
4965:
4728:for stopping times
4664:and, consequently,
4564:
4479:
4422:
4386:
4297:Doob's inequalities
4105:
3899:
3652:
3505:change of variables
3483:quadratic variation
3411:
3366:
3025:-integrable. Then,
2770:
2735:
2679:holds. That is, if
2489:
2367:quadratic variation
2353:change of variables
2275:
1996:
1914:
1862:
1834:
1759:
1741:
1671:
1629:
1497:
1482:
1415:
1348:
1304:
1230:
960:
411:
383:
283:quadratic variation
279:change of variables
161:
8218:Probability theory
8098:Skorokhod integral
8068:Malliavin calculus
7651:Korn-Kreer-Lenssen
7535:Time series models
7498:Pitman–Yor process
6942:Skorokhod integral
6879:Dirichlet integral
6866:Improper integrals
6809:Reduction formulas
6744:Regulated integral
6709:Hellinger integral
6544:Øksendal, Bernt K.
6321:Mathematics portal
6283:
6134:
5953:
5904:
5780:
5753:
5683:Langevin equations
5607:
5566:
5515:
5463:Malliavin calculus
5363:
5320:
5305:
5073:
5027:
4996:
4966:
4951:
4834:
4699:simple predictable
4654:
4550:
4512:is integrable and
4485:
4460:
4408:
4367:
4278:is finite for all
4137:
4091:
4013:is finite for all
3975:locally integrable
3932:
3885:
3794:{\displaystyle f,}
3791:
3768:
3638:
3464:
3397:
3352:
3261:
3102:
2784:
2756:
2721:
2658:Girsanov's theorem
2643:
2567:
2518:
2475:
2339:
2261:
2230:
2141:
2061:
1982:
1945:
1900:
1848:
1820:
1799:
1745:
1727:
1692:
1657:
1615:
1572:
1510:
1483:
1468:
1401:
1366:
1334:
1333:
1290:
1273:
1216:
1120:
1044:
989:
946:
898:
802:
781:is independent of
747:
705:
668:
617:
582:
439:
397:
369:
349:defined before as
189:
147:
87:
8312:Integral calculus
8284:
8283:
8238:Signal processing
7957:Doob's upcrossing
7952:Doob's martingale
7916:Engelbert–Schmidt
7859:Donsker's theorem
7793:Feller-continuous
7661:Rendleman–Bartter
7451:Dirichlet process
7368:Branching process
7337:Telegraph process
7230:Geometric process
7210:Empirical process
7200:Diffusion process
7056:Branching process
7051:Bernoulli process
7001:
7000:
6904:Frullani integral
6874:Gaussian integral
6824:Laplace transform
6799:Inverse functions
6789:Partial fractions
6714:Khinchin integral
6674:Lebesgue integral
6410:978-1-4939-2867-5
6265:
6226:
6167:
6103:
6055:
6035:
6023:
5992:
5702:
4640:
4477:
4384:
3997:square integrable
3814:Local martingales
3664:
3544:, it states that
2868:, if and only if
2822:= 1 / (1 + |
2519:
2503:
2456:be a sequence of
2229:
1520:which holds when
1318:
990:
974:
908:is a sequence of
16:(Redirected from
8319:
8258:Machine learning
8145:Usual hypotheses
8028:Girsanov theorem
8013:Dynkin's formula
7778:Continuous paths
7686:Actuarial models
7626:Garman–Kohlhagen
7596:Black–Karasinski
7591:Black–Derman–Toy
7578:Financial models
7444:Fields and other
7373:Gaussian process
7322:Sigma-martingale
7126:Additive process
7028:
7021:
7014:
7005:
6849:Trapezoidal rule
6834:Laplace's method
6734:Pfeffer integral
6694:Darboux integral
6689:Daniell integral
6684:Bochner integral
6679:Burkill integral
6669:Riemann integral
6646:
6639:
6632:
6623:
6614:
6596:
6578:
6560:
6539:
6528:
6517:
6492:
6471:
6453:
6413:
6399:(2nd ed.),
6391:
6377:(1st ed.),
6376:
6323:
6318:
6317:
6292:
6290:
6289:
6284:
6279:
6278:
6266:
6264:
6263:
6262:
6261:
6247:
6246:
6245:
6229:
6227:
6219:
6214:
6213:
6204:
6203:
6188:
6187:
6175:
6174:
6169:
6168:
6160:
6148:Stratonovich SDE
6143:
6141:
6140:
6135:
6130:
6129:
6117:
6116:
6104:
6102:
6101:
6100:
6087:
6086:
6073:
6069:
6068:
6058:
6056:
6048:
6043:
6042:
6037:
6036:
6028:
6024:
6022:
6021:
6020:
6007:
5999:
5994:
5993:
5985:
5976:has to be used:
5971:
5962:
5960:
5959:
5954:
5949:
5948:
5913:
5911:
5910:
5905:
5900:
5899:
5887:
5886:
5871:
5870:
5849:
5848:
5836:
5835:
5822:
5821:
5809:
5808:
5789:
5787:
5786:
5781:
5779:
5778:
5762:
5760:
5759:
5754:
5749:
5748:
5739:
5738:
5723:
5722:
5710:
5709:
5704:
5703:
5695:
5681:(SDEs), such as
5668:
5650:
5646:
5640:
5630:
5629:
5616:
5614:
5613:
5608:
5606:
5605:
5596:
5590:
5589:
5579:
5574:
5562:
5561:
5549:
5548:
5532:
5524:
5522:
5521:
5516:
5500:
5494:
5486:
5432:
5426:
5405:
5380:
5372:
5370:
5369:
5364:
5359:
5355:
5354:
5353:
5333:
5328:
5318:
5313:
5299:
5291:
5287:
5286:
5285:
5276:
5275:
5252:
5240:
5230:
5229:
5220:
5219:
5210:
5204:
5203:
5182:
5165:
5139:
5133:
5127:
5121:
5115:
5101:
5088:
5082:
5080:
5079:
5074:
5063:
5059:
5051:
5050:
5040:
5035:
5021:
5005:
5003:
5002:
4997:
4992:
4988:
4979:
4974:
4964:
4959:
4945:
4937:
4933:
4932:
4931:
4922:
4921:
4898:
4886:
4868:
4859:
4853:
4843:
4841:
4840:
4835:
4827:
4826:
4814:
4813:
4798:
4797:
4780:
4771:
4770:
4748:
4742:
4733:
4727:
4687:
4679:
4673:
4663:
4661:
4660:
4655:
4647:
4643:
4642:
4641:
4633:
4627:
4626:
4608:
4607:
4590:
4579:
4575:
4574:
4573:
4563:
4558:
4529:
4517:
4511:
4500:
4494:
4492:
4491:
4486:
4484:
4480:
4478:
4470:
4468:
4448:
4437:
4433:
4432:
4431:
4421:
4416:
4399:
4391:
4387:
4385:
4377:
4375:
4355:
4340:
4336:
4330:
4324:
4318:
4312:
4294:
4288:
4281:
4277:
4273:
4267:
4260:
4258:
4231:
4222:
4215:
4211:
4209:
4198:
4192:
4174:
4164:
4152:
4146:
4144:
4143:
4138:
4133:
4129:
4115:
4114:
4104:
4099:
4085:
4077:
4073:
4072:
4071:
4062:
4061:
4038:
4022:
4016:
4012:
4008:
3986:
3972:
3964:
3955:
3945:
3941:
3939:
3938:
3933:
3909:
3908:
3898:
3893:
3877:
3871:
3865:
3859:
3853:
3843:
3833:
3827:
3820:local martingale
3800:
3798:
3797:
3792:
3777:
3775:
3774:
3769:
3764:
3763:
3754:
3753:
3741:
3740:
3721:
3720:
3708:
3707:
3691:
3686:
3665:
3657:
3651:
3646:
3630:
3629:
3617:
3616:
3606:
3601:
3580:
3579:
3554:
3543:
3537:
3531:
3525:
3510:
3473:
3471:
3470:
3465:
3463:
3462:
3438:
3437:
3424:
3423:
3410:
3405:
3393:
3392:
3379:
3378:
3365:
3360:
3348:
3347:
3338:
3337:
3325:
3324:
3315:
3314:
3298:
3292:
3270:
3268:
3267:
3262:
3248:
3247:
3213:
3205:
3199:
3193:
3184:
3180:
3159:
3153:
3147:
3142:
3131:
3111:
3109:
3108:
3103:
3052:
3046:
3040:
3030:
3024:
3018:
3012:
3006:
2992:
2973:
2963:
2947:
2912:
2901:
2895:
2877:
2867:
2853:
2847:
2841:
2835:
2829:
2827:
2816:
2810:
2793:
2791:
2790:
2785:
2769:
2764:
2745:
2744:
2734:
2729:
2713:
2707:
2702:
2691:
2652:
2650:
2649:
2644:
2636:
2635:
2634:
2633:
2610:
2609:
2608:
2607:
2590:
2589:
2588:
2587:
2566:
2565:
2564:
2552:
2551:
2539:
2538:
2517:
2488:
2483:
2467:
2455:
2446:
2436:
2418:
2411:finite variation
2408:
2402:local martingale
2399:
2385:
2364:
2348:
2346:
2345:
2340:
2335:
2334:
2321:
2320:
2308:
2307:
2295:
2294:
2274:
2269:
2257:
2256:
2244:
2243:
2231:
2222:
2209:
2208:
2196:
2195:
2183:
2182:
2170:
2169:
2150:
2148:
2147:
2142:
2137:
2136:
2118:
2117:
2101:
2095:
2082:
2076:
2070:
2068:
2067:
2062:
2045:
2041:
2040:
2029:
2021:
2020:
2011:
2010:
1995:
1990:
1974:
1969:-integrable and
1968:
1962:
1954:
1952:
1951:
1946:
1934:
1933:
1924:
1923:
1913:
1908:
1896:
1895:
1882:
1881:
1872:
1871:
1861:
1856:
1833:
1828:
1812:
1808:
1806:
1805:
1800:
1782:
1777:
1776:
1767:
1758:
1753:
1740:
1735:
1715:
1709:
1701:
1699:
1698:
1693:
1681:
1680:
1670:
1665:
1653:
1652:
1639:
1638:
1628:
1623:
1611:
1610:
1598:
1597:
1565:
1559:
1544:
1525:
1519:
1517:
1516:
1511:
1509:
1505:
1496:
1491:
1481:
1476:
1462:
1454:
1450:
1449:
1444:
1440:
1439:
1438:
1425:
1424:
1414:
1409:
1390:
1375:
1373:
1372:
1367:
1358:
1357:
1347:
1342:
1332:
1303:
1298:
1282:
1280:
1279:
1274:
1259:
1258:
1249:
1248:
1229:
1224:
1201:
1195:
1189:
1184:with respect to
1183:
1177:
1170:
1156:
1129:
1127:
1126:
1121:
1113:
1112:
1111:
1110:
1087:
1086:
1085:
1084:
1067:
1066:
1065:
1064:
1043:
1042:
1041:
1026:
1025:
1013:
1012:
988:
959:
954:
935:
931:
926:with respect to
925:
919:
907:
905:
904:
899:
894:
893:
865:right-continuous
862:
852:
822:
811:
809:
808:
803:
801:
800:
795:
794:
780:
756:
754:
753:
748:
746:
745:
740:
739:
725:
714:
712:
711:
706:
704:
703:
698:
697:
683:
677:
675:
674:
669:
667:
666:
661:
660:
646:
637:
632:, and a process
631:
626:
624:
623:
618:
616:
615:
610:
609:
591:
589:
588:
583:
575:
567:
566:
551:
550:
545:
544:
531:
530:
504:
483:
466:
448:
446:
445:
440:
435:
434:
421:
420:
410:
405:
382:
377:
365:
364:
348:
326:
316:
260:
256:
248:
234:
218:
204:
198:
196:
195:
190:
185:
184:
171:
170:
160:
155:
143:
142:
84:
60:
56:
50:
21:
8327:
8326:
8322:
8321:
8320:
8318:
8317:
8316:
8287:
8286:
8285:
8280:
8262:
8223:Queueing theory
8166:
8108:Skorokhod space
7971:
7962:Kunita–Watanabe
7933:
7899:Sanov's theorem
7869:Ergodic theorem
7842:
7838:Time-reversible
7756:
7719:Queueing models
7713:
7709:Sparre–Anderson
7699:Cramér–Lundberg
7680:
7666:SABR volatility
7572:
7529:
7481:Boolean network
7439:
7425:Renewal process
7356:
7305:Non-homogeneous
7295:Poisson process
7185:Contact process
7148:Brownian motion
7118:Continuous time
7112:
7106:Maximal entropy
7037:
7032:
7002:
6997:
6973:Integration Bee
6946:
6913:
6860:
6856:Risch algorithm
6794:Euler's formula
6754:
6748:
6729:Pettis integral
6661:
6655:
6650:
6620:
6612:
6599:
6594:
6581:
6576:
6563:
6558:
6542:
6537:
6520:
6474:
6469:
6456:
6451:
6438:
6411:
6394:
6389:
6368:
6364:
6359:
6319:
6312:
6309:
6267:
6253:
6248:
6234:
6230:
6205:
6192:
6179:
6157:
6152:
6151:
6118:
6105:
6092:
6078:
6074:
6060:
6059:
6025:
6012:
6008:
6000:
5978:
5977:
5969:
5964:
5940:
5923:
5922:
5891:
5878:
5859:
5840:
5827:
5813:
5800:
5792:
5791:
5770:
5765:
5764:
5740:
5727:
5714:
5692:
5687:
5686:
5675:
5667:
5660:
5652:
5648:
5642:
5636:
5623:
5618:
5597:
5581:
5553:
5540:
5535:
5534:
5526:
5507:
5506:
5496:
5488:
5482:
5479:
5460:
5452:
5428:
5422:
5415:
5401:
5387:Doléans measure
5375:
5345:
5304:
5300:
5277:
5267:
5257:
5253:
5243:
5242:
5236:
5223:
5222:
5213:
5212:
5206:
5197:
5188:
5178:
5170:. The standard
5141:
5135:
5129:
5123:
5117:
5103:
5097:
5084:
5042:
5026:
5022:
5012:
5011:
4950:
4946:
4923:
4913:
4903:
4899:
4889:
4888:
4880:
4874:
4864:
4855:
4845:
4818:
4805:
4775:
4762:
4751:
4750:
4744:
4740:
4735:
4729:
4726:
4707:
4702:
4694:
4688:is integrable.
4681:
4675:
4665:
4628:
4618:
4599:
4595:
4591:
4565:
4534:
4530:
4520:
4519:
4513:
4508:
4502:
4496:
4453:
4449:
4423:
4404:
4400:
4360:
4356:
4343:
4342:
4338:
4332:
4326:
4320:
4314:
4307:
4290:
4283:
4279:
4275:
4269:
4262:
4256:
4251:
4249:
4238:
4233:
4227:
4217:
4213:
4207:
4202:
4200:
4194:
4187:
4184:
4166:
4163:
4154:
4148:
4106:
4090:
4086:
4063:
4053:
4043:
4039:
4029:
4028:
4018:
4014:
4010:
4004:
3993:
3978:
3966:
3960:
3947:
3943:
3900:
3880:
3879:
3873:
3867:
3861:
3855:
3845:
3835:
3829:
3823:
3816:
3811:
3780:
3779:
3755:
3745:
3732:
3712:
3693:
3621:
3608:
3571:
3557:
3556:
3545:
3539:
3533:
3527:
3512:
3508:
3497:
3491:
3454:
3429:
3412:
3384:
3367:
3339:
3329:
3316:
3306:
3301:
3300:
3294:
3288:
3277:
3239:
3216:
3215:
3207:
3201:
3195:
3189:
3182:
3166:
3161:
3155:
3149:
3140:
3135:
3133:
3125:
3120:
3119:. Suppose that
3055:
3054:
3048:
3042:
3032:
3026:
3020:
3014:
3008:
3002:
2975:
2971:
2965:
2962:
2954:
2949:
2945:
2919:
2903:
2897:
2879:
2875:
2869:
2855:
2849:
2843:
2837:
2831:
2823:
2818:
2812:
2802:
2736:
2716:
2715:
2709:
2700:
2695:
2693:
2685:
2680:
2619:
2614:
2599:
2594:
2573:
2568:
2556:
2543:
2524:
2470:
2469:
2461:
2454:
2448:
2438:
2432:
2421:Brownian motion
2414:
2404:
2387:
2381:
2375:
2360:
2326:
2312:
2299:
2286:
2248:
2232:
2200:
2187:
2174:
2161:
2153:
2152:
2128:
2109:
2104:
2103:
2097:
2091:
2078:
2072:
2012:
2002:
2001:
1997:
1977:
1976:
1970:
1964:
1958:
1925:
1915:
1887:
1873:
1863:
1815:
1814:
1810:
1768:
1722:
1721:
1711:
1705:
1672:
1644:
1630:
1602:
1589:
1584:
1583:
1561:
1546:
1539:
1532:
1521:
1467:
1463:
1430:
1416:
1400:
1396:
1395:
1391:
1381:
1380:
1349:
1285:
1284:
1250:
1240:
1211:
1210:
1207:
1197:
1191:
1185:
1179:
1172:
1162:
1158:
1152:
1096:
1091:
1076:
1071:
1050:
1045:
1033:
1017:
998:
941:
940:
938:random variable
933:
927:
921:
913:
885:
877:
876:
858:
848:
839:, that is as a
833:
813:
788:
783:
782:
779:
770:
758:
733:
728:
727:
724:
716:
691:
686:
685:
679:
654:
649:
648:
644:
639:
633:
629:
603:
598:
597:
552:
538:
511:
510:
495:
485:
472:
454:
426:
412:
356:
351:
350:
344:
341:
335:, Chapter IV).
322:
312:
305:
261:and constructs
258:
254:
244:
237:random variable
232:
225:Brownian motion
214:
200:
176:
162:
134:
129:
128:
106:Brownian motion
70:
62:
58:
52:
48:
42:
28:
23:
22:
15:
12:
11:
5:
8325:
8323:
8315:
8314:
8309:
8304:
8299:
8289:
8288:
8282:
8281:
8279:
8278:
8273:
8271:List of topics
8267:
8264:
8263:
8261:
8260:
8255:
8250:
8245:
8240:
8235:
8230:
8228:Renewal theory
8225:
8220:
8215:
8210:
8205:
8200:
8195:
8193:Ergodic theory
8190:
8185:
8183:Control theory
8180:
8174:
8172:
8168:
8167:
8165:
8164:
8163:
8162:
8157:
8147:
8142:
8137:
8132:
8127:
8126:
8125:
8115:
8113:Snell envelope
8110:
8105:
8100:
8095:
8090:
8085:
8080:
8075:
8070:
8065:
8060:
8055:
8050:
8045:
8040:
8035:
8030:
8025:
8020:
8015:
8010:
8005:
8000:
7995:
7990:
7985:
7979:
7977:
7973:
7972:
7970:
7969:
7964:
7959:
7954:
7949:
7943:
7941:
7935:
7934:
7932:
7931:
7912:Borel–Cantelli
7901:
7896:
7891:
7886:
7881:
7876:
7871:
7866:
7861:
7856:
7850:
7848:
7847:Limit theorems
7844:
7843:
7841:
7840:
7835:
7830:
7825:
7820:
7815:
7810:
7805:
7800:
7795:
7790:
7785:
7780:
7775:
7770:
7764:
7762:
7758:
7757:
7755:
7754:
7749:
7744:
7739:
7734:
7729:
7723:
7721:
7715:
7714:
7712:
7711:
7706:
7701:
7696:
7690:
7688:
7682:
7681:
7679:
7678:
7673:
7668:
7663:
7658:
7653:
7648:
7643:
7638:
7633:
7628:
7623:
7618:
7613:
7608:
7603:
7598:
7593:
7588:
7582:
7580:
7574:
7573:
7571:
7570:
7565:
7560:
7555:
7550:
7545:
7539:
7537:
7531:
7530:
7528:
7527:
7522:
7517:
7516:
7515:
7510:
7500:
7495:
7490:
7485:
7484:
7483:
7478:
7468:
7466:Hopfield model
7463:
7458:
7453:
7447:
7445:
7441:
7440:
7438:
7437:
7432:
7427:
7422:
7417:
7412:
7411:
7410:
7405:
7400:
7395:
7385:
7383:Markov process
7380:
7375:
7370:
7364:
7362:
7358:
7357:
7355:
7354:
7352:Wiener sausage
7349:
7347:Wiener process
7344:
7339:
7334:
7329:
7327:Stable process
7324:
7319:
7317:Semimartingale
7314:
7309:
7308:
7307:
7302:
7292:
7287:
7282:
7277:
7272:
7267:
7262:
7260:Jump diffusion
7257:
7252:
7247:
7242:
7237:
7235:Hawkes process
7232:
7227:
7222:
7217:
7215:Feller process
7212:
7207:
7202:
7197:
7192:
7187:
7182:
7180:Cauchy process
7177:
7176:
7175:
7170:
7165:
7160:
7155:
7145:
7144:
7143:
7133:
7131:Bessel process
7128:
7122:
7120:
7114:
7113:
7111:
7110:
7109:
7108:
7103:
7098:
7093:
7083:
7078:
7073:
7068:
7063:
7058:
7053:
7047:
7045:
7039:
7038:
7033:
7031:
7030:
7023:
7016:
7008:
6999:
6998:
6996:
6995:
6994:
6993:
6988:
6980:
6975:
6970:
6968:Gabriel's horn
6965:
6960:
6954:
6952:
6948:
6947:
6945:
6944:
6939:
6934:
6929:
6923:
6921:
6915:
6914:
6912:
6911:
6906:
6901:
6896:
6895:
6894:
6889:
6881:
6876:
6870:
6868:
6862:
6861:
6859:
6858:
6853:
6852:
6851:
6846:
6844:Simpson's rule
6836:
6831:
6826:
6821:
6816:
6811:
6806:
6804:Changing order
6801:
6796:
6791:
6786:
6781:
6780:
6779:
6774:
6769:
6758:
6756:
6750:
6749:
6747:
6746:
6741:
6736:
6731:
6726:
6721:
6716:
6711:
6706:
6701:
6696:
6691:
6686:
6681:
6676:
6671:
6665:
6663:
6657:
6656:
6651:
6649:
6648:
6641:
6634:
6626:
6619:
6618:
6615:
6610:
6597:
6592:
6579:
6574:
6561:
6556:
6540:
6535:
6518:
6472:
6467:
6454:
6450:978-0849377150
6449:
6436:
6417:Hagen Kleinert
6414:
6409:
6392:
6387:
6365:
6363:
6360:
6358:
6357:
6355:Wiener process
6352:
6350:Semimartingale
6347:
6342:
6337:
6332:
6326:
6325:
6324:
6308:
6305:
6282:
6277:
6274:
6270:
6260:
6256:
6251:
6244:
6241:
6237:
6233:
6225:
6222:
6217:
6212:
6208:
6202:
6199:
6195:
6191:
6186:
6182:
6178:
6173:
6166:
6163:
6133:
6128:
6125:
6121:
6115:
6112:
6108:
6099:
6095:
6091:
6085:
6081:
6077:
6072:
6067:
6063:
6054:
6051:
6046:
6041:
6034:
6031:
6019:
6015:
6011:
6006:
6003:
5997:
5991:
5988:
5967:
5952:
5947:
5943:
5939:
5936:
5933:
5930:
5903:
5898:
5894:
5890:
5885:
5881:
5877:
5874:
5869:
5866:
5862:
5858:
5855:
5852:
5847:
5843:
5839:
5834:
5830:
5825:
5820:
5816:
5812:
5807:
5803:
5799:
5777:
5773:
5752:
5747:
5743:
5737:
5734:
5730:
5726:
5721:
5717:
5713:
5708:
5701:
5698:
5674:
5671:
5665:
5656:
5604:
5600:
5595:
5588:
5584:
5578:
5573:
5569:
5565:
5560:
5556:
5552:
5547:
5543:
5514:
5478:
5475:
5459:
5456:
5451:
5448:
5444:Bichteler 2002
5413:
5391:submartingales
5362:
5358:
5352:
5348:
5344:
5341:
5338:
5332:
5327:
5323:
5317:
5312:
5308:
5303:
5298:
5294:
5290:
5284:
5280:
5274:
5270:
5266:
5263:
5260:
5256:
5251:
5205:exists, where
5072:
5069:
5066:
5062:
5058:
5055:
5049:
5045:
5039:
5034:
5030:
5025:
5020:
4995:
4991:
4987:
4984:
4978:
4973:
4969:
4963:
4958:
4954:
4949:
4944:
4940:
4936:
4930:
4926:
4920:
4916:
4912:
4909:
4906:
4902:
4897:
4878:
4833:
4830:
4825:
4821:
4817:
4812:
4808:
4804:
4801:
4796:
4793:
4790:
4787:
4784:
4779:
4774:
4769:
4765:
4761:
4758:
4738:
4716:
4705:
4693:
4690:
4653:
4650:
4646:
4639:
4636:
4631:
4625:
4621:
4617:
4614:
4611:
4606:
4602:
4598:
4594:
4589:
4585:
4582:
4578:
4572:
4568:
4562:
4557:
4553:
4549:
4546:
4543:
4540:
4537:
4533:
4528:
4506:
4483:
4476:
4473:
4467:
4463:
4459:
4456:
4452:
4447:
4443:
4440:
4436:
4430:
4426:
4420:
4415:
4411:
4407:
4403:
4398:
4394:
4390:
4383:
4380:
4374:
4370:
4366:
4363:
4359:
4354:
4350:
4254:
4241:
4236:
4232:is written as
4205:
4183:
4177:
4159:
4136:
4132:
4128:
4125:
4122:
4119:
4113:
4109:
4103:
4098:
4094:
4089:
4084:
4080:
4076:
4070:
4066:
4060:
4056:
4052:
4049:
4046:
4042:
4037:
3992:
3989:
3931:
3928:
3925:
3922:
3919:
3916:
3913:
3907:
3903:
3897:
3892:
3888:
3815:
3812:
3810:
3807:
3790:
3787:
3767:
3762:
3758:
3752:
3748:
3744:
3739:
3735:
3731:
3728:
3724:
3719:
3715:
3711:
3706:
3703:
3700:
3696:
3690:
3685:
3682:
3679:
3676:
3673:
3669:
3663:
3660:
3655:
3650:
3645:
3641:
3637:
3633:
3628:
3624:
3620:
3615:
3611:
3605:
3600:
3597:
3594:
3590:
3586:
3583:
3578:
3574:
3570:
3567:
3564:
3493:Main article:
3490:
3487:
3461:
3457:
3453:
3450:
3447:
3444:
3441:
3436:
3432:
3428:
3422:
3419:
3415:
3409:
3404:
3400:
3396:
3391:
3387:
3383:
3377:
3374:
3370:
3364:
3359:
3355:
3351:
3346:
3342:
3336:
3332:
3328:
3323:
3319:
3313:
3309:
3276:
3273:
3272:
3271:
3260:
3257:
3254:
3251:
3246:
3242:
3238:
3235:
3232:
3229:
3226:
3223:
3186:
3164:
3138:
3123:
3112:
3101:
3098:
3095:
3092:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
2994:
2969:
2960:
2952:
2942:
2939:semimartingale
2918:
2915:
2873:
2783:
2780:
2777:
2773:
2768:
2763:
2759:
2755:
2752:
2749:
2743:
2739:
2733:
2728:
2724:
2698:
2683:
2642:
2639:
2632:
2629:
2626:
2622:
2617:
2613:
2606:
2602:
2597:
2593:
2586:
2583:
2580:
2576:
2571:
2563:
2559:
2555:
2550:
2546:
2542:
2537:
2534:
2531:
2527:
2522:
2516:
2513:
2510:
2506:
2502:
2499:
2496:
2492:
2487:
2482:
2478:
2450:
2429:Lévy processes
2379:semimartingale
2374:
2371:
2338:
2333:
2329:
2325:
2319:
2315:
2311:
2306:
2302:
2298:
2293:
2289:
2285:
2282:
2279:
2273:
2268:
2264:
2260:
2255:
2251:
2247:
2242:
2239:
2235:
2228:
2225:
2219:
2216:
2213:
2207:
2203:
2199:
2194:
2190:
2186:
2181:
2177:
2173:
2168:
2164:
2160:
2140:
2135:
2131:
2127:
2124:
2121:
2116:
2112:
2060:
2057:
2054:
2051:
2048:
2044:
2039:
2035:
2032:
2028:
2024:
2019:
2015:
2009:
2005:
2000:
1994:
1989:
1985:
1944:
1941:
1938:
1932:
1928:
1922:
1918:
1912:
1907:
1903:
1899:
1894:
1890:
1886:
1880:
1876:
1870:
1866:
1860:
1855:
1851:
1847:
1844:
1841:
1837:
1832:
1827:
1823:
1798:
1795:
1792:
1789:
1785:
1781:
1775:
1771:
1766:
1762:
1757:
1752:
1748:
1744:
1739:
1734:
1730:
1691:
1688:
1685:
1679:
1675:
1669:
1664:
1660:
1656:
1651:
1647:
1643:
1637:
1633:
1627:
1622:
1618:
1614:
1609:
1605:
1601:
1596:
1592:
1568:Ricker wavelet
1531:
1528:
1508:
1504:
1501:
1495:
1490:
1486:
1480:
1475:
1471:
1466:
1461:
1457:
1453:
1448:
1443:
1437:
1433:
1429:
1423:
1419:
1413:
1408:
1404:
1399:
1394:
1389:
1365:
1362:
1356:
1352:
1346:
1341:
1337:
1331:
1328:
1325:
1321:
1317:
1314:
1311:
1307:
1302:
1297:
1293:
1272:
1269:
1266:
1263:
1257:
1253:
1247:
1243:
1239:
1236:
1233:
1228:
1223:
1219:
1205:
1196:is said to be
1160:
1119:
1116:
1109:
1106:
1103:
1099:
1094:
1090:
1083:
1079:
1074:
1070:
1063:
1060:
1057:
1053:
1048:
1040:
1036:
1032:
1029:
1024:
1020:
1016:
1011:
1008:
1005:
1001:
997:
993:
987:
984:
981:
977:
973:
970:
967:
963:
958:
953:
949:
897:
892:
888:
884:
855:Wiener process
832:
829:
799:
793:
775:
762:
744:
738:
720:
702:
696:
665:
659:
642:
638:is adapted if
614:
608:
581:
578:
574:
570:
565:
562:
559:
555:
549:
543:
537:
534:
529:
524:
521:
518:
493:
438:
433:
429:
425:
419:
415:
409:
404:
400:
396:
393:
390:
386:
381:
376:
372:
368:
363:
359:
340:
337:
303:
229:semimartingale
188:
183:
179:
175:
169:
165:
159:
154:
150:
146:
141:
137:
110:Wiener process
92:, named after
66:
38:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8324:
8313:
8310:
8308:
8305:
8303:
8300:
8298:
8295:
8294:
8292:
8277:
8274:
8272:
8269:
8268:
8265:
8259:
8256:
8254:
8251:
8249:
8246:
8244:
8241:
8239:
8236:
8234:
8231:
8229:
8226:
8224:
8221:
8219:
8216:
8214:
8211:
8209:
8206:
8204:
8201:
8199:
8196:
8194:
8191:
8189:
8186:
8184:
8181:
8179:
8176:
8175:
8173:
8169:
8161:
8158:
8156:
8153:
8152:
8151:
8148:
8146:
8143:
8141:
8138:
8136:
8133:
8131:
8130:Stopping time
8128:
8124:
8121:
8120:
8119:
8116:
8114:
8111:
8109:
8106:
8104:
8101:
8099:
8096:
8094:
8091:
8089:
8086:
8084:
8081:
8079:
8076:
8074:
8071:
8069:
8066:
8064:
8061:
8059:
8056:
8054:
8051:
8049:
8046:
8044:
8041:
8039:
8036:
8034:
8031:
8029:
8026:
8024:
8021:
8019:
8016:
8014:
8011:
8009:
8006:
8004:
8001:
7999:
7996:
7994:
7991:
7989:
7986:
7984:
7981:
7980:
7978:
7974:
7968:
7965:
7963:
7960:
7958:
7955:
7953:
7950:
7948:
7945:
7944:
7942:
7940:
7936:
7929:
7925:
7921:
7920:Hewitt–Savage
7917:
7913:
7909:
7905:
7904:Zero–one laws
7902:
7900:
7897:
7895:
7892:
7890:
7887:
7885:
7882:
7880:
7877:
7875:
7872:
7870:
7867:
7865:
7862:
7860:
7857:
7855:
7852:
7851:
7849:
7845:
7839:
7836:
7834:
7831:
7829:
7826:
7824:
7821:
7819:
7816:
7814:
7811:
7809:
7806:
7804:
7801:
7799:
7796:
7794:
7791:
7789:
7786:
7784:
7781:
7779:
7776:
7774:
7771:
7769:
7766:
7765:
7763:
7759:
7753:
7750:
7748:
7745:
7743:
7740:
7738:
7735:
7733:
7730:
7728:
7725:
7724:
7722:
7720:
7716:
7710:
7707:
7705:
7702:
7700:
7697:
7695:
7692:
7691:
7689:
7687:
7683:
7677:
7674:
7672:
7669:
7667:
7664:
7662:
7659:
7657:
7654:
7652:
7649:
7647:
7644:
7642:
7639:
7637:
7634:
7632:
7629:
7627:
7624:
7622:
7619:
7617:
7614:
7612:
7609:
7607:
7604:
7602:
7601:Black–Scholes
7599:
7597:
7594:
7592:
7589:
7587:
7584:
7583:
7581:
7579:
7575:
7569:
7566:
7564:
7561:
7559:
7556:
7554:
7551:
7549:
7546:
7544:
7541:
7540:
7538:
7536:
7532:
7526:
7523:
7521:
7518:
7514:
7511:
7509:
7506:
7505:
7504:
7503:Point process
7501:
7499:
7496:
7494:
7491:
7489:
7486:
7482:
7479:
7477:
7474:
7473:
7472:
7469:
7467:
7464:
7462:
7461:Gibbs measure
7459:
7457:
7454:
7452:
7449:
7448:
7446:
7442:
7436:
7433:
7431:
7428:
7426:
7423:
7421:
7418:
7416:
7413:
7409:
7406:
7404:
7401:
7399:
7396:
7394:
7391:
7390:
7389:
7386:
7384:
7381:
7379:
7376:
7374:
7371:
7369:
7366:
7365:
7363:
7359:
7353:
7350:
7348:
7345:
7343:
7340:
7338:
7335:
7333:
7330:
7328:
7325:
7323:
7320:
7318:
7315:
7313:
7310:
7306:
7303:
7301:
7298:
7297:
7296:
7293:
7291:
7288:
7286:
7283:
7281:
7278:
7276:
7273:
7271:
7268:
7266:
7263:
7261:
7258:
7256:
7253:
7251:
7250:Itô diffusion
7248:
7246:
7243:
7241:
7238:
7236:
7233:
7231:
7228:
7226:
7225:Gamma process
7223:
7221:
7218:
7216:
7213:
7211:
7208:
7206:
7203:
7201:
7198:
7196:
7193:
7191:
7188:
7186:
7183:
7181:
7178:
7174:
7171:
7169:
7166:
7164:
7161:
7159:
7156:
7154:
7151:
7150:
7149:
7146:
7142:
7139:
7138:
7137:
7134:
7132:
7129:
7127:
7124:
7123:
7121:
7119:
7115:
7107:
7104:
7102:
7099:
7097:
7096:Self-avoiding
7094:
7092:
7089:
7088:
7087:
7084:
7082:
7081:Moran process
7079:
7077:
7074:
7072:
7069:
7067:
7064:
7062:
7059:
7057:
7054:
7052:
7049:
7048:
7046:
7044:
7043:Discrete time
7040:
7036:
7029:
7024:
7022:
7017:
7015:
7010:
7009:
7006:
6992:
6989:
6987:
6984:
6983:
6981:
6979:
6976:
6974:
6971:
6969:
6966:
6964:
6961:
6959:
6958:Basel problem
6956:
6955:
6953:
6951:Miscellaneous
6949:
6943:
6940:
6938:
6935:
6933:
6930:
6928:
6925:
6924:
6922:
6920:
6916:
6910:
6907:
6905:
6902:
6900:
6897:
6893:
6890:
6888:
6885:
6884:
6882:
6880:
6877:
6875:
6872:
6871:
6869:
6867:
6863:
6857:
6854:
6850:
6847:
6845:
6842:
6841:
6840:
6837:
6835:
6832:
6830:
6827:
6825:
6822:
6820:
6817:
6815:
6812:
6810:
6807:
6805:
6802:
6800:
6797:
6795:
6792:
6790:
6787:
6785:
6782:
6778:
6775:
6773:
6770:
6768:
6767:Trigonometric
6765:
6764:
6763:
6760:
6759:
6757:
6751:
6745:
6742:
6740:
6737:
6735:
6732:
6730:
6727:
6725:
6722:
6720:
6717:
6715:
6712:
6710:
6707:
6705:
6704:Haar integral
6702:
6700:
6697:
6695:
6692:
6690:
6687:
6685:
6682:
6680:
6677:
6675:
6672:
6670:
6667:
6666:
6664:
6658:
6654:
6647:
6642:
6640:
6635:
6633:
6628:
6627:
6624:
6616:
6613:
6611:0-521-77593-0
6607:
6603:
6598:
6595:
6593:3-540-57622-3
6589:
6585:
6580:
6577:
6575:3-540-00313-4
6571:
6567:
6562:
6559:
6557:3-540-04758-1
6553:
6549:
6545:
6541:
6538:
6536:3-540-28328-5
6532:
6527:
6526:
6519:
6516:
6512:
6508:
6504:
6500:
6496:
6491:
6486:
6483:(1): 011123,
6482:
6478:
6473:
6470:
6468:0-387-97655-8
6464:
6460:
6455:
6452:
6446:
6442:
6437:
6434:
6430:
6429:981-238-107-4
6426:
6422:
6418:
6415:
6412:
6406:
6402:
6398:
6393:
6390:
6388:0-521-81129-5
6384:
6380:
6375:
6374:
6367:
6366:
6361:
6356:
6353:
6351:
6348:
6346:
6343:
6341:
6340:Otto calculus
6338:
6336:
6333:
6331:
6328:
6327:
6322:
6316:
6311:
6306:
6304:
6302:
6298:
6297:colored noise
6293:
6280:
6275:
6272:
6268:
6258:
6254:
6242:
6239:
6235:
6223:
6220:
6215:
6210:
6206:
6200:
6197:
6193:
6189:
6184:
6180:
6176:
6171:
6164:
6161:
6149:
6144:
6131:
6126:
6123:
6119:
6113:
6110:
6106:
6097:
6093:
6083:
6079:
6070:
6065:
6052:
6049:
6044:
6039:
6032:
6029:
6017:
6013:
6004:
5995:
5989:
5986:
5975:
5970:
5945:
5941:
5934:
5931:
5928:
5919:
5917:
5896:
5892:
5888:
5883:
5879:
5872:
5867:
5864:
5860:
5856:
5845:
5841:
5832:
5828:
5818:
5814:
5805:
5801:
5775:
5771:
5750:
5745:
5741:
5735:
5732:
5728:
5724:
5719:
5715:
5711:
5706:
5699:
5696:
5684:
5680:
5672:
5670:
5664:
5659:
5655:
5645:
5639:
5634:
5627:
5621:
5602:
5598:
5586:
5582:
5576:
5571:
5567:
5563:
5558:
5554:
5550:
5545:
5541:
5530:
5512:
5504:
5499:
5492:
5485:
5476:
5474:
5472:
5468:
5464:
5457:
5455:
5449:
5447:
5445:
5441:
5437:
5431:
5425:
5420:
5416:
5409:
5404:
5398:
5396:
5392:
5388:
5382:
5378:
5360:
5356:
5350:
5342:
5336:
5330:
5325:
5321:
5315:
5310:
5306:
5301:
5292:
5288:
5282:
5272:
5268:
5264:
5261:
5254:
5239:
5234:
5227:
5217:
5209:
5201:
5195:
5191:
5186:
5181:
5175:
5173:
5169:
5164:
5160:
5156:
5152:
5148:
5144:
5138:
5132:
5126:
5120:
5114:
5110:
5106:
5100:
5094:
5092:
5087:
5070:
5064:
5060:
5056:
5053:
5047:
5043:
5037:
5032:
5028:
5023:
5009:
4993:
4989:
4985:
4982:
4976:
4971:
4967:
4961:
4956:
4952:
4947:
4938:
4934:
4928:
4918:
4914:
4910:
4907:
4900:
4885:
4881:
4872:
4867:
4861:
4858:
4852:
4848:
4831:
4823:
4819:
4815:
4810:
4806:
4799:
4791:
4788:
4785:
4772:
4767:
4763:
4759:
4756:
4747:
4741:
4732:
4724:
4720:
4715:
4712:
4708:
4700:
4691:
4689:
4685:
4678:
4672:
4668:
4648:
4644:
4637:
4634:
4623:
4615:
4609:
4604:
4600:
4592:
4583:
4580:
4576:
4570:
4560:
4555:
4547:
4544:
4541:
4531:
4516:
4509:
4499:
4481:
4474:
4471:
4465:
4457:
4450:
4441:
4438:
4434:
4428:
4418:
4413:
4409:
4401:
4392:
4388:
4381:
4378:
4372:
4364:
4357:
4348:
4335:
4329:
4323:
4317:
4310:
4305:
4300:
4298:
4293:
4286:
4272:
4265:
4257:
4248:
4244:
4239:
4230:
4224:
4220:
4208:
4197:
4190:
4181:
4178:
4176:
4173:
4169:
4162:
4157:
4151:
4134:
4130:
4123:
4117:
4111:
4107:
4101:
4096:
4092:
4087:
4078:
4074:
4068:
4058:
4054:
4050:
4047:
4040:
4026:
4021:
4007:
4002:
3998:
3990:
3988:
3985:
3981:
3976:
3970:
3963:
3957:
3954:
3950:
3929:
3923:
3917:
3911:
3905:
3901:
3895:
3890:
3886:
3876:
3870:
3864:
3858:
3852:
3848:
3842:
3838:
3832:
3826:
3822:property. If
3821:
3813:
3808:
3806:
3804:
3788:
3785:
3765:
3760:
3750:
3746:
3742:
3737:
3733:
3726:
3717:
3713:
3704:
3701:
3698:
3694:
3688:
3683:
3680:
3677:
3674:
3671:
3667:
3661:
3658:
3653:
3648:
3643:
3639:
3635:
3626:
3622:
3613:
3609:
3603:
3598:
3595:
3592:
3588:
3584:
3576:
3572:
3565:
3562:
3552:
3548:
3542:
3536:
3530:
3523:
3519:
3515:
3506:
3502:
3496:
3486:
3484:
3480:
3475:
3459:
3451:
3448:
3445:
3439:
3434:
3430:
3426:
3420:
3417:
3413:
3407:
3402:
3398:
3394:
3389:
3385:
3381:
3375:
3372:
3368:
3362:
3357:
3353:
3349:
3344:
3340:
3334:
3330:
3326:
3321:
3317:
3311:
3307:
3297:
3291:
3286:
3282:
3274:
3255:
3249:
3244:
3240:
3236:
3230:
3227:
3224:
3211:
3204:
3198:
3192:
3187:
3179:
3175:
3171:
3167:
3158:
3152:
3146:
3141:
3130:
3126:
3118:
3117:
3113:
3099:
3096:
3090:
3087:
3081:
3075:
3072:
3069:
3063:
3060:
3051:
3045:
3039:
3035:
3029:
3023:
3017:
3011:
3005:
3000:
2999:
2998:Associativity
2995:
2991:
2987:
2983:
2979:
2972:
2959:
2955:
2943:
2940:
2936:
2932:
2931:
2930:
2928:
2924:
2916:
2914:
2910:
2906:
2900:
2896:. The set of
2894:
2890:
2886:
2882:
2872:
2866:
2862:
2858:
2852:
2846:
2840:
2834:
2826:
2821:
2815:
2809:
2805:
2799:
2797:
2781:
2778:
2775:
2771:
2766:
2761:
2757:
2750:
2747:
2741:
2737:
2731:
2726:
2722:
2712:
2706:
2701:
2690:
2686:
2678:
2673:
2671:
2667:
2663:
2659:
2653:
2640:
2630:
2627:
2624:
2620:
2615:
2611:
2604:
2600:
2595:
2584:
2581:
2578:
2574:
2569:
2561:
2557:
2553:
2548:
2544:
2540:
2535:
2532:
2529:
2525:
2520:
2508:
2500:
2497:
2494:
2490:
2485:
2480:
2476:
2465:
2459:
2453:
2445:
2441:
2437:the integral
2435:
2430:
2426:
2423:, which is a
2422:
2417:
2413:process
2412:
2407:
2403:
2398:
2394:
2390:
2384:
2380:
2372:
2370:
2368:
2363:
2358:
2354:
2349:
2336:
2331:
2327:
2323:
2317:
2313:
2304:
2300:
2287:
2283:
2280:
2277:
2271:
2266:
2262:
2253:
2249:
2233:
2226:
2223:
2217:
2214:
2211:
2205:
2201:
2192:
2188:
2175:
2171:
2166:
2162:
2158:
2133:
2129:
2122:
2119:
2114:
2110:
2100:
2094:
2089:
2084:
2083:-integrable.
2081:
2075:
2058:
2052:
2049:
2046:
2042:
2033:
2030:
2022:
2017:
2013:
2007:
2003:
1998:
1992:
1987:
1983:
1973:
1967:
1961:
1955:
1942:
1939:
1936:
1930:
1926:
1920:
1916:
1910:
1905:
1901:
1897:
1892:
1888:
1884:
1878:
1874:
1868:
1864:
1858:
1853:
1849:
1845:
1842:
1839:
1835:
1830:
1825:
1821:
1793:
1790:
1787:
1773:
1769:
1760:
1755:
1750:
1746:
1737:
1732:
1728:
1719:
1714:
1708:
1702:
1689:
1686:
1683:
1677:
1673:
1667:
1662:
1658:
1654:
1649:
1645:
1641:
1635:
1631:
1625:
1620:
1616:
1612:
1607:
1603:
1599:
1594:
1590:
1581:
1577:
1569:
1564:
1557:
1553:
1549:
1542:
1536:
1530:Itô processes
1529:
1527:
1524:
1506:
1502:
1499:
1493:
1488:
1484:
1478:
1473:
1469:
1464:
1455:
1451:
1446:
1441:
1435:
1431:
1427:
1421:
1417:
1411:
1406:
1402:
1397:
1392:
1379:
1363:
1360:
1354:
1350:
1344:
1339:
1335:
1323:
1315:
1312:
1309:
1305:
1300:
1295:
1291:
1270:
1264:
1261:
1255:
1245:
1241:
1237:
1234:
1226:
1221:
1217:
1208:
1200:
1194:
1188:
1182:
1175:
1168:
1165:
1155:
1150:
1146:
1142:
1137:
1135:
1130:
1117:
1107:
1104:
1101:
1097:
1092:
1088:
1081:
1077:
1072:
1061:
1058:
1055:
1051:
1046:
1038:
1034:
1030:
1022:
1018:
1014:
1009:
1006:
1003:
999:
991:
979:
971:
968:
965:
961:
956:
951:
947:
939:
930:
924:
917:
911:
890:
886:
874:
870:
866:
861:
856:
851:
846:
842:
838:
830:
828:
826:
820:
816:
797:
778:
774:
769:
765:
761:
742:
723:
719:
700:
682:
663:
645:
636:
627:
612:
595:
579:
568:
563:
560:
557:
547:
532:
522:
508:
503:
499:
492:
488:
482:
479:
475:
470:
465:
461:
457:
452:
436:
431:
427:
423:
417:
413:
407:
402:
398:
394:
391:
388:
384:
379:
374:
370:
366:
361:
357:
347:
338:
336:
334:
330:
325:
320:
315:
310:
306:
299:
298:Black–Scholes
295:
291:
286:
284:
280:
277:, which is a
276:
271:
268:
264:
252:
247:
242:
238:
230:
226:
222:
217:
213:generated by
212:
208:
203:
186:
181:
177:
173:
167:
163:
157:
152:
148:
144:
139:
135:
126:
121:
119:
115:
111:
107:
103:
99:
95:
91:
82:
78:
74:
69:
65:
55:
46:
41:
37:
34:Itô integral
32:
19:
8188:Econometrics
8150:Wiener space
8038:Itô integral
7939:Inequalities
7828:Self-similar
7798:Gauss–Markov
7788:Exchangeable
7768:Càdlàg paths
7704:Risk process
7656:LIBOR market
7525:Random graph
7520:Random field
7332:Superprocess
7270:Lévy process
7265:Jump process
7240:Hunt process
7076:Markov chain
6927:Itô integral
6926:
6762:Substitution
6753:Integration
6601:
6583:
6565:
6547:
6529:, Springer,
6524:
6480:
6477:Phys. Rev. E
6476:
6458:
6440:
6420:
6396:
6372:
6294:
6150:which reads
6145:
5965:
5920:
5676:
5662:
5657:
5653:
5643:
5637:
5625:
5619:
5528:
5497:
5490:
5483:
5480:
5471:Nualart 2006
5467:Wiener space
5461:
5453:
5436:Protter 2004
5429:
5423:
5418:
5411:
5407:
5402:
5399:
5383:
5376:
5237:
5232:
5225:
5215:
5207:
5199:
5193:
5189:
5179:
5176:
5167:
5162:
5158:
5154:
5150:
5146:
5142:
5136:
5130:
5124:
5118:
5112:
5108:
5104:
5098:
5095:
5091:localization
5085:
4883:
4876:
4865:
4862:
4856:
4850:
4846:
4745:
4736:
4730:
4722:
4718:
4713:
4710:
4703:
4698:
4695:
4683:
4676:
4670:
4666:
4514:
4504:
4497:
4333:
4327:
4321:
4315:
4308:
4303:
4301:
4291:
4284:
4270:
4263:
4252:
4246:
4242:
4234:
4228:
4225:
4218:
4203:
4195:
4188:
4185:
4179:
4171:
4167:
4160:
4155:
4149:
4027:states that
4025:Itô isometry
4024:
4019:
4005:
4003:martingales
3996:
3994:
3983:
3979:
3968:
3961:
3958:
3952:
3948:
3874:
3868:
3862:
3856:
3850:
3846:
3840:
3836:
3830:
3824:
3817:
3550:
3546:
3540:
3534:
3528:
3521:
3517:
3513:
3498:
3476:
3295:
3289:
3278:
3209:
3202:
3196:
3190:
3177:
3173:
3169:
3162:
3156:
3150:
3144:
3136:
3128:
3121:
3114:
3049:
3043:
3037:
3033:
3027:
3021:
3015:
3009:
3003:
2996:
2989:
2985:
2981:
2977:
2967:
2957:
2950:
2920:
2908:
2904:
2898:
2892:
2888:
2884:
2880:
2870:
2864:
2860:
2856:
2850:
2844:
2838:
2832:
2824:
2819:
2813:
2807:
2803:
2800:
2710:
2704:
2696:
2688:
2681:
2674:
2654:
2463:
2451:
2443:
2439:
2433:
2415:
2405:
2396:
2392:
2388:
2382:
2376:
2361:
2355:formula and
2350:
2098:
2092:
2085:
2079:
2073:
1971:
1965:
1959:
1956:
1712:
1706:
1703:
1575:
1573:
1562:
1555:
1551:
1547:
1540:
1522:
1378:Itô isometry
1203:
1198:
1192:
1186:
1180:
1173:
1166:
1163:
1153:
1138:
1131:
928:
922:
915:
859:
849:
845:Riemann sums
834:
818:
814:
776:
772:
767:
763:
759:
721:
717:
680:
640:
634:
596:
501:
497:
490:
486:
480:
477:
473:
463:
459:
455:
450:
345:
343:The process
342:
329:Riemann sums
323:
319:clairvoyance
313:
308:
301:
287:
272:
263:Riemann sums
245:
215:
201:
122:
90:Itô calculus
89:
88:
80:
76:
72:
67:
63:
53:
44:
39:
35:
18:Itô integral
8233:Ruin theory
8171:Disciplines
8043:Itô's lemma
7818:Predictable
7493:Percolation
7476:Potts model
7471:Ising model
7435:White noise
7393:Differences
7255:Itô process
7195:Cox process
7091:Loop-erased
7086:Random walk
6777:Weierstrass
6401:Birkhaueser
6335:Itô's lemma
5974:Itô's lemma
5438:). Also, a
3965:is that if
3803:Itô's lemma
3495:Itô's lemma
3489:Itô's lemma
2670:local times
2088:Itô's lemma
2077:are called
1576:Itô process
1145:local times
932:up to time
275:Itô's lemma
8291:Categories
8243:Statistics
8023:Filtration
7924:Kolmogorov
7908:Blumenthal
7833:Stationary
7773:Continuous
7761:Properties
7646:Hull–White
7388:Martingale
7275:Local time
7163:Fractional
7141:pure birth
6892:incomplete
6755:techniques
6362:References
5651:to obtain
5533:such that
4341:such that
4331:, but not
4274:such that
4261:. For any
4153:such that
4009:such that
3501:chain rule
2917:Properties
2458:partitions
2425:martingale
2357:chain rule
1171:for every
910:partitions
211:filtration
94:Kiyosi Itô
8155:Classical
7168:Geometric
7158:Excursion
6662:integrals
6660:Types of
6653:Integrals
6490:0707.2234
6433:PDF-files
6250:∂
6232:∂
6216:−
6207:ξ
6165:˙
6090:∂
6076:∂
6062:∂
6033:˙
6010:∂
6002:∂
5990:˙
5918:is used.
5889:−
5873:δ
5861:δ
5854:⟩
5829:ξ
5802:ξ
5798:⟨
5772:ξ
5742:ξ
5700:˙
5583:α
5568:∫
5513:α
5347:⟩
5340:⟨
5307:∫
5265:⋅
5068:∞
5029:∫
4953:∫
4911:⋅
4816:−
4773:≡
4760:⋅
4652:∞
4610:⋅
4581:≤
4561:∗
4545:⋅
4439:≤
4419:∗
4393:≤
4093:∫
4051:⋅
3942:for each
3927:∞
3887:∫
3668:∑
3589:∑
3421:−
3399:∫
3376:−
3354:∫
3250:⋅
3228:⋅
3143:| ≤
3097:⋅
3073:⋅
3064:⋅
2758:∫
2754:→
2723:∫
2703:| ≤
2628:−
2612:−
2582:−
2558:π
2554:∈
2533:−
2521:∑
2515:∞
2512:→
2477:∫
2314:σ
2292:′
2263:σ
2241:′
2238:′
2202:μ
2180:′
2056:∞
2034:μ
2014:σ
1984:∫
1927:μ
1902:∫
1875:σ
1850:∫
1822:∫
1809:for each
1797:∞
1770:μ
1747:σ
1729:∫
1674:μ
1659:∫
1632:σ
1617:∫
1470:∫
1403:∫
1336:∫
1330:∞
1327:→
1292:∫
1268:→
1238:−
1218:∫
1105:−
1089:−
1059:−
1035:π
1031:∈
1007:−
992:∑
986:∞
983:→
948:∫
887:π
594:σ-algebra
561:≥
520:Ω
509:is given
399:∫
395:≡
371:∫
241:variation
149:∫
8276:Category
8160:Abstract
7694:Bühlmann
7300:Compound
6982:Volumes
6887:complete
6784:By parts
6546:(2003),
6515:17677426
6419:(2004).
6307:See also
5624:[0,
5527:[0,
5489:[0,
5417:: |
5228:⟩
5224:⟨
5218:⟩
5214:⟨
5202:⟩
5198:⟨
4201:E(|
4186:For any
3148:, where
2462:[0,
1718:Lebesgue
1560:, where
914:[0,
812:for all
339:Notation
104:such as
98:calculus
7783:Ergodic
7671:Vašíček
7513:Poisson
7173:Meander
6986:Washers
6495:Bibcode
5972:, then
5503:adapted
4319:,
4210:|)
2925:) and (
2828:|)
2714:, then
1580:adapted
1566:is the
873:adapted
285:terms.
251:adapted
209:to the
207:adapted
8123:Tanaka
7808:Mixing
7803:Markov
7676:Wilkie
7641:Ho–Lee
7636:Heston
7408:Super-
7153:Bridge
7101:Biased
6991:Shells
6608:
6590:
6572:
6554:
6533:
6513:
6465:
6447:
6427:
6407:
6385:
5763:where
5379:< ∞
4337:or on
4287:> 1
4259:|
4250:|
4191:> 1
4001:càdlàg
3946:, and
3485:term.
3154:is an
3134:|
3001:. Let
2935:càdlàg
2694:|
2427:, and
2400:for a
1704:Here,
1169:< ∞
869:càdlàg
199:where
7976:Tools
7752:M/M/c
7747:M/M/1
7742:M/G/1
7732:Fluid
7398:Local
6772:Euler
6485:arXiv
5628:]
5531:]
5493:]
5006:By a
4721:>
4674:is a
4282:. If
4240:= sup
3977:then
3532:from
3520:,...,
2830:then
2466:]
936:is a
918:]
863:is a
853:is a
296:(see
235:is a
108:(see
75:) = (
7928:Lévy
7727:Bulk
7611:Chen
7403:Sub-
7361:Both
6606:ISBN
6588:ISBN
6570:ISBN
6552:ISBN
6531:ISBN
6511:PMID
6463:ISBN
6445:ISBN
6425:ISBN
6405:ISBN
6383:ISBN
5914:and
5389:for
5134:and
5065:<
4882:) =
4875:Var(
4789:>
4734:and
4649:<
4302:The
3924:<
3293:and
3194:and
3132:and
2984:) =
2891:) ·
2878:and
2836:and
2692:and
2668:and
2409:and
2053:<
1794:<
1545:and
1143:and
592:The
267:mesh
116:and
49:blue
7508:Cox
6503:doi
6303:).
5921:If
5525:on
5473:).
5446:).
5235:of
4854:in
4686:· )
4311:≥ 1
4266:≥ 1
4221:= 1
3973:is
3971:· )
3872:is
3805:).
3538:to
3516:= (
3503:or
3047:is
3031:is
3019:be
2948:is
2929:):
2887:= (
2876:= 0
2848:is
2505:lim
2460:of
1963:be
1574:An
1558:−5)
1543:= 0
1320:lim
1176:≥ 0
976:lim
912:of
871:),
843:of
827:).
821:≥ 0
726:is
647:is
288:In
249:is
100:to
83:)/2
59:red
47:) (
8293::
7926:,
7922:,
7918:,
7914:,
7910:,
6509:,
6501:,
6493:,
6481:76
6479:,
6403:,
6381:,
5661:−
5622:∈
5410:·
5196:+
5192:=
5161:·
5157:+
5153:·
5149:=
5145:·
5111:+
5107:=
4860:.
4849:·
4709:=
4669:·
4505:M*
4299:.
4235:M*
4170:·
4158:·
3982:·
3951:·
3849:·
3839:·
3212:·
3208:=
3176:·
3172:→
3168:·
3127:→
3044:JK
3036:·
3007:,
2980:·
2976:Δ(
2961:t−
2956:−
2913:.
2889:KH
2883:·
2863:=
2859:·
2839:KH
2806:·
2798:.
2687:→
2672:.
2442:·
2395:+
2391:=
2369:.
1972:Hμ
1960:Hσ
1550:=
1167:ds
1136:.
771:−
500:·
496:=
489:−
481:dX
476:=
474:dY
462:·
458:=
120:.
79:−
7930:)
7906:(
7027:e
7020:t
7013:v
6645:e
6638:t
6631:v
6505::
6497::
6487::
6281:.
6276:l
6273:m
6269:g
6259:m
6255:x
6243:l
6240:k
6236:g
6224:2
6221:1
6211:l
6201:l
6198:k
6194:g
6190:+
6185:k
6181:h
6177:=
6172:k
6162:x
6132:.
6127:l
6124:m
6120:g
6114:m
6111:k
6107:g
6098:l
6094:x
6084:k
6080:x
6071:y
6066:2
6053:2
6050:1
6045:+
6040:j
6030:x
6018:j
6014:x
6005:y
5996:=
5987:y
5968:k
5966:x
5951:)
5946:k
5942:x
5938:(
5935:y
5932:=
5929:y
5902:)
5897:2
5893:t
5884:1
5880:t
5876:(
5868:l
5865:k
5857:=
5851:)
5846:2
5842:t
5838:(
5833:l
5824:)
5819:1
5815:t
5811:(
5806:k
5776:j
5751:,
5746:l
5736:l
5733:k
5729:g
5725:+
5720:k
5716:h
5712:=
5707:k
5697:x
5666:0
5663:M
5658:t
5654:M
5649:t
5644:B
5638:M
5631:(
5626:T
5620:t
5603:s
5599:B
5594:d
5587:s
5577:t
5572:0
5564:+
5559:0
5555:M
5551:=
5546:t
5542:M
5529:T
5498:B
5491:T
5484:M
5434:(
5430:X
5424:t
5419:H
5414:t
5412:X
5408:H
5403:X
5377:E
5361:,
5357:]
5351:s
5343:M
5337:d
5331:2
5326:s
5322:H
5316:t
5311:0
5302:[
5297:E
5293:=
5289:]
5283:2
5279:)
5273:t
5269:M
5262:H
5259:(
5255:[
5250:E
5238:M
5226:M
5216:M
5208:N
5200:M
5194:N
5190:M
5180:M
5168:X
5163:A
5159:H
5155:M
5151:H
5147:X
5143:H
5137:A
5131:M
5125:A
5119:M
5113:A
5109:M
5105:X
5099:X
5086:B
5071:,
5061:]
5057:s
5054:d
5048:2
5044:H
5038:t
5033:0
5024:[
5019:E
4994:.
4990:]
4986:s
4983:d
4977:2
4972:s
4968:H
4962:t
4957:0
4948:[
4943:E
4939:=
4935:]
4929:2
4925:)
4919:t
4915:B
4908:H
4905:(
4901:[
4896:E
4884:t
4879:t
4877:B
4866:B
4857:H
4851:X
4847:H
4832:.
4829:)
4824:T
4820:X
4811:t
4807:X
4803:(
4800:A
4795:}
4792:T
4786:t
4783:{
4778:1
4768:t
4764:X
4757:H
4746:A
4739:T
4737:F
4731:T
4725:}
4723:T
4719:t
4717:{
4714:1
4711:A
4706:t
4704:H
4684:H
4682:(
4677:p
4671:M
4667:H
4645:]
4638:2
4635:p
4630:)
4624:t
4620:]
4616:M
4613:[
4605:2
4601:H
4597:(
4593:[
4588:E
4584:C
4577:]
4571:p
4567:)
4556:t
4552:)
4548:M
4542:H
4539:(
4536:(
4532:[
4527:E
4515:H
4510:)
4507:t
4503:(
4498:M
4482:]
4475:2
4472:p
4466:t
4462:]
4458:M
4455:[
4451:[
4446:E
4442:C
4435:]
4429:p
4425:)
4414:t
4410:M
4406:(
4402:[
4397:E
4389:]
4382:2
4379:p
4373:t
4369:]
4365:M
4362:[
4358:[
4353:E
4349:c
4339:t
4334:M
4328:p
4322:C
4316:c
4309:p
4292:p
4285:p
4280:t
4276:E
4271:M
4264:p
4255:s
4253:M
4247:t
4245:≤
4243:s
4237:t
4229:M
4219:p
4214:t
4206:t
4204:M
4196:p
4189:p
4180:p
4172:M
4168:H
4161:t
4156:H
4150:M
4135:.
4131:]
4127:]
4124:M
4121:[
4118:d
4112:2
4108:H
4102:t
4097:0
4088:[
4083:E
4079:=
4075:]
4069:2
4065:)
4059:t
4055:M
4048:H
4045:(
4041:[
4036:E
4020:M
4015:t
4011:E
4006:M
3984:M
3980:H
3969:H
3967:(
3962:M
3953:M
3949:H
3944:t
3930:,
3921:]
3918:M
3915:[
3912:d
3906:2
3902:H
3896:t
3891:0
3875:M
3869:H
3863:M
3857:M
3851:M
3847:H
3841:M
3837:H
3831:H
3825:M
3789:,
3786:f
3766:.
3761:t
3757:]
3751:j
3747:X
3743:,
3738:i
3734:X
3730:[
3727:d
3723:)
3718:t
3714:X
3710:(
3705:j
3702:,
3699:i
3695:f
3689:n
3684:1
3681:=
3678:j
3675:,
3672:i
3662:2
3659:1
3654:+
3649:i
3644:t
3640:X
3636:d
3632:)
3627:t
3623:X
3619:(
3614:i
3610:f
3604:n
3599:1
3596:=
3593:i
3585:=
3582:)
3577:t
3573:X
3569:(
3566:f
3563:d
3553:)
3551:X
3549:(
3547:f
3541:R
3535:R
3529:f
3524:)
3522:X
3518:X
3514:X
3509:n
3460:t
3456:]
3452:Y
3449:,
3446:X
3443:[
3440:+
3435:s
3431:X
3427:d
3418:s
3414:Y
3408:t
3403:0
3395:+
3390:s
3386:Y
3382:d
3373:s
3369:X
3363:t
3358:0
3350:+
3345:0
3341:Y
3335:0
3331:X
3327:=
3322:t
3318:Y
3312:t
3308:X
3296:Y
3290:X
3259:]
3256:X
3253:[
3245:2
3241:H
3237:=
3234:]
3231:X
3225:H
3222:[
3210:H
3203:X
3197:Y
3191:X
3183:t
3178:X
3174:H
3170:X
3165:n
3163:H
3157:X
3151:J
3145:J
3139:n
3137:H
3129:H
3124:n
3122:H
3100:X
3094:)
3091:K
3088:J
3085:(
3082:=
3079:)
3076:X
3070:K
3067:(
3061:J
3050:X
3038:X
3034:K
3028:J
3022:X
3016:K
3010:K
3004:J
2990:X
2988:Δ
2986:H
2982:X
2978:H
2970:t
2968:X
2966:Δ
2958:X
2953:t
2951:X
2946:t
2941:.
2911:)
2909:X
2907:(
2905:L
2899:X
2893:X
2885:Y
2881:K
2874:0
2871:Y
2865:Y
2861:X
2857:H
2851:X
2845:H
2833:K
2825:H
2820:K
2814:H
2808:X
2804:H
2782:,
2779:X
2776:d
2772:H
2767:t
2762:0
2751:X
2748:d
2742:n
2738:H
2732:t
2727:0
2711:J
2705:J
2699:n
2697:H
2689:H
2684:n
2682:H
2641:.
2638:)
2631:1
2625:i
2621:t
2616:X
2605:i
2601:t
2596:X
2592:(
2585:1
2579:i
2575:t
2570:H
2562:n
2549:i
2545:t
2541:,
2536:1
2530:i
2526:t
2509:n
2501:=
2498:X
2495:d
2491:H
2486:t
2481:0
2464:t
2452:n
2449:π
2444:X
2440:H
2434:H
2416:A
2406:M
2397:A
2393:M
2389:X
2383:X
2362:f
2337:.
2332:t
2328:B
2324:d
2318:t
2310:)
2305:t
2301:X
2297:(
2288:f
2284:+
2281:t
2278:d
2272:2
2267:t
2259:)
2254:t
2250:X
2246:(
2234:f
2227:2
2224:1
2218:+
2215:t
2212:d
2206:t
2198:)
2193:t
2189:X
2185:(
2176:f
2172:=
2167:t
2163:Y
2159:d
2139:)
2134:t
2130:X
2126:(
2123:f
2120:=
2115:t
2111:Y
2099:X
2093:f
2080:X
2074:H
2059:.
2050:s
2047:d
2043:)
2038:|
2031:H
2027:|
2023:+
2018:2
2008:2
2004:H
1999:(
1993:t
1988:0
1966:B
1943:.
1940:s
1937:d
1931:s
1921:s
1917:H
1911:t
1906:0
1898:+
1893:s
1889:B
1885:d
1879:s
1869:s
1865:H
1859:t
1854:0
1846:=
1843:X
1840:d
1836:H
1831:t
1826:0
1811:t
1791:s
1788:d
1784:)
1780:|
1774:s
1765:|
1761:+
1756:2
1751:s
1743:(
1738:t
1733:0
1713:B
1707:B
1690:.
1687:s
1684:d
1678:s
1668:t
1663:0
1655:+
1650:s
1646:B
1642:d
1636:s
1626:t
1621:0
1613:+
1608:0
1604:X
1600:=
1595:t
1591:X
1563:ψ
1556:t
1554:(
1552:ψ
1548:σ
1541:μ
1523:H
1507:]
1503:s
1500:d
1494:2
1489:s
1485:H
1479:t
1474:0
1465:[
1460:E
1456:=
1452:]
1447:2
1442:)
1436:s
1432:B
1428:d
1422:s
1418:H
1412:t
1407:0
1398:(
1393:[
1388:E
1364:B
1361:d
1355:n
1351:H
1345:t
1340:0
1324:n
1316:=
1313:B
1310:d
1306:H
1301:t
1296:0
1271:0
1265:s
1262:d
1256:2
1252:)
1246:n
1242:H
1235:H
1232:(
1227:t
1222:0
1206:n
1204:H
1199:B
1193:H
1187:B
1181:H
1174:t
1164:H
1161:0
1159:∫
1154:H
1118:.
1115:)
1108:1
1102:i
1098:t
1093:B
1082:i
1078:t
1073:B
1069:(
1062:1
1056:i
1052:t
1047:H
1039:n
1028:]
1023:i
1019:t
1015:,
1010:1
1004:i
1000:t
996:[
980:n
972:=
969:B
966:d
962:H
957:t
952:0
934:t
929:B
923:H
916:t
896:}
891:n
883:{
867:(
860:H
850:B
823:(
819:t
817:,
815:s
798:t
792:F
777:t
773:B
768:s
766:+
764:t
760:B
743:t
737:F
722:t
718:B
701:t
695:F
681:B
664:t
658:F
643:t
641:X
635:X
630:t
613:t
607:F
580:.
577:)
573:P
569:,
564:0
558:t
554:)
548:t
542:F
536:(
533:,
528:F
523:,
517:(
502:X
498:H
494:0
491:Y
487:Y
478:H
467:(
464:X
460:H
456:Y
451:t
437:,
432:s
428:X
424:d
418:s
414:H
408:t
403:0
392:X
389:d
385:H
380:t
375:0
367:=
362:t
358:Y
346:Y
331:(
324:H
314:H
309:t
304:t
302:H
259:t
255:t
246:H
233:t
219:(
216:X
202:H
187:,
182:s
178:X
174:d
168:s
164:H
158:t
153:0
145:=
140:t
136:Y
85:.
81:t
77:B
73:B
71:(
68:t
64:Y
57:(
54:B
45:B
43:(
40:t
36:Y
20:)
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