Stochastic processes

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processes

Stochastic processes Filtration Stopping time References

Stochastic Processes

80-646-08 Calcul stochastique

Geneviève Gauthier

HEC Montréal

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processes

Stochastic processes

Filtration Stopping time References

Stochastic processes

De…nition

Let(Ω,F)be a measurable space. A stochastic process X = f^X^t ^:^t 2 T g

is a family of random variables, all built on the same

measurable space(Ω,F) whereT represents a set of indices.

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processes

Stochastic processes Filtration De…nitions Example

Stopping time References

Filtration I

De…nitions

De…nition

A familyF=fF^t ^:^t 2 T g^of ^σ algebras on Ωis a …ltration on the measurable space(_Ω,F) if

(F1) 8^t 2 T^,F^t F^,

(F2) 8^t¹^,^t² 2 T ^{such that}^t¹ ^t²^,F^t1 F^t2.

De…nition

A stochastic processX =f^X^t ^:^t 2 T g is said to be adapted to the …ltrationF=fF^t ^:^t2 T g ^if

8^t 2 T^,^X^t ^is F^t measurable.

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processes

Filtration II

De…nitions

De…nition

The …ltrationF=fF^t ^:^t 2 T g is said to be generated by the stochastic processX = f^X^t ^:^t 2 T g^if

8^t 2 T^,F^t =σf^X^s ^:^s 2 T^,^s ^tg^.

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processes

Filtration I

Example

Example¹. Let’s assume that the sample space is Ω=f^ω¹^,^ω²^,^ω³^,^ω⁴g^{and that} T =f^0,^1,^2,³g^{. The} stochastic processX = f^X^t ^:^t 2 f^0,^1,^2,³gg represents the evolution of a stock price,Xt = the stock price at close of market on thet th day, while time t =0 represents today.

ω X₀(ω) X₁(ω) X₂(ω) X₃(ω) ω1 1 0,50 1 0,50 ω2 1 0,50 1 0,50

ω3 1 2 1 1

ω4 1 2 2 2

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processes

Filtration II

Example

Question. What is the …ltration generated by this stochastic process?

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processes

Filtration III

Example

ω X₀(ω) X₁(ω) X₂(ω) X₃(ω) ω1 1 0,50 1 0,50 ω2 1 0,50 1 0,50

ω3 1 2 1 1

ω4 1 2 2 2

Answer.

F⁰ = σf^X⁰g=f?^,^Ωg^,

F¹ = σf^X⁰^,^X¹g=σff^ω¹^,^ω²g^,f^ω³^,^ω⁴gg^, F² = σf^X⁰^,^X¹^,^X²g= σff^ω¹^,^ω²g^,f^ω³g^,f^ω⁴gg^, F³ = σf^X⁰^,^X¹^,^X²^,^X³g=σff^ω¹^,^ω²g^,f^ω³g^,f^ω⁴gg^.

Note that anyσ algebraF containing the sub-σ algebraF3 makeX₀, X1,X2 andX3 F measurable.

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processes

Filtration IV

Example

Recall that

F⁰ = σf^X⁰g=f?^,^Ωg^,

F1 = σf^X⁰^,^X¹g=σff^ω¹^,^ω²g^,f^ω³^,^ω⁴gg^, F2 = σf^X⁰^,^X¹^,^X²g=σff^ω¹^,^ω²g^,f^ω³g^,f^ω⁴gg^, F3 = σf^X⁰^,^X¹^,^X²^,^X³g=σff^ω¹^,^ω²g^,f^ω³g^,f^ω⁴gg^. Interpretation.Ωrepresents states of nature. Xt(ω_i)represents the stock price at timet if it is thei i th state of nature that has occurred. At time 0 (today), we know with certitude the stock price and we cannot identify which of the states of nature has occurred. That’s why the sub-σ algebra F0 is the trivialσ algebra, since it doesn’t contain any information.

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processes

Filtration V

Example

Recall that

F1=σf^X0,X₁g=σff^ω1,ω2g^,f^ω3,ω4gg^.

1 At timet=1, we know a bit more. Indeed, if we observe a stock price of 0.50, then we know that the state of nature that has occurred is ω₁ orω₂ but certainly notω₃ orω₄. As a result, we can deduce that the stock price for the following two periods (t=2 and t=3) will be 1 and 0.50 dollar respectively.

2 On the contrary, if at timet=1, we observe a stock price of 2 dollars, then we know that the state of nature that has occurred is eitherω3 orω4. We can deduce from there that the stock price won’t fall back under the one-dollar level: because, after observing the process at timet=1, we’ll be able to determine whether event f^ω1,ω₂g^{or event}f^ω3,ω₄ghas happened,

F1=σff^ω¹^,^ω²g^,f^ω³^,^ω⁴gg^.

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processes

Filtration VI

Example

Recall that

F2=σf^X⁰^,^X¹^,^X²g=σff^ω¹^,^ω²g^,f^ω³g^,f^ω⁴gg^.

Let’s now assume that in timet=2, we observe a price of one dollar. A frequently made mistake is to conclude that the sub-σ algebra associated with that time isσff^ω1,ω₂,ω₃g^,f^ω4gg since by observingX₂ we are able to distinguish between the eventsf^ω¹^,^ω²^,^ω³g^andf^ω⁴g^{. That} would be true if we were just beginning to observe the process, which is not the case. We must take into account the information obtained since timet=0. But the paths(X0(ω),X1(ω),X2(ω))enable us to distinguish between the three following events: f^ω1,ω2g^,f^ω3g^andf^ω4g^{. Indeed,} after observing the prices until time two, we will know with certitude which state of natureω has occurred, unless we have observed path(1,¹₂,1), in which case we’ll be unable to distinguish between states of natureω1 and ω₂.

1Throughout this chapter, we go further into an example initiated in Stochastic Calculus, A Tool for Financeby Daniel Dufresne.

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processes

Stochastic processes Filtration Stopping time

De…nition Example Transformations First passage References

Stopping time

Introduction

We will realize how very useful the concept of stopping time is when we will attempt to price American-style derivative

products. The main role of stopping times is to help determine the time when the option holder will exercise his or her right.

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processes

Stopping time

De…nition

Let(Ω,F)be a measurable space such that Card(Ω)< _∞ and equipped with the …ltrationF= fF^t ^:^t 2 f^0,^{1, ...}gg^{. A} stopping time τis a (_Ω,F) random variable that takes its values inf^0,^{1, ...}gand is such that

f^ω2 Ω:τ(ω) tg 2 F^t ^{for all} ^t 2 f^0,^{1, ...}g^. ⁽¹⁾ Exercise. Show that the condition (1) above is equivalent to

f^ω2 ^Ω^:^τ(ω) =tg 2 F^t ^{for all} ^t 2 f^0,^{1, ...}g^.

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processes

Stopping time I

Example

Example. Let’s return to the example described earlier: X represents a stock price.

ω X0(ω) X1(ω) X2(ω) X3(ω) ω1 1 0,50 1 0,50 ω2 1 0,50 1 0,50

ω3 1 2 1 1

ω4 1 2 2 2

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processes

Stopping time II

Example

We had determined that the …ltration containing the information revealed by the process at each time is

F⁰ = σf^X⁰g=f?^,Ωg^,

F¹ = σf^X⁰^,^X¹g=σff^ω¹^,^ω²g^,f^ω³^,^ω⁴gg^, F² = σf^X⁰^,^X¹^,^X²g= σff^ω¹^,^ω²g^,f^ω³g^,f^ω⁴gg^, F³ = σf^X⁰^,^X¹^,^X²^,^X³g=σff^ω¹^,^ω²g^,f^ω³g^,f^ω⁴gg^.

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processes

Stopping time III

Example

Recall that:

ω X0(ω) X1(ω) X2(ω) X3(ω)

ω1 1 0,50 1 0,50 ω₂ 1 0,50 1 0,50

ω3 1 2 1 1

ω₄ 1 2 2 2

We won’t sell our stocks today (t =0)but we will sell them as soon as the price is greater than or equal to 1.

The random time representing that situation is τ(ω1) =2, τ(ω2) =2, τ(ω3) =1 and τ(ω4) =1.

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processes

Stopping time IV

Example

Such a random variable truly is a stopping time since

f^ω 2^Ω^:^τ(ω) =0g = ?2 F⁰^,

f^ω 2^Ω^:^τ(ω) =1g = f^ω³^,^ω⁴g 2 F¹^, f^ω 2^Ω^:^τ(ω) =2g = f^ω¹^,^ω²g 2 F²^, f^ω 2Ω:τ(ω) =3g = ?2 F³^.

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processes

Stopping time V

Example Recall that:

ω X₀(ω) X₁(ω) X₂(ω) X₃(ω)

ω₁ 1 0,50 1 0,50 ω2 1 0,50 1 0,50

ω₃ 1 2 1 1

ω4 1 2 2 2

Let’s now consider the random time τ modelling the following situation: we’ll buy stock as soon as it enables us to make a pro…t later.

Such a random value takes values τ (ω1) =1,

τ (ω2) =1, τ (ω3) =0 and τ (ω4) =0. τ is not a stopping time since

f^ω 2^Ω^:^τ (ω) =0g=f^ω³^,^ω⁴g2 F^/ ⁰^.

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processes

Stopping time VI

Example

Intuitively, the time τ when one makes a decision is a stopping time if the decision is made based on the

information available at that time. In the case of stopping times, using a crystal ball is prohibited.

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processes

Stopping time I

Stopping time transformation

Theorem

Let(_Ω,F), be a measurable space such that Card(_Ω)<_∞ and equipped with the …ltrationF= fF^t ^:^t 2 f^0,^{1, ...}gg^{. If} the random variablesτ1 andτ2 are stopping times with respect to the …ltrationF, then τ1^^τ² ^minf^τ¹^,^τ²g ^and

τ1_^τ² ^maxf^τ¹^,^τ²gare also stopping times.

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processes

Stopping time II

Stopping time transformation

Proof of the theorem. Ifτ is a stopping time, then 8^t 2 f^0,^{1, ...}g

f^ω2^Ω^:^τ(ω) tg 2 F^t^. 8^k 2 f^0,^{1, ...}g^,

f^ω 2Ω:τ1(ω)^^τ²(ω) kg

= f^ω 2Ω:τ1(ω) k or τ2(ω) kg

= f_|^ω 2Ω:τ_{z1(ω) kg_}

2F^k

[ f_|^ω2Ω:τ_{z2(ω) kg_}

2F^k

2 F^k^.

Exercise. Prove the above result for τ1_^τ²^.

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processes

Stopping time I

First passage time

De…nition

Let(_Ω,F)be a measurable space such that Card(_Ω)< _∞ and equipped with the …ltrationF= fF^t ^:^t 2 f^0,^{1, ...}gg^. X = f^X^t ^:^t 2 f^0,^{1, ...}gg represents a stochastic process adapted to that …ltration. LetB R a subset of the real numbers. We de…ne the time until the stochastic processX

…rst enters the setB as

τB(ω) =minf^t 2 f^0,^{1, ...}g^:^X^t(ω)2^Bg^. If it happened that the patht !^X^t(ω)never hits the set B then we de…neτB(ω) =∞.

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processes

Stopping time II

First passage time

Theorem

The random variableτB is a stopping time.

Proof of the theorem. Since Card(Ω)<_∞, then 8^t 2 f^0,^{1, ...}g^,^X^t can only take a …nite number of values.

Let’s denote them by

x₁⁽^t⁾ <...<x_m⁽^t_t⁾. 8^t 2 f^0,^{1, ...}g^,

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processes

Stopping time III

First passage time

f^ω2^Ω^:^τB(ω) =tg

= f^ω2^Ω^:^X⁰(ω)2/^{B, ...,}^X^t ¹(ω)2/^B^,^X^t(ω)2^Bg

=

t\1 k=0

f^ω2^Ω^:^Xk(ω)2/^Bg

!

\ f^ω2^Ω^:^Xt(ω)2^Bg

= 0 B@

t\1 k=0

[ x_i^(k)/2B

n

ω2^Ω^:^Xk(ω) =x_i^(k)o 1 CA

\ 0 B@ ^[

x_i^(t)2B

n

ω2Ω:Xt(ω) =x_i^(t)o1 CA

2 F^t

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processes

Stopping time IV

First passage time

since

t\1 k=0

[ x_i^(k)2/B

n

ω2^Ω^:^Xk(ω) =x_i^(k⁾o

| {z }

2FksinceXis adapted.

| {z }

2Fk FtsinceFkis aσ algebra

| {z }

2FtsinceFtis aσ algebra.

\ ^[

x_i^(t)2B

n

ω2^Ω^:^X^t(ω) =x_i^(t)o

| {z }

2FtsinceXtisFt measurable.

| {z }

2FtsinceFtis aσ algebra.

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processes

Stochastic processes Filtration Stopping time References

References

BILLINGSLEY, Patrick (1986). Probability and Measure, Second Edition, Wiley, New York.

DUFRESNE, Daniel (1996). Stochastic Calculus, A Tool for Finance, Department of Mathematics and Statistics, Université de Montréal.

KARLIN Samuel and TAYLOR Howard M. (1975). A First Course in Stochastic Processes, Second Edition, Academic Press, New York.