Stochastic di¤erential equation (SDE) and Ito’s lemma

(1)

Introduction Itô’s lemma Solutions to

SDEs

Stochastic di¤erential equation (SDE) and Ito’s lemma

80-646-08 Stochastic Calculus I

Geneviève Gauthier

HEC Montréal

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Introduction Equations Ordinary di¤erential equations Introduction to SDE Itô’s lemma Solutions to SDEs

Ordinary di¤erential eq. I

When we model some situations, we don’t know a priori which function to use, since we only know the local behavior of our system.

For example, assume that

f ( t ) represents a commodity price at time t . We write

f ( t +

_∆

t ) f ( t ) =

µ ∆

t f ( t ) where

µ

0 in order to mean that the variation f ( t +

∆

t ) f ( t ) of the commodity price over a time period is is proportional to the length

∆

t of the time period considered as well as the commodity price f ( t ) at the start of the period, i.e.

µ

∆

t f ( t )

,µ

being a constant.

(3)

Ordinary di¤erential eq. II

By dividing both sides of the equality by

∆

t, we obtain f ( t +

_∆

t ) f ( t )

∆

t =

µ

f ( t )

.

Let’s now take the limit when

∆

t tends to zero:

d

dt f ( t ) = lim

∆t!⁰

f ( t +

_∆

t ) f ( t )

∆

t

= lim

∆t!⁰µ

f ( t )

=

µ

f ( t )

.

(4)

Ordinary di¤erential eq. III

Recall that we consider the equation d

dt f ( t ) =

µ

f ( t )

.

The notation commonly used for di¤erential equations allows us to rewrite the equation above as:

d f ( t ) =

µ

f ( t ) dt

.

(1) Note that, technically speaking, the object d f ( t ) is not well-de…ned. The latter equation is only a notation to express that ”the derivative of the function is proportional to the function itself”, i.e.

_dt^d

f ( t ) =

µ

f ( t )

.

The unknown, in that equation, is the function f . We are

looking for the functions that satisfy that equality.

(5)

Ordinary di¤erential eq. IV

Recall that we are studying the equation

d f ( t ) =

µ

f ( t ) dt

.

(1) It is possible to show that the function de…ned for all t 2

^R

^as

f ( t ) = ce

^µt,

where c is any constant, (2) satis…es equation (1).

Indeed, in such a case, d

dt f ( t ) = ^d

dt ce

^µt

=

µce^µt

=

µf

( t )

.

(6)

Ordinary di¤erential eq. V

The initial condition helps determine the constant c . We know commodity price f

₀

today. As a consequence,

f

₀

= f ( 0 ) = ce

^µ ⁰

= c

,

which yields that the commodity price at time t is f ( t ) = f

₀

e

^µt.

In this example, knowing the in…nitesimal behavior of the

commodity price ( d f ( t ) =

µ

f ( t ) dt ) and the initial

price f

0

is su¢ cient to determine accurately the price at

any time.

(7)

Ordinary di¤erential eq. VI

Recall that we are considering the equation

d f ( t ) =

µ

f ( t ) dt

.

(1) equation (1) is an example of ordinary di¤erential equation and it behaves very charmingly since there exists at least one function f which satis…es equation (1) and, in addition, it is possible to show that this function is necessarily of the form described in (2) :

f ( t ) = f

₀

e

^µt.

(8)

Ordinary di¤erential eq. VII

There exist ordinary di¤erential equations much less nice.

For example,

d f ( t ) = ^f ( t )

t

²

dt

.

(3)

The solution to that equation has form

f ( t ) = ce

¹^t

where c is a constant.

We must now specify c by using the initial condition.

But f ( 0 ) = 0 whatever c

,

which implies that (3)

possesses an in…nity of solutions when f ( 0 ) = 0 and

possesses none when f ( 0 ) 6 = 0.

(9)

Introduction I

Stochastic di¤erential equations

Now assume that the stochastic process S = f ^S

^t ^:

^t ⁰ g represents the evolution of a risky asset price.

We don’t know, in general, the law that governs such a

process, but we may have an idea of its local behavior.

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Introduction II

For example, over a short time interval of length ∆

t, it is possible that such a price tends to vary proportionally to the period length and the asset price at the beginning of the period. We write, to begin with,

S

_t+_∆t

S

t

=

µ

S

t ∆

t.

If, in general, prices increase, then

µ

is a positive constant

and if the prices tend to decrease, then

µ

is negative.

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Introduction III

S

_t+_∆t

S

t

=

µ

S

t ∆

t

There is however a problem with the latter equation: we are not certain that the price varies proportionally to the period length and the asset price, we only claim that it has a tendency to do so.

Therefore, an unpredictable error needs to be incorporated into our equation.

We can however control the magnitude of such a random

error.

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Introduction IV

For example, we can assume that it depends on the asset

price at the beginning of the period.

Indeed, we observe that, the higher the price, the more the risky asset price can diverge from the trend.

Moreover, the random error must also depend on the length of the time interval considered: the longer the interval, the greater the chance that the price diverges from the trend.

That is why we add a stochastic term to our initial equation.

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Introduction V

The stochastic term to be added to our initial equation leads us to the equation

S

_t₊_∆_t

S

_t

=

µ

S

_t ∆

t +

σ

S

_t

p

∆

t

ξ_t

(4) where

σ is a positive constant and

ξ_t is a random variable with distributionN(0,1) independent fromf^S^u ^:⁰ ^u ^tg^.

The latter condition is important, since we must not be

able to predict the error

ξ_t

from observing the behavior of

the risky asset price prior to date t.

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Introduction VI

Such an equation is random and must be satis…ed by

”almost” every

ω, which is to say that

Prhn

ω2Ω:St+∆t(ω) St(ω) =µSt(ω) _∆t+σSt(ω) p

∆tξ_t(ω)^oi

worth one.

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Introduction VII

With respect to the magnitude of the random error, note that p

∆

t

ξ_t

is N ( 0,

∆

t ) -distributed. Moreover,

Eh σSt

p∆tξ_t j^σf^S^u ^:^u2 f^0,^{∆t, ...,}^tggⁱ = σSt

p∆tE[ξ_t]

= 0,

E σSt

p∆tξ_t ² j^σf^S^u ^:^u2 f^0,^{∆t, ...,}^tgg = σ²S_t²∆tEh ξ²_t

i

= σ²S_t²∆t,

which implies that the conditional standard deviation of the error term is

σSt

p

∆

t .

Thus the longer the time interval

∆

t or the higher the

security price S

_t

, the greater the standard deviation of the

random error.

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Introduction VIII

This implies that the values that the random error may

take are more dispersed around its expectation (which is

zero).

(17)

Introduction IX

Recall that

S

_t₊_∆_t

S

_t

=

µ

S

_t ∆

t +

σ

S

_t

p

∆

t

ξ_t.

(4) Let’s rewrite equation (4) for the next period:

S

_t₊₂_∆_t

S

_t₊_∆_t

=

µ

S

_t₊_∆_t ∆

t +

σ

S

_t₊_∆_t

p

∆

t

ξ_t₊_∆_t.

If we don’t want to be able to predict the error

ξ_t₊_∆_t

, the latter must be independent from

f ^S

^u ^:

^u 2 f ^0,

∆

t

, ...,

t +

_∆

t gg ^.

For that reason, we introduce the Brownian motion since it is a Gaussian process, the increments of which are mutually independent:

S

_t₊_∆_t

S

_t

=

µ

S

_t ∆

t +

σ

S

_t

( W

_t₊_∆_t

W

_t

)

.

(5) Note that the law of W

_t+_∆t

W

t

is the same as the law of p

∆

tξ

_t

: both are N ( 0,

∆

t ) -distributed.

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Introduction X

Let W = f ^W

^t ^:

^t ⁰ g be a Brownian motion built on a

…ltered probability space (

Ω,

F

^,^F^,^P

) such that the

…ltration

F

is the one generated by the Brownian motion, plus it includes all zero-probability events, i.e. for all t 0,

F

^t

=

σ

( N ^and ^W

^s ^:

⁰ ^s ^t )

.

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Introduction XI

The risky asset price today ( t = 0 ) is known with certainty. S

₀

is thus ( ?

^,^Ω

) measurable and therefore F

⁰

measurable. Let’s go back to equation (5) and let’s set t = 0.

S_∆t= S0

|{z}

F0 measurable

+ µS0 ∆t

| {z }

F0 measurable

+ σS0

|{z}

F0 measurable

(W_∆t W0)

| {z }

F∆t measurable indpendent fromF0

| {z }

F∆t measurable

We observe that S

_∆_t

is F

∆t

measurable.

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Introduction XII

We can show by induction that S

_n_∆_t

is F

ⁿ∆t

measurable, for any positive integer n.

Indeed, let’s assume there exists k 2 f ^0, ^1, ^{2, ...} g ^{such that} S

_k _∆_t

is F

k ∆t

measurable. Then

S₍_k₊₁₎_∆t

= Sk∆t

| {z } Fk∆t measurable

+ µSk∆t ∆t

| {z }

Fk∆t measurable

+ σSk∆t

| {z } Fk∆t measurable

W₍k+1)∆t Wk∆t

| {z }

F(k+1)∆t measurable independent fromFk∆t

| {z }

F(k+1)∆t measurable

which implies that S

₍_k₊₁₎_∆_t

is F

(k+1)_∆t

measurable.

Our process S lives on the same …ltered probability space

as the Brownian motion W that we have used to build

that process.

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Introduction XIII

Let’s go back to equation (5)

S

_t₊_∆_t

S

_t

=

µ

S

_t ∆

t +

σ

S

_t

( W

_t₊_∆_t

W

_t

)

.

When time intervals of length

∆

t become of in…nitesimal length, we obtain an equation of the type

dS

_t

=

µ

S

_t

dt +

σ

S

_t

dW

_t.

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Introduction XIV

The latter equation is an example of stochastic di¤erential equation and it should raise a few questions:

1 the termσ S_t dW_t is not well de…ned, particularly if we recall that the Brownian motion paths are nowhere di¤erentiable!

2 does a solution to that equation exist? and

3 if that solution exists, is it unique and how can it be found?

Note that the solution to a stochastic di¤erential equation

is not, as is the case for ordinary di¤erential equations, a

function, but rather a stochastic process.

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Introduction XV

In order to answer those questions, let’s consider a stochastic di¤erential equation in a more general form

dX ( t ) = b ( X ( t )

,

t )

| {z }

drift coe¢ cient

dt + a ( X ( t )

,

t )

| {z }

di¤usion coe¢ cient

dW ( t )

.

(6)

where the functions a

:R

[ 0,

∞

) !

^R

^and

b

:R

[ 0,

∞

) !

R

are measurable functions.

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Introduction XVI

What do we mean by a ( X ( t )

,

t ) dW ( t ) ? We haven’t de…ned that term. Actually, Equation (6) is the di¤erential form of the integral equation:

X(t) =X(0) + Zt

0 b(X(u),u) du+ Zt

0 a(X(u),u) dW(u)

and we now know the meaning of the term

Z _t

0

a ( X ( u )

,

u ) dW ( u )

.

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Introduction XVII

There doesn’t always exist a solution to that equation and we will see a few results which give the conditions that b and a functions must meet for a solution to exist.

To answer questions (2) and (3), we need a few additional

tools.

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Introduction Itô’s lemma Fundamental th.

of calculus Itô (version 1) Itô (version 2)

Example Itô process Quadratic variation Itô (version 3)

Example Itô (version 4) Taylor Solution to an SDE Quadratic covariation Multiplication rule

Example Multidimensional Itô

Itô process Quadratic covariation Itô’s lemma Example Solutions to SDEs

Fundamental theorem of calculus I

Itô’s lemma is the stochastic equivalent of the fundamental theorem of calculus. It will allow us to determine the stochastic di¤erential equation satis…ed by some given stochastic processes.

Theorem

The

fundamental theorem of calculus

stipulates that, if

df

dx :R

!

^R

represents the derivative of the function f

:R

!

R

, then

f ( b ) f ( a ) =

Z b a

df

dx ( x ) dx

. Richard R. Goldberg, Theorem 7.8A, page 205.

(27)

Fundamental theorem of calculus II

Example. if

f ( x ) = x

²

, a = 0 and b = t, then t

²

=

Z _t

0

2x dx

.

Is such a rule still valid in the context of stochastic calculus? Is

W

_t²

=

Z t 0

2W

_s

dW

_s ?

(7)

We have observed, when constructing the stochastic

integral, that the paths of the process

(28)

Fundamental theorem of calculus III

n

R

t

0

W

_s

dW

_s :

0 t T

o

could be negative at some times

-1,4 -1,2 -1 -0,8 -0,6 -0,4 -0,2 0 0,2 0,4 0,6

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

t

(29)

Fundamental theorem of calculus IV

Recall: is

W

_t²

=

Z t 0

2W

s

dW

s

? (7)

But the left side of equality (7) is necessarily non-negative,

while the right side can take negative values. There is a

contradiction and we conclude that equation (7) is false.

(30)

First version I

Itô’s lemma

There exists, in the framework of stochastic calculus, an

equivalent of the fundamental theorem of calculus that

was established by K. Itô. Note that a term is added.

(31)

First version II

Itô’s lemma

Theorem

Itô’s lemma (…rst version). Let W be a Brownian motion

built on the …ltered probability space (

Ω,

F

^,^F^,^P

) and let f

:R

!

R

be a function, the …rst two derivatives of which exist and are continuous. Then 8 ⁰ ^t ^{T ,}

f ( W

t

) f ( W

0

)

^P

=

^a.s.

Z t 0

df

dw ( W

s

) dW

s

+ ¹ 2

Z t 0

d

²

f

dw

²

( W

s

) ds.

(8) In its di¤erential form, equation (8) is written

df ( W

t

) = ^df

dw ( W

t

) dW

t

+ ¹ 2

d

²

f

dw

²

( W

t

) dt.

(32)

First version III

Itô’s lemma

For example, if

f ( x ) = x

²,

then

f ( W

_t

) = W

_t²

and f ( W

₀

) = W

₀²

= 0 and

df

dw ( W

_s

) = 2W

_s

and d

²

f

dw

²

( W

_s

) = 2.

By substituting into Itô’s equation, we obtain W

_t² ^P

=

^a.s.

2

Z t 0

W

_s

dW

_s

+

Z t 0

1ds = 2

Z t

0

W

_s

dW

_s

+ t which implies

Z t 0

W

s

dW

s

= ^W

t2

t

2

.

(33)

Idea of the proof I

Itô’s lemma

Let’s assume a division of time 0 = t

₀

< t

₁

<

...

< t

_n

= t.

One can assume that t

i

t

i 1

= t/n.

By expanding f as a Taylor series about point x

₀

, one …nds f ( x ) f ( x

0

) = f

⁰

( x

0

) ( x x

0

) + ¹

2 f

⁰⁰

(

ξ

) ( x x

0

)

²

where min ( x

₀,

x )

ξ

max ( x

₀,

x )

.

By applying that result to random points ( x = W

_t_i

and x

₀

= W

_t_i ₁

) ,

f (W_t_i) f (W_t_i ₁) =f⁰(W_t_i ₁) (W_t_i W_t_i ₁) +¹

2f⁰⁰(ξ_i) (W_t_i W_t_i ₁)²

where min ( W

_t_i ₁,

W

_t_i

)

ξ_i

max ( W

_t_i ₁,

W

_t_i

)

.

(34)

Idea of the proof II

Itô’s lemma

f ( W

t

) f ( W

0

)

=

∑

n i=1

( f ( W

t_i

) f ( W

t_i ₁

))

=

∑

n i=1

f

⁰

( W

_t_i ₁

) ( W

_t_i

W

_t_i ₁

) + ¹ 2

∑

n i=1

f

⁰⁰

(

ξ_i

) ( W

_t_i

W

_t_i ₁

)

²

By taking the limit when n !

∞

on both sides,

f ( W

t

) f ( W

0

) = lim

n!∞

∑

n i=1

f

⁰

( W

t_i ₁

) ( W

t_i

W

t_i ₁

) + ¹

2 lim

n!∞

∑

n i=1

f

⁰⁰

(

ξ_i

) ( W

_t_i

W

_t_i ₁

)

².

(35)

Idea of the proof III

Itô’s lemma

The objective is to show that

n

lim

!∞

∑

n i=1

f

⁰

( W

ti 1

) ( W

ti

W

ti 1

) =

Z t 0

f

⁰

( W

s

) dW

s

et lim

n!∞

∑

n i=1

f

⁰⁰

(

ξ_i

) ( W

_t_i

W

_t_i ₁

)

²

=

Z t 0

f

⁰⁰

( W

_s

) ds.

(36)

Idea of the proof IV

Itô’s lemma

First,

Z _t

0

f

⁰

( W

_s

) dW

_s

=

∑

n i=1

Z _t_i

t_i ₁

f

⁰

( W

_s

) dW

_s

which implies that

Z t 0

f

⁰

( W

_s

) dW

_s

∑

n i=1

f

⁰

( W

_t_i ₁

) ( W

_t_i

W

_t_i ₁

)

=

∑

n i=1

Z _t_i

t_i ₁

f

⁰

( W

_s

) dW

_s

∑

n i=1

Z _t_i

t_i ₁

f

⁰

( W

_t_i ₁

) dW

_t

=

∑

n i=1

Z ti

ti 1

f

⁰

( W

s

) f

⁰

( W

t_i ₁

) dW

s n!∞

! ^0.

(37)

Idea of the proof V

Itô’s lemma

Second,

Z _t

0 f⁰⁰(Ws)ds

∑

n i=1

f⁰⁰(ξ_i) (Wti Wti 1)²

=

∑

n i=1

Z _t_i ti 1

f⁰⁰(Ws)ds

∑

n i=1

=

∑

n i=1

Z _t_i ti 1

f⁰⁰(Ws) f⁰⁰(ξ_i) ds

+

∑

n i=1

Z_t_i ti 1

f⁰⁰(ξ_i)ds

∑

n i=1

=

∑

n i=1

Z _t_i ti 1

f⁰⁰(Ws) f⁰⁰(ξ_i) ds

∑

n i=1

f⁰⁰(ξ_i) (Wti Wti 1)² (t_i t_i ₁)

(38)

Idea of the proof VI

Itô’s lemma

The …rst term

∑ⁿi=1

R

t_i

t_i ₁

( f

⁰⁰

( W

_s

) f

⁰⁰

(

ξ_i

)) ds converges to zero since min ( W

_t_i ₁,

W

_t_i

)

ξ_i

max ( W

_t_i ₁,

W

_t_i

) implies that

ξ_i

W

_s

! ^0. ^f

⁰⁰

being continuous, we deduce that f

⁰⁰

(

ξ_i

) f

⁰⁰

( W

s

) ! ^0.

The second term

∑ⁿi=1

f

⁰⁰

(

ξ_i

) ( W

_t_i

W

_t_i ₁

)

²

( t

_i

t

_i ₁

) tends also to 0 since

Eh

( W

ti

W

ti 1

)

²

( t

i

t

i 1

)

ⁱ

= 0

Varh

( W

_t_i

W

_t_i ₁

)

²

( t

_i

t

_i ₁

)

ⁱ

= ( t

_i

t

_i ₁

) ! ^0.

Such a proof is only roughly sketched since we have

omitted to specify some technical details. (ref. R. Durrett)

(39)

Second version I

Itô’s lemma

The proof given by K. Itô applies to much more complex

situations that the one described above. Here is a …rst

generalization:

(40)

Second version II

Itô’s lemma

Theorem

Itô’s lemma (second version). Let W be a Brownian motion

built on the …ltered probability space (

_Ω,

F

^,F,P

) and let f

:R

[ 0,

∞

) !

R

be a function, the …rst and second partial derivatives of which exist and are continuous. Then

8 ⁰ ^t ^{T ,}

f ( W

_t,

t ) f ( W

₀,

0 )

=

Z _t

0

∂f

∂t

( W

_s,

s ) + ¹ 2

∂²

f

∂w²

( W

_s,

s ) ds +

Z t 0

∂f

∂w

( W

_s,

s ) dW

_s

In its di¤erential form, we have

df (Wt,t) = ^∂f

∂t (Wt,t) +¹ 2

∂²f

∂w² (Wt,t) dt+ ^∂f

∂w (Wt,t) dWt.

(41)

Example I

The Black-Scholes model

The stochastic process S = f ^S

^t ^:

⁰ ^t ^T g ^represents the price evolution of a risky asset where

S

_t

= S

₀

exp

µ σ²

2 t +

σW_t ,

µ

and

σ

being constants, and W represents a standard Brownian motion.

Since W

_t

is N ( 0, t ) -distributed,

µ σ²

2 t +

σW_t

follows a distribution N

µ σ²

2 t,

σ²

t

and S

t

is lognormally distributed.

(42)

Example II

The Black-Scholes model An intermission: the moments of a lognormally-distributed random variable

IfZ represents a standard normal random variable and ifa andbare constants then

E[exp[b+aZ]] =exp b+^a

2

2 . So, ifS(0)is independent from the Brownian motion,

E[S(t)] = E[S(0)]E exp µ σ²

2 t+σWt

= E[S(0)]exp[µt]

Eh

S²(t)ⁱ = Eh

S²(0)ⁱE exp 2 µ σ²

2 t+2σWt

= Eh

S²(0)ⁱexph

2µt+σ²ti

Var[S(t)] = Eh

S²(0)ⁱexph

2µt+σ²ti

E²[S(0)]exp[2µt].

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Example III

Using Itô’s lemma, it is possible to show that the stochastic process S de…ned as

S

_t

= S

₀

exp

µ σ²

2 t +

σW_t

is a solution to the integral equation

S

_t

S

₀

=

µ Z _t

0

S

_u

du +

σ Z _t

0

S

_u

dW

_u.

(44)

Example IV

Recall that

S

_t

= S

₀

exp

µ σ²

2 t +

σW_t .

Indeed, if f ( w

,

t ) = S

₀

exp

h

µ ^σ

2

t +

σwi ,

then

f(Wt,t) =St

f(W₀,0) =S₀

∂f

∂t (w,t) = µ ^σ

2

2 f (w,t)

∂f

∂w (w,t) =σf (w,t)

∂²f

∂w² (w,t) =σ²f(w,t).

(45)

Example V

Thus,

St S₀

= f(Wt,t) f(W0,0)

= Z_t

0

∂f

∂w (Wu,u)dWu+ Z_t

0

∂f

∂t (Wu,u) +¹ 2

∂²f

∂w²(Wu,u) du

= Zt

0 σf(Wu,u)dWu+ Z t

0 µ σ²

2 f (Wu,u) +¹

2σ²f(Wu,u) du

= Z_t

0 σf(Wu,u)dWu+ Z _t

0 µf(Wu,u)du

= Z_t

0 σSudWu+ Z_t

0 µSudu.

(46)

Example VI

We have just proved that the stochastic process S = f ^S

^t ^:

⁰ ^t ^T g ^where

S

t

= S

0

exp

µ σ²

2 t +

σWt

is a solution to the stochastic di¤erential equation

dS

t

=

µSt

dt +

σSt

dW

t.

Stochastic di¤erential equation (SDE) and Ito’s lemma