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Martingales
Denition
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Martingales80-646-08
Stochastic Calculus I
Geneviève Gauthier
HEC Montr éal
Martingales
DenitionLemma 1Examplelemma 2
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
DenitionOn the ltered probability space (Ω,F ,F,P), where F is theltration fFt : t 2 f0, 1, 2, . . .gg, the stochastic process
M = fMt : t 2 f0, 1, 2, . . .gg
is a discrete-time martingale if
(M1) 8t 2 f0, 1, 2, . . .g, EP [jMt j] < ∞;(M2) 8t 2 f0, 1, 2, . . .g, Mt is Ftmeasurable;(M3) 8s, t 2 f0, 1, 2, . . .g such that s < t, EP [Mt jFs ] = Ms .
Martingales
DenitionLemma 1Examplelemma 2
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
MartingaleConstant expectation process
Lemma. Let M = fMt : t 2 f0, 1, 2, . . .gg be a martingalebuilt on the ltered probability space (Ω,F ,F,P). Then
8t 2 f1, 2, . . .g , EP [Mt ] = EP [M0] .
Proof of the lemma. 8t 2 f1, 2, . . .g,
EP [Mt ] = EPhEP [Mt jF0 ]
iby (EC3)
= EP [M0] by (M3) .
Interpretation. A martingale is a stochastic process that,on average, is constant. This doesnt mean however thatsuch a process varies little, since the variance VarP [Mt ],at every time, can be innite.
Martingales
DenitionLemma 1Examplelemma 2
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Example IExample. Let fξt : t 2 f1, 2, . . .gg be a sequence of(Ω,F )independent and identically distributed randomvariables with respect to the measure P and such that
EP [ξt ] = 0 and EPξ2t< ∞.
Lets dene
F0 = f?,Ωg ;8t 2 f1, 2, . . .g ,Ft = σ fξs : s 2 f1, . . . , tgg ;
and
M0 = 0, Mt =t
∑s=1
ξs .
The stochastic process M is a martingale on the space(Ω,F ,F,P).
Martingales
DenitionLemma 1Examplelemma 2
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Example II
Indeed,
EP [jMt j] = EP
" t∑s=1 ξs
#
t
∑s=1
EP [jξs j] t
∑s=1
qEPξ2s< ∞
where the second inequality comes from the fact that, for anyrandom variable,
0 Var [jX j] = EhjX j2
i (E [jX j])2 ) E [jX j]
rEhjX j2
i.
Given the selected ltration, M is adapted (which is to say that8t 2 f0, 1, 2, . . .g, Mt is Ftmesurable).
Martingales
DenitionLemma 1Examplelemma 2
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Example III
Lastly, 8s, t 2 f0, 1, 2, . . .g such that s < t,
EP [Mt jFs ]
= EP
"Ms +
t
∑u=s+1
ξu jFs
#
= EP [Ms jFs ] +t
∑u=s+1
EP [ξu jFs ]
= Ms +t
∑u=s+1
EP [ξu jFs ]| z =EP[ξu ]
from (EC1) since Ms is Fs measurable, and
from (EC7) since ξu is independent from ξ1, . . . , ξs .= Ms .
Martingales
DenitionLemma 1Examplelemma 2
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Martingale I
LemmaIn the denition of a martingale, the condition (M3) isequivalent to
M3* 8t 2 f1, 2, . . .g, EP [Mt jFt1 ] = Mt1.
Proof of the lemma. Clearly, (M3)) (M3)since (M3) isonly a special case of (M3). Indeed, it is su¢ cient to denes = t 1.So we must show that (M3)) (M3). This can be proved byinduction. Intuitively, if s < t then
EP [Mt jFs ] = EPhEP [Mt jFt1 ] jFs
ifrom (EC3),
= EP [Mt1 jFs ] from (M3)
Martingales
DenitionLemma 1Examplelemma 2
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Martingale II
But if s < t 1, then we can use the same logic again, and weget
EP [Mt1 jFs ] = EPhEP [Mt1 jFt2 ] jFs
ifrom (EC3),
= EP [Mt2 jFs ] from (M3).
Now just substitute this result into the rst equation:
EP [Mt jFs ] = EP [Mt2 jFs ] .
By iterating such an algorithm, we will eventually obtain
EP [Mt jFs ] = EP [Ms+1 jFs ]= Ms from (M3).
Martingales
Denition
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Example I
ω ξ1 ξ2 ξ3 P Q
ω1 1 1 1 18
12
ω2 1 1 1 18
114
ω3 1 1 1 18
114
ω4 1 1 1 18
114
ω ξ1 ξ2 ξ3 P Q
ω5 1 1 1 18
114
ω6 1 1 1 18
114
ω7 1 1 1 18
114
ω8 1 1 1 18
114
On the sample space Ω = fω1, . . . ,ω8g, we will use theσ-algebra F = the set of all events in Ω. The ltration F ismade up of the σ-subalgebras
F0 = f?,Ωg ,F1 = σ fξ1g = σ ffω1,ω2,ω3,ω4g , fω5,ω6,ω7,ω8gg ,F2 = σ fξ1, ξ2g = σ ffω1,ω2g , fω3,ω4g , fω5,ω6g , fω7,ω8gg ,F3 = σ fξ1, ξ2, ξ3g = F .
Martingales
Denition
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Example II
The stochastic process M, built on the ltered measurablespace (Ω,F ,F) is dened as follows :
M0 = 0,
M1 = ξ1M2 = ξ1 + ξ2 and
M3 = ξ1 + ξ2 + ξ3.
By construction, M is adapted to the ltration F.
Martingales
Denition
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Example III
M = fMt : t 2 f0, 1, 2, 3gg is a martingale on (Ω,F ,F,P).
Indeed, condition (M2) is already veried since M isFadapted.Condition (M1) is also satised since 8t 2 f0, 1, 2, 3g,
EP [jMt j] EP [jξ1j] + EP [jξ2j] + EP [jξ3j] = 3.
Lets verify condition (M3).
Martingales
Denition
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Example IV
EP [M1 jF0 ] = EP [M1 ] from (EC4)
= 0
= M0
8ω 2 fω1,ω2,ω3,ω4g ,
EP [M2 jF1 ] (ω) =112
2 1
8 2 1
8+ 0 1
8+ 0 1
8
= 1
M1 (ω) = 1;
8ω 2 fω5,ω6,ω7,ω8g ,
EP [M2 jF1 ] (ω) =112
0 1
8+ 0 1
8+ 2 1
8+ 2 1
8
= 1
M1 (ω) = 1
Martingales
Denition
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Example V
8ω 2 fω1,ω2g ,
EP [M3 jF2 ] (ω) =114
3 1
8 1 1
8
= 2 and M2 (ω) = 2;
8ω 2 fω3,ω4g ,
EP [M3 jF2 ] (ω) =114
1 1
8+ 1 1
8
= 0 and M2 (ω) = 0;
8ω 2 fω5,ω6g ,
EP [M3 jF2 ] (ω) =114
1 1
8+ 1 1
8
= 0 and M2 (ω) = 0;
8ω 2 fω7,ω8g ,
EP [M3 jF2 ] (ω) =114
1 1
8+ 3 1
8
= 2 and M2 (ω) = 2.
Martingales
Denition
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Example VI
By contrast M = fMt : t 2 f0, 1, 2, 3gg is not a martingale on(Ω,F ,F,Q). Indeed,
EQ [M1 jF0 ]= EQ [M1] from (EC4)
=12+114+114+114+114+114+114+114
= 614
6= 0
= M0.
Martingales
Denition
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Example VII
Conclusion. For a stochastic process, the property ofbeing a martingale depends on the ltration and on themeasure. Thats why the notation (F,P)martingalemay sometimes be seen.
Martingales
Denition
Example
StoppedprocessDenitionExampleTheorem
OptionalStoppingTheorem
Markovianprocess
Denition
DenitionThe stochastic process X and the stopping time τ are built onthe same ltered measurable space (Ω,F ,F). The stochasticprocess X τdened by
X τt (ω) = Xt^τ(ω) (ω) (1)
is called a stopped process with stopping time τ.
Martingales
Denition
Example
StoppedprocessDenitionExampleTheorem
OptionalStoppingTheorem
Markovianprocess
Example I
ω X0 X1 X2 X3 τ X τ0 X τ
1 X τ2 X τ
3
ω1 1 12 1 1
2 0 1 1 1 1
ω2 1 12 1 1
2 3 1 12 1 1
2
ω3 1 2 1 1 1 1 2 2 2
ω4 1 2 2 1 1 1 2 2 2
Martingales
Denition
Example
StoppedprocessDenitionExampleTheorem
OptionalStoppingTheorem
Markovianprocess
Stopped martingale ITheorem
TheoremIf the martingale M and the stopping time τ are built on thesame ltered probability space (Ω,F ,F,P) then the stoppedprocess Mτ is also a martingale on that space.
Martingales
Denition
Example
StoppedprocessDenitionExampleTheorem
OptionalStoppingTheorem
Markovianprocess
Stopped martingale IITheorem
Proof of the theorem. The key to the proof is to express Mτt
in terms of the components of the process M.
Mτ0 = M0
and 8t 2 f1, 2, . . .g , Mτt = Mτ
t
t1∑k=0
Ifτ=kg + Ifτtg
!
=t1∑k=0
Ifτ=kgMτt + IfτtgM
τt
=t1∑k=0
Ifτ=kgMk + IfτtgMt .
Martingales
Denition
Example
StoppedprocessDenitionExampleTheorem
OptionalStoppingTheorem
Markovianprocess
Stopped martingale IIITheorem
Verifying condition (M1) :
EP [jMτ0 j] = EP [jM0j] < ∞
and 8t 2 f1, 2, . . .g ,
EP [jMτt j] = EP
"t1∑k=0
Ifτ=kgMk + IfτtgMt
#
t1∑k=0
EPIfτ=kgMk
+ EPIfτtgMt
t1∑k=0
EP [jMk j] + EP [jMt j] < ∞
since, M being a martingale, we have that 8t 2 f0, 1, 2, . . .g,EP [jMt j] < ∞.
Martingales
Denition
Example
StoppedprocessDenitionExampleTheorem
OptionalStoppingTheorem
Markovianprocess
Stopped martingale IVTheorem
Verifying condition (M2) :
Mτ0 = M0 is F0 measurable. (2)
Now, 8t 2 f1, 2, . . .g,
Mτt =
t1∑k=0
Ifτ=kg| z Fk measurablesince fτ=kg2Fk
Mk|zFk measurablesince M is adapted.| z
Ft measurable since k<t)FkFt
+ Ifτtg| z Ft1 measurable
sincefτtg=fτt1gc2Ft1
Mt|zFt measurablesince M is adapted.
is Ftmeasurable.
Martingales
Denition
Example
StoppedprocessDenitionExampleTheorem
OptionalStoppingTheorem
Markovianprocess
Stopped martingale VTheorem
Verifying condition (M3) : 8t 2 f1, 2, . . .g ,
Mτt Mτ
t1
=
t1∑k=0
Ifτ=kgMk + IfτtgMt
!
t2∑k=0
Ifτ=kgMk + Ifτt1gMt1
!= Ifτ=t1gMt1 + IfτtgMt Ifτt1gMt1
= IfτtgMt Ifτt1g Ifτ=t1g
Mt1
= IfτtgMt IfτtgMt1
since, fτ = t 1g and fτ tg being disjoint,Ifτ=t1g + Ifτtg = Ifτ=t1g[fτtg = Ifτt1g.
= Ifτtg (Mt Mt1) .
Martingales
Denition
Example
StoppedprocessDenitionExampleTheorem
OptionalStoppingTheorem
Markovianprocess
Stopped martingale VITheorem
As a consequence, since Ifτtg is Ft1measurable
EP [Mτt jFt1 ]Mτ
t1= EP [Mτ
t Mτt1 jFt1 ]
= EPIfτtg (Mt Mt1) jFt1
= IfτtgE
P [Mt Mt1 jFt1 ]
= Ifτtg
EP [Mt jFt1 ] EP [Mt1 jFt1 ]
= Ifτtg (Mt1 Mt1) = 0
henceEP [Mτ
t jFt1 ] = Mτt1 .
Martingales
Denition
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Optional Stopping Theorem
Theorem(Optional Stopping Theorem). Let
X = fXt : t 2 f0, 1, 2, . . .gg
be a process built on the ltered probability space(Ω,F ,F,P), where F is the ltration fFt : t 2 f0, 1, 2, . . .gg.Lets assume that the stochastic process X is Fadapted andthat it is integrable, i.e. EP [jXt j] < ∞. Then X is amartingale if and only if
EP [Xτ] = EP [X0]
for any bounded stopping time τ, i.e for any given stoppingtime τ, there exists a constant b such that
8ω 2 Ω , 0 τ (ω) b.
Martingales
Denition
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Optional Stopping Theorem IProof of the theorem
First part. Lets assume that X is a martingale and letsshow that, in such a case, EP [Xτ] = EP [X0] for anybounded stopping time.
Let τ be any bounded stopping time. Then, there exists aconstant b such that 8ω 2 Ω, 0 τ (ω) b. As a
Martingales
Denition
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Optional Stopping Theorem IIProof of the theorem
consequence,
EP [Xτ ] = EP
"b
∑k=0
XkIfτ=kg
#
= EP
"b
∑k=0
Xk
Ifτkg Ifτk+1g#
=b
∑k=0
EPhXkIfτkg
i
b
∑k=0
EPhXkIfτk+1g
i= EP [X0 ] +
b
∑k=1
EPhXkIfτkg
ib1∑k=0
EPhXkIfτk+1g
isince Ifτ0g = IΩ = 1 and Ifτb+1g = I? = 0.
= EP [X0 ] +b
∑k=1
EPhXkIfτkg
i
b
∑k=1
EP[Xk1Ifτkg
Martingales
Denition
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Optional Stopping Theorem IIIProof of the theorem
= EP [X0 ] +b
∑k=1
EPh(Xk Xk1) Ifτkg
i= EP [X0 ] +
b
∑k=1
EPhEPh(Xk Xk1) Ifτkg jFk1
iifrom (EC3),
= EP [X0 ] +b
∑k=1
EPhIfτkgEP [Xk Xk1 jFk1 ]
ifrom (EC6),
= EP [X0 ]
since, X being a martingale,
EP [Xk Xk1 jFk1 ] = EP [Xk jFk1 ] EP [Xk1 jFk1 ]= Xk1 Xk1 = 0.
Martingales
Denition
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Optional Stopping Theorem IVProof of the theorem
Second part. Lets now assume that, for any boundedstopping time τ, EP [Xτ] = EP [X0] and lets show that, insuch a case, the adapted and integrable stochastic processis a martingale.
By hypothesis, X already satises conditions (M1) and (M2).The only thing left to verify is that 8s, t 2 f0, 1, 2, . . .g suchthat s < t, EP [Xt jFs ] = Xs .
So, lets set s and t 2 f0, 1, 2, . . .g such that s < t.We denote by P s =
nA(s)1 , . . . ,A(s)ns
othe nite
partition generated by Fs .
Martingales
Denition
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Optional Stopping Theorem VProof of the theorem
For any i 2 f1, . . . , nsg we build a random time :
Si (ω) =
8><>:s if ω 2 A(s)i
t if ω /2 A(s)i .
Si is a stopping time (obviously bounded) since8u 2 f0, 1, 2, . . .g ,
fω 2 Ω : Si (ω) = ug =
8>>>>><>>>>>:
A(s)i if u = s 2 FsA(s)i
cif u = t 2 Fs Ft
? otherwise 2 F0 Fu
.
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Denition
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Optional Stopping Theorem VIProof of the theorem
So, by hypothesis, we have that
EP [XSi ] = EP [X0] .
Besides, since the random time τt dened as 8ω 2 Ω,τt (ω) = t is also a bounded stopping time, we have, again byhypothesis, that
EP [Xt ] = EP [Xτt ] = EP [X0]
henceEP [XSi ] = EP [Xt ] .
Martingales
Denition
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Optional Stopping Theorem VIIProof of the theorem
As a consequence, 8i 2 f1, . . . , nsg,
0 = EP [Xt ] EP [XSi ]
= EP [Xt XSi ]
= EP
(Xt XSi ) I
A(s)i+ (Xt XSi ) I
A(s)ic
= EP
(Xt Xs ) I
A(s)i+ (Xt Xt ) I
A(s)ic
= EPh(Xt Xs ) I
A(s)i
i= ∑
ω2A(s)i
(Xt (ω) Xs (ω))P (ω)
hence
∑ω2A(s)i
Xt (ω)P (ω) = ∑ω2A(s)i
Xs (ω)P (ω) .
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Denition
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Optional Stopping Theorem VIIIProof of the theorem
Now we can conclude the proof, since
EP [Xt jFs ] =ns
∑i=1
IA(s)i
PA(s)i
∑ω2A(s)i
Xt (ω)P (ω)
=ns
∑i=1
IA(s)i
PA(s)i
∑ω2A(s)i
Xs (ω)P (ω)
= EP [Xs jFs ]= Xs .
Martingales
Denition
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Markovian process IDenition
DenitionA stochastic process X = fXt : t 2 T g, where T is a set ofindicesa, is said to be markovian if, for anyt1 < t2 < . . . < tn 2 T , the conditional distribution of Xtngiven Xt1 , . . . ,Xtn1 is equal to the conditional distribution ofXtn given Xtn1 , i.e. for any x1, . . . , xn 2 R,
P [Xtn xn jXt1 = x1, . . . ,Xtn1 = xn1 ]= P [Xtn xn jXtn1 = xn1 ] .
aExamples: T = f0, 1, 2, . . .g, T = f0, 1, 2, . . . ,T g where T is apositive integer, T = [0,T ] where T is a positive real number, T = [0,∞),etc.
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OptionalStoppingTheorem
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Markovian process IIDenition
Intuitively, if we assume that t are temporal indices, theprocess X is markovian if its distribution in the future, giventhe present and the past, only depends on the present.
A Markov chain is then a memoryless randomphenomenon: the distribution of an observation tocome, given our present knowledge of the system andits whole history, is the same when only its presentstate is known.1
The set of values that the process may take is called the statespace of X and we denote it by EX .
1Jean Vaillancourt.
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Stoppedprocess
OptionalStoppingTheorem
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Markovian process IExample
Example. We throw a dice repeatedly.The random variable ξn represents the number of pointsobtained on the n th throw.The stochastic process X represents the total cumulativenumber of points obtained at any time, i.e. for any naturalinteger t,
Xt =t
∑n=1
ξn.
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Denition
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Markovian process IIExample
X is a Markovian process. Indeed,
Xt =t
∑n=1
ξn =t1∑n=1
ξn + ξt = Xt1 + ξt .
But the outcome of the t th throw of dice, ξt , is independentfrom the results obtained on the rst t 1 th throws,σ fξn : n 2 f1, . . . , t 1gg. As a consequence, the distributionof X t depends on the past of the stochastic process,σ fξn : n 2 f1, . . . , t 1gg , through σ fXt1g only.
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Denition
Example
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OptionalStoppingTheorem
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Markovian processRemark
Question. Why do we need a probability space? Wouldnta measurable space have been su¢ cient?
Answer. A probability measure is required to ensure thatthe independence property is satised.
Martingales
Denition
Example
Stoppedprocess
OptionalStoppingTheorem
Markovianprocess
Markovian processRandom walk
The stochastic process X , built on the probability space(Ω,F ,P), is a random walk if it admits the representation
X0 = 0 et 8t 2 f1, 2, . . .g , Xt =t
∑n=1
ξn
where the sequence fξt : t 2 f1, 2, . . .gg is made up ofindependent and identically distributed random variables.Random walks are Markovian processes.
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