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The theorem only asserts the existence of the representation and does not help to find it explicitly; it is possible in many cases to determine the form of the representation using
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and it is easy to check that the strategy is self-financing: the change in the value of the portfolio only depends on the change of the asset prices
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1360:(1976) "Stochastic Integrals for Martingales of a Jump Process with Partially Accessible Jump Times",
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The martingale representation theorem can be used to establish the existence of a
87:
823:{\displaystyle \int _{0}^{T}\varphi _{t}^{2}\sigma _{t}^{2}\,dt<\infty }
1326:{\displaystyle \left(dV_{t}=\varphi _{t}\,dS_{t}+\psi _{t}\,dB_{t}\right)}
554:{\displaystyle E(X|{\mathcal {G}}_{t})=E(X)+\int _{0}^{t}C_{s}\,dB_{s}.}
1354:
Montin, Benoît. (2002) "Stochastic
Processes Applied in Finance"
1362:
1229:{\displaystyle V_{T}=\varphi _{T}S_{T}+\psi _{T}B_{T}=C_{T}=X}
915:{\displaystyle N_{t}=N_{0}+\int _{0}^{t}\varphi _{s}\,dM_{s}.}
15:
218:{\displaystyle (\Omega ,{\mathcal {F}},{\mathcal {F}}_{t},P)}
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477:
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1067:
is the stock price discounted by the bond price to time
440:{\displaystyle X=E(X)+\int _{0}^{\infty }C_{s}\,dB_{s}.}
704:{\displaystyle \left(N_{t}\right)_{0\leq t<\infty }}
620:{\displaystyle \left(M_{t}\right)_{0\leq t<\infty }}
1029:{\displaystyle \psi _{t}B_{t}=C_{t}-\varphi _{t}Z_{t}}
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1114:is the expected payoff of the option at time
8:
755:, unique up to sets of measure 0, such that
925:The replicating strategy is defined to be:
711:is any other Q-martingale, there exists an
288:random variable measurable with respect to
1343:Backward stochastic differential equation
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66:Learn how and when to remove this message
312:{\displaystyle {\mathcal {G}}_{\infty }}
29:This article includes a list of general
102:with respect to this Brownian motion.
86:states that a random variable that is
7:
357:{\displaystyle {\mathcal {G}}_{t},}
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612:
405:
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249:{\displaystyle {\mathcal {G}}_{t}}
176:
35:it lacks sufficient corresponding
14:
1141:, the value of the portfolio is:
627:is a Q-martingale process, whose
84:martingale representation theorem
112:Similar theorems also exist for
20:
956:units of the stock at the time
728:{\displaystyle {\mathcal {F}}}
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98:can be written in terms of an
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657:is always non-zero. Then, if
949:{\displaystyle \varphi _{t}}
650:{\displaystyle \sigma _{t}}
1399:
830:with probability one, and
165:filtered probability space
116:on filtrations induced by
748:{\displaystyle \varphi }
573:strategy. Suppose that
50:more precise citations.
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1137:At the expiration day
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565:Application in finance
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319:, then there exists a
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1107:{\displaystyle C_{t}}
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1060:{\displaystyle Z_{t}}
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152:{\displaystyle B_{t}}
1383:Probability theorems
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258:augmented filtration
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90:with respect to the
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834:can be written as:
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321:predictable process
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120:, for example, by
107:Malliavin calculus
80:probability theory
1378:Martingale theory
1127:{\displaystyle t}
1080:{\displaystyle t}
286:square integrable
273:{\displaystyle B}
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46:this article by
37:inline citations
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161:Brownian motion
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96:Brownian motion
94:generated by a
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42:Please help to
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450:Consequently,
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163:on a standard
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118:jump processes
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260:generated by
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122:Markov chains
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100:Itô integral
83:
77:
62:
56:October 2011
53:
34:
114:martingales
48:introducing
1372:Categories
1349:References
629:volatility
364:such that
92:filtration
88:measurable
31:references
1296:ψ
1269:φ
1189:ψ
1166:φ
1008:φ
1004:−
972:ψ
938:φ
887:φ
872:∫
818:∞
794:σ
779:φ
764:∫
743:φ
697:∞
688:≤
639:σ
613:∞
604:≤
511:∫
406:∞
397:∫
326:which is
305:∞
177:Ω
128:Statement
1337:See also
225:and let
571:hedging
328:adapted
256:be the
44:improve
1040:where
82:, the
33:, but
963:hold
960:, and
929:hold
284:is a
280:. If
159:be a
1087:and
815:<
694:<
610:<
132:Let
124:.
78:In
1374::
1333:.
1134:.
109:.
1320:)
1314:t
1310:B
1306:d
1300:t
1292:+
1287:t
1283:S
1279:d
1273:t
1265:=
1260:t
1256:V
1252:d
1248:(
1224:X
1221:=
1216:T
1212:C
1208:=
1203:T
1199:B
1193:T
1185:+
1180:T
1176:S
1170:T
1162:=
1157:T
1153:V
1139:T
1122:t
1100:t
1096:C
1075:t
1053:t
1049:Z
1022:t
1018:Z
1012:t
999:t
995:C
991:=
986:t
982:B
976:t
958:t
942:t
910:.
905:s
901:M
897:d
891:s
881:t
876:0
868:+
863:0
859:N
855:=
850:t
846:N
832:N
812:t
809:d
803:2
798:t
788:2
783:t
773:T
768:0
721:F
691:t
685:0
680:)
675:t
671:N
667:(
643:t
607:t
601:0
596:)
591:t
587:M
583:(
549:.
544:s
540:B
536:d
530:s
526:C
520:t
515:0
507:+
504:)
501:X
498:(
495:E
492:=
489:)
484:t
478:G
471:|
467:X
464:(
461:E
435:.
430:s
426:B
422:d
416:s
412:C
401:0
393:+
390:)
387:X
384:(
381:E
378:=
375:X
352:,
347:t
341:G
324:C
299:G
282:X
268:B
242:t
236:G
213:)
210:P
207:,
202:t
196:F
190:,
185:F
180:,
174:(
145:t
141:B
69:)
63:(
58:)
54:(
40:.
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