80-646-08 Calcul stochastique GeneviŁve...
Transcript of 80-646-08 Calcul stochastique GeneviŁve...
Stochasticprocesses
Stochasticprocesses
Filtration
Stopping time
References
Stochastic Processes80-646-08
Calcul stochastique
Geneviève Gauthier
HEC Montréal
Stochasticprocesses
Stochasticprocesses
Filtration
Stopping time
References
Stochastic processesDenition
DenitionLet (Ω,F ) be a measurable space. A stochastic process
X = fXt : t 2 T g
is a family of random variables, all built on the samemeasurable space (Ω,F ) where T represents a set of indices.
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Stochasticprocesses
FiltrationDenitionsExample
Stopping time
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Filtration IDenitions
DenitionA family F = fFt : t 2 T g of σalgebras on Ω is a ltrationon the measurable space (Ω,F ) if
(F1) 8t 2 T , Ft F ,(F2) 8t1, t2 2 T such that t1 t2, Ft1 Ft2 .
DenitionA stochastic process X = fXt : t 2 T g is said to be adaptedto the ltration F = fFt : t 2 T g if
8t 2 T , Xt is Ft measurable.
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FiltrationDenitionsExample
Stopping time
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Filtration IIDenitions
DenitionThe ltration F = fFt : t 2 T g is said to be generated by thestochastic process X = fXt : t 2 T g if
8t 2 T , Ft = σ fXs : s 2 T , s tg .
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FiltrationDenitionsExample
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Filtration IExample
Example1. Lets assume that the sample space isΩ = fω1,ω2,ω3,ω4g and that T = f0, 1, 2, 3g. Thestochastic process X = fXt : t 2 f0, 1, 2, 3gg represents theevolution of a stock price, Xt = the stock price at close ofmarket on the t th day, while time t = 0 represents today.
ω X0 (ω) X1 (ω) X2 (ω) X3 (ω)
ω1 1 0, 50 1 0, 50ω2 1 0, 50 1 0, 50ω3 1 2 1 1ω4 1 2 2 2
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FiltrationDenitionsExample
Stopping time
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Filtration IIExample
Question. What is the ltration generated by this stochasticprocess?
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FiltrationDenitionsExample
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Filtration IIIExample
ω X0 (ω) X1 (ω) X2 (ω) X3 (ω)
ω1 1 0, 50 1 0, 50ω2 1 0, 50 1 0, 50ω3 1 2 1 1ω4 1 2 2 2
Answer.
F0 = σ fX0g = f?,Ωg ,F1 = σ fX0,X1g = σ ffω1,ω2g , fω3,ω4gg ,F2 = σ fX0,X1,X2g = σ ffω1,ω2g , fω3g , fω4gg ,F3 = σ fX0,X1,X2,X3g = σ ffω1,ω2g , fω3g , fω4gg .
Note that any σalgebra F containing the sub-σalgebra F3 make X0,X1, X2 and X3 Fmeasurable.
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FiltrationDenitionsExample
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Filtration IVExample
Recall that
F0 = σ fX0g = f?,Ωg ,F1 = σ fX0,X1g = σ ffω1,ω2g , fω3,ω4gg ,F2 = σ fX0,X1,X2g = σ ffω1,ω2g , fω3g , fω4gg ,F3 = σ fX0,X1,X2,X3g = σ ffω1,ω2g , fω3g , fω4gg .
Interpretation. Ω represents states of nature. Xt (ωi ) represents the stockprice at time t if it is the i i th state of nature that has occurred. At time0 (today), we know with certitude the stock price and we cannot identifywhich of the states of nature has occurred. Thats why the sub-σalgebraF0 is the trivial σalgebra, since it doesnt contain any information.
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Stochasticprocesses
FiltrationDenitionsExample
Stopping time
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Filtration VExample
Recall that
F1 = σ fX0,X1g = σ ffω1,ω2g , fω3,ω4gg .
1 At time t = 1, we know a bit more. Indeed, if we observe a stockprice of 0.50, then we know that the state of nature that hasoccurred is ω1 or ω2 but certainly not ω3 or ω4. As a result, we candeduce that the stock price for the following two periods (t = 2 andt = 3) will be 1 and 0.50 dollar respectively.
2 On the contrary, if at time t = 1, we observe a stock price of 2dollars, then we know that the state of nature that has occurred iseither ω3 or ω4. We can deduce from there that the stock pricewont fall back under the one-dollar level: because, after observingthe process at time t = 1, well be able to determine whether eventfω1,ω2g or event fω3,ω4g has happened,F1 = σ ffω1,ω2g , fω3,ω4gg.
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FiltrationDenitionsExample
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Filtration VIExample
Recall that
F2 = σ fX0,X1,X2g = σ ffω1,ω2g , fω3g , fω4gg .
Lets now assume that in time t = 2, we observe a price of one dollar. Afrequently made mistake is to conclude that the sub-σalgebra associatedwith that time is σ ffω1,ω2,ω3g , fω4gg since by observing X2 we areable to distinguish between the events fω1,ω2,ω3g and fω4g. Thatwould be true if we were just beginning to observe the process, which isnot the case. We must take into account the information obtained sincetime t = 0. But the paths (X0(ω),X1(ω),X2(ω)) enable us to distinguishbetween the three following events: fω1,ω2g, fω3g and fω4g. Indeed,after observing the prices until time two, we will know with certitude whichstate of nature ω has occurred, unless we have observed path (1, 12 , 1), inwhich case well be unable to distinguish between states of nature ω1 andω2.
1Throughout this chapter, we go further into an example initiated inStochastic Calculus, A Tool for Finance by Daniel Dufresne.
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Stopping timeDenitionExampleTransformationsFirst passage
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Stopping timeIntroduction
We will realize how very useful the concept of stopping time iswhen we will attempt to price American-style derivativeproducts. The main role of stopping times is to help determinethe time when the option holder will exercise his or her right.
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Stopping timeDenitionExampleTransformationsFirst passage
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Stopping timeDenition
DenitionLet (Ω,F ) be a measurable space such that Card (Ω) < ∞and equipped with the ltration F = fFt : t 2 f0, 1, ...gg. Astopping time τ is a (Ω,F )random variable that takes itsvalues in f0, 1, ...g and is such that
fω 2 Ω : τ (ω) tg 2 Ft for all t 2 f0, 1, ...g . (1)
Exercise. Show that the condition (1) above is equivalent to
fω 2 Ω : τ (ω) = tg 2 Ft for all t 2 f0, 1, ...g .
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Stopping timeDenitionExampleTransformationsFirst passage
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Stopping time IExample
Example. Lets return to the example described earlier: Xrepresents a stock price.
ω X0 (ω) X1 (ω) X2 (ω) X3 (ω)
ω1 1 0, 50 1 0, 50ω2 1 0, 50 1 0, 50ω3 1 2 1 1ω4 1 2 2 2
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Filtration
Stopping timeDenitionExampleTransformationsFirst passage
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Stopping time IIExample
We had determined that the ltration containing theinformation revealed by the process at each time is
F0 = σ fX0g = f?,Ωg ,F1 = σ fX0,X1g = σ ffω1,ω2g , fω3,ω4gg ,F2 = σ fX0,X1,X2g = σ ffω1,ω2g , fω3g , fω4gg ,F3 = σ fX0,X1,X2,X3g = σ ffω1,ω2g , fω3g , fω4gg .
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Stopping timeDenitionExampleTransformationsFirst passage
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Stopping time IIIExample
Recall that:
ω X0 (ω) X1 (ω) X2 (ω) X3 (ω)
ω1 1 0, 50 1 0, 50ω2 1 0, 50 1 0, 50ω3 1 2 1 1ω4 1 2 2 2
We wont sell our stocks today (t = 0) but we will sellthem as soon as the price is greater than or equal to 1.
The random time representing that situation isτ (ω1) = 2, τ (ω2) = 2, τ (ω3) = 1 and τ (ω4) = 1.
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Stopping timeDenitionExampleTransformationsFirst passage
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Stopping time IVExample
Such a random variable truly is a stopping time since
fω 2 Ω : τ (ω) = 0g = ? 2 F0,fω 2 Ω : τ (ω) = 1g = fω3,ω4g 2 F1,fω 2 Ω : τ (ω) = 2g = fω1,ω2g 2 F2,fω 2 Ω : τ (ω) = 3g = ? 2 F3.
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Stopping timeDenitionExampleTransformationsFirst passage
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Stopping time VExample
Recall that:
ω X0 (ω) X1 (ω) X2 (ω) X3 (ω)
ω1 1 0, 50 1 0, 50ω2 1 0, 50 1 0, 50ω3 1 2 1 1ω4 1 2 2 2
Lets now consider the random time τ modelling thefollowing situation: well buy stock as soon as it enablesus to make a prot later.
Such a random value takes values τ (ω1) = 1,τ (ω2) = 1, τ (ω3) = 0 and τ (ω4) = 0. τ is not astopping time since
fω 2 Ω : τ (ω) = 0g = fω3,ω4g /2 F0.
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Stopping timeDenitionExampleTransformationsFirst passage
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Stopping time VIExample
Intuitively, the time τ when one makes a decision is astopping time if the decision is made based on theinformation available at that time. In the case of stoppingtimes, using a crystal ball is prohibited.
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Stopping timeDenitionExampleTransformationsFirst passage
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Stopping time IStopping time transformation
TheoremLet (Ω,F ), be a measurable space such that Card (Ω) < ∞and equipped with the ltration F = fFt : t 2 f0, 1, ...gg. Ifthe random variables τ1 and τ2 are stopping times with respectto the ltration F, then τ1 ^ τ2 min fτ1, τ2g andτ1 _ τ2 max fτ1, τ2g are also stopping times.
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Stopping time IIStopping time transformation
Proof of the theorem. If τ is a stopping time, then8t 2 f0, 1, ...g
fω 2 Ω : τ (ω) tg 2 Ft .
8k 2 f0, 1, ...g ,
fω 2 Ω : τ1 (ω) ^ τ2 (ω) kg= fω 2 Ω : τ1 (ω) k or τ2 (ω) kg= fω 2 Ω : τ1 (ω) kg| z
2Fk
[ fω 2 Ω : τ2 (ω) kg| z 2Fk
2 Fk .
Exercise. Prove the above result for τ1 _ τ2.
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Stopping timeDenitionExampleTransformationsFirst passage
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Stopping time IFirst passage time
DenitionLet (Ω,F ) be a measurable space such that Card (Ω) < ∞and equipped with the ltration F = fFt : t 2 f0, 1, ...gg.X = fXt : t 2 f0, 1, ...gg represents a stochastic processadapted to that ltration. Let B R a subset of the realnumbers. We dene the time until the stochastic process Xrst enters the set B as
τB (ω) = min ft 2 f0, 1, ...g : Xt (ω) 2 Bg .
If it happened that the path t ! Xt (ω) never hits the set Bthen we dene τB (ω) = ∞.
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Stopping time IIFirst passage time
TheoremThe random variable τB is a stopping time.
Proof of the theorem. Since Card (Ω) < ∞, then8t 2 f0, 1, ...g, Xt can only take a nite number of values.Lets denote them by
x (t)1 < ... < x (t)mt .
8t 2 f0, 1, ...g,
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Stopping timeDenitionExampleTransformationsFirst passage
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Stopping time IIIFirst passage time
fω 2 Ω : τB (ω) = tg= fω 2 Ω : X0 (ω) /2 B , ...,Xt1 (ω) /2 B ,Xt (ω) 2 Bg
=
t1\k=0
fω 2 Ω : Xk (ω) /2 Bg!\ fω 2 Ω : Xt (ω) 2 Bg
=
0B@t1\k=0
[x (k )i /2B
nω 2 Ω : Xk (ω) = x
(k )i
o1CA\
0B@ [x (t)i 2B
nω 2 Ω : Xt (ω) = x
(t)i
o1CA2 Ft
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Stopping time IVFirst passage time
since
t1\k=0
[x (k )i /2B
nω 2 Ω : Xk (ω) = x
(k )i
o| z
2Fk since X is adapted.| z 2FkFt since Fk is a σalgebra| z 2Ft since Ft is a σalgebra.
\[
x (t)i 2B
nω 2 Ω : Xt (ω) = x
(t)i
o| z 2Ft since Xt is Ftmeasurable.| z
2Ft since Ft is a σalgebra.
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References
BILLINGSLEY, Patrick (1986). Probability and Measure,Second Edition, Wiley, New York.
DUFRESNE, Daniel (1996). Stochastic Calculus, A Toolfor Finance, Department of Mathematics and Statistics,Université de Montréal.
KARLIN Samuel and TAYLOR Howard M. (1975). A FirstCourse in Stochastic Processes, Second Edition, AcademicPress, New York.