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Page 1: Stochastic calculus : Introduction (II)homepages.ulb.ac.be/~cazizieh/.../CalculSto_racc_Chap1_2_trans_CAZ.pdf · definition of the stochastic integral (or more precisely from the

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Stochastic calculus - II

Stochastic calculus : Introduction (II)

VUB

VUB Stochastic calculus : Introduction (II)

Page 2: Stochastic calculus : Introduction (II)homepages.ulb.ac.be/~cazizieh/.../CalculSto_racc_Chap1_2_trans_CAZ.pdf · definition of the stochastic integral (or more precisely from the

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

Stochastic differential

Introduction of the differential notation :b = b(s, ω) adapted integrable processσ = σ(s, ω) adapted square-integrable process

We can hence construct a new process X (t) by integration of b andσ :

X (t) = X0 +

∫ t

0b(s)ds +

∫ t

0σ(s)dB(s)

We say that X is an Ito process, and we use the notation

dX (t) = b(t)dt + σ(t)dB(t)

VUB Stochastic calculus : Introduction (II)

Page 3: Stochastic calculus : Introduction (II)homepages.ulb.ac.be/~cazizieh/.../CalculSto_racc_Chap1_2_trans_CAZ.pdf · definition of the stochastic integral (or more precisely from the

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

Stochastic differential : calculation rules

Other rules than in the classical calculus :

Classical calculus :

12

f 2(t) =

∫ t

0f (s)df (s)

Stochastic calculus :

12

B2(t) ??=

∫ t

0B(s)dB(s) ??

VUB Stochastic calculus : Introduction (II)

Page 4: Stochastic calculus : Introduction (II)homepages.ulb.ac.be/~cazizieh/.../CalculSto_racc_Chap1_2_trans_CAZ.pdf · definition of the stochastic integral (or more precisely from the

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

Stochastic differential : calculation rules

We have seen before that∫ t

0B(s) dB(s) =

12

B2(t)− 12

t ⇔ 2∫ t

0B(s) dB(s) +

∫ t

0ds = B2(t)

which means by using differential notations :

d(B2(t)) = 2B(t)dB(t) + dt

→ the usual chain rule does not work anymore...

VUB Stochastic calculus : Introduction (II)

Page 5: Stochastic calculus : Introduction (II)homepages.ulb.ac.be/~cazizieh/.../CalculSto_racc_Chap1_2_trans_CAZ.pdf · definition of the stochastic integral (or more precisely from the

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

Ito formula : The most simple case

Theorem (Ito formula : The most simple case)

Let f : R→ R ∈ C2(R), then a.s. ∀t ∈ [0,T ]

f (Bt ) = f (0) +

∫ t

0f ′ (Bs) dBs +

12

∫ t

0f ′′ (Bs) ds. (1)

In differential notations :

d(f (Bt )) = f ′(Bt )dBt +12

f ′′(Bt )dt

VUB Stochastic calculus : Introduction (II)

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

Ito formula : The most simple case*

Proof1) Let us first consider the case where f has a compact support. Forti = it

n for 0 ≤ i ≤ n, we consider :

f (Bt )− f (0) =n∑

i=1

f (Bti )− f

(Bti−1

)such that f (Bti )− f

(Bti−1

)sufficiently small for applying a Taylor

development of order 2.

VUB Stochastic calculus : Introduction (II)

Page 7: Stochastic calculus : Introduction (II)homepages.ulb.ac.be/~cazizieh/.../CalculSto_racc_Chap1_2_trans_CAZ.pdf · definition of the stochastic integral (or more precisely from the

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

Ito formula : The most simple case*

If f ∈ C2 has a compact support, then ∀x , y ∈ R :

f (y)− f (x)

= (y − x) f ′ (x) +12

(y − x)2 f ′′ (x) + r (x , y)

where

|r (x , y)| ≤ (y − x)2 h (x , y)

where h is uniformly continuous, bounded and h (x , x) = 0 ∀x .

VUB Stochastic calculus : Introduction (II)

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

Ito formula : The most simple case*

Then :

f (Bt )− f (0) = An + Bn + Cn

with

An =n∑

i=1f ′(Bti−1

) (Bti − Bti−1

)Bn = 1

2

n∑i=1

f ′′(Bti−1

) (Bti − Bti−1

)2

|Cn| ≤n∑

i=1

(Bti − Bti−1

)2 h(Bti−1 ,Bti

).

The continuity of f ′ and the Riemann representation imply that

AnP→∫ t

0 f ′ (Bs) dBs.

VUB Stochastic calculus : Introduction (II)

Page 9: Stochastic calculus : Introduction (II)homepages.ulb.ac.be/~cazizieh/.../CalculSto_racc_Chap1_2_trans_CAZ.pdf · definition of the stochastic integral (or more precisely from the

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

Ito formula : The most simple case*

We reformulate Bn = 12

n∑i=1

f ′′(Bti−1

) (Bti − Bti−1

)2 as :

Bn =12

n∑i=1

f ′′(Bti−1

)(ti − ti−1)

+12

n∑i=1

f ′′(Bti−1

)(Bti − Bti−1

)2 − (ti − ti−1).

Thanks to the continuity of f ′′ (Bs (ω)) with respect to variable s, thefirst term converges to an ordinary integral for all ω

limn→∞

n∑i=1

f ′′(Bti−1

)(ti − ti−1) =

∫ t0 f ′′ (Bs) ds.

VUB Stochastic calculus : Introduction (II)

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

Ito formula : The most simple case*

Let us denote the second term by∧Bn. Then by orthogonality of the different

terms :

IE

[∧B

2

n

]

=14

n∑i=1

IE[f ′′(Bti−1

)2(

Bti − Bti−1

)2−(ti − ti−1)2]

≤ 14∥∥f ′′∥∥2∞

n∑i=1

IE[(

Bti − Bti−1

)2 − (ti − ti−1)2]

=t2

2n∥∥f ′′∥∥2∞

where we used the fact that Bti − Bti−1 ∼ N (0, t/n) . Hence

Var((

Bti − Bti−1

)2)

= 2t2

n2 .

VUB Stochastic calculus : Introduction (II)

Page 11: Stochastic calculus : Introduction (II)homepages.ulb.ac.be/~cazizieh/.../CalculSto_racc_Chap1_2_trans_CAZ.pdf · definition of the stochastic integral (or more precisely from the

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

Ito formula : The most simple case*Since the quadratic convergence implies convergence in probability, we

hence get∧Bn

P→ 0

Let us now concentrate on

|Cn| ≤n∑

i=1

(Bti − Bti−1

)2 h(Bti−1 ,Bti

).

Cauchy-Schwartz inequality (for the L2 scalar product) implies that :

IE [|Cn|] ≤n∑

i=1

IE[(

Bti − Bti−1

)4]1/2

IE[h2 (Bti−1 ,Bti

)]1/2

and we remark that :

a) Bti − Bti−1 v N (0, t/n) and hence :

IE[(

Bti − Bti−1

)4]

= 3t2

n2 .

VUB Stochastic calculus : Introduction (II)

Page 12: Stochastic calculus : Introduction (II)homepages.ulb.ac.be/~cazizieh/.../CalculSto_racc_Chap1_2_trans_CAZ.pdf · definition of the stochastic integral (or more precisely from the

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

Ito formula : The most simple case*

b) h (x , x) = 0 ∀xwith moreover h uniformly continuous, which means that ∀ε > 0 ∃δ = δ (ε) such that |h (x , y)| ≤ ε for all x , y with |x − y | ≤ δ.

Hence

IE[h2 (Bti−1 ,Bti

)]≤ ε2 + ‖h‖2

∞ P(∣∣Bti − Bti−1

∣∣ ≥ δ)≤ ε2 + ‖h‖2

∞ δ−2IE[∣∣Bti − Bti−1

∣∣2]= ε2 + ‖h‖2

∞ δ−2 tn.

VUB Stochastic calculus : Introduction (II)

Page 13: Stochastic calculus : Introduction (II)homepages.ulb.ac.be/~cazizieh/.../CalculSto_racc_Chap1_2_trans_CAZ.pdf · definition of the stochastic integral (or more precisely from the

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

Ito formula : The most simple case*

Results a) et b) imply that

IE [|Cn|] ≤ n(

3t2

n2

)1/2 (ε2 + ‖h‖2

∞ δ−2 tn

)1/2

In consequence ∀ε > 0,

limn→∞

IE [|Cn|] ≤√

3tε if n→∞

and hence we have convergence in L1 :

IE [|Cn|]→ 0

which implies convergence in probability :

CnP→ 0.

VUB Stochastic calculus : Introduction (II)

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

Ito formula : The most simple case*

Let us consider sub-sequences Anj ,Bnj and Cnj which converge a.s. 1.Then (1) is satisfied almost surely for all t (i.e. for each t, there existsa subset Ωt of probability 1 such that the equality is satisfied on thatsubset - the subset depending on t).

We conclude that there exists a subset Ω of proba 1 such that theequality holds for all rational numbers. As both sides of (1) arecontinuous, we conclude that there exists Ω0 with P(Ω0) = 1 on which(1) is true for all t ∈ [0,T ].

1. this is possible as if Xn → X in proba, we can extract a sub-sequence whichconverges a.s

VUB Stochastic calculus : Introduction (II)

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

Ito formula : The most simple case*2) Let us now treat the general case :We recall that for all f ∈ C2 (R), there exists fM ∈ C2 with compact supportwith

f (x) = fM (x), f ′ (x) = f ′M (x) and f ′′ (x) = f ′′M (x) for all |x | ≤ M.

We know that

fM (Bt )− fM (0) =

=

∫ t

0f ′M (Bs) dBs +

12

∫ t

0f ′′M (Bs) ds a.s..

Let us take τM = min t : |Bt | ≥ M or t ≥ T . Then for all ω ∈ s ≤ τM, wehave

f ′ (Bs) = f ′M (Bs)

and by persistence of the identity in L2LOC , we have that∫ t

0 f ′ (Bs) dBs =∫ t

0 f ′M (Bs) dBs

for almost all ω ∈ t ≤ τM .

VUB Stochastic calculus : Introduction (II)

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

Ito formula : The most simple case*

It is clear that for ω ∈ t ≤ τM, we have∫ t0 f ′′ (Bs) ds =

∫ t0 f′′

M (Bs) ds

and

f (Bt ) = fM (Bt )

Hence for almost all ω ∈ t ≤ τM :

f (Bt )− f (0) =

∫ t

0f ′ (Bs) dBs +

12

∫ t

0f ′′ (Bs) ds (2)

Because τM → T for M →∞ with probability 1, (2) is verified withprobability 1.

VUB Stochastic calculus : Introduction (II)

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

Ito formula : Example

We directly see that by applying the formula to f (x) = x2, we get :∫ t

0BsdBs =

12

B2t − t/2.

We recover the result obtained previously by departing from thedefinition of the stochastic integral (or more precisely from theRiemann representation).Ito formula allows however a more direct calculation, applicable in ageneral framework.

Remark :We see from what preceeds that ”(∆iB)2 behaves like ∆i t” i.e.,“(dBt )

2 = dt”.

VUB Stochastic calculus : Introduction (II)

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

Ito formula with time and space variables

Theorem

For each function f ∈ C1,2 (R+ × R), we have the representation

f (t ,Bt ) =f (0,0) +

∫ t

0

∂f∂x

(s,Bs) dBs

+

∫ t

0

∂f∂t

(s,Bs) ds +12

∫ t

0

∂2f∂x2 (s,Bs) ds a.s.

VUB Stochastic calculus : Introduction (II)

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

Ito formula with time and space variables*ProofBy the same localisation argument, we can work without loss ofgenerality with f with a compact support in R+ × R.Now, we use the Taylor development of f (t , y) at point (s, x) :

f (t , y)− f (s, x) = (t − s)∂f∂t

(s, x) + (y − x)∂f∂x

(s, x)

+12

(y − x)2 ∂2

∂x2 f (s, x) + r (s, t , x , y)

with

|r (s, t , x , y)| ≤

≤ (y − x)2 h (x , y , s, t) + (t − s) k (x , y , s, t)

where h and k are bounded and uniformly continuous functions whichvanish if x = y and s = t .

VUB Stochastic calculus : Introduction (II)

Page 20: Stochastic calculus : Introduction (II)homepages.ulb.ac.be/~cazizieh/.../CalculSto_racc_Chap1_2_trans_CAZ.pdf · definition of the stochastic integral (or more precisely from the

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

Ito formula with time and space variables*

We apply this Taylor development to the telescoping sums :f (t ,Bt )− f (0,0) . We get then 4 terms, from which one converges tozero in expectation and 3 terms to the integrals of the theorem.The final steps are similar to those of the proof of Thm 1.1 (slide 5).

VUB Stochastic calculus : Introduction (II)

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

General unidimensional Ito formula

Let X (t) be an Ito process, having the representation :

dXt = b(t)dt + σ(t)dB(t)

with b and σ measurable, adapted and satisfying the ad-hocintegrability properties (cf. above).

Definition

We define the integral∫ t

0 f (ω, s)dXs by∫ t

0f (ω, s)dXs =

∫ t

0f (ω, s)b(ω, s)ds+

∫ t

0f (ω, s)σ(ω, s)dBs

where we assume f (ω, s) adapted and f (ω, s)b(ω, s) ∈ L1(dt) withprobability 1 and f (ω, s)σ(ω, s) ∈ L2

LOC .

VUB Stochastic calculus : Introduction (II)

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

General unidimensional Ito formula

Theorem (Ito formula for an Ito process)

Let f ∈ C1,2 (R+ × R)

and let Xt , t ∈ [0,T ] be an Ito process :

dXt = b(t)dt + σ(t)dB(t)

Then Yt = f (t ,Xt ) is differentiable :

Yt = f (t ,Xt ) = f (0, 0) +

∫ t

0

∂f∂t

(s,Xs) ds +

∫ t

0

∂f∂x

(s,Xs) dXs

+12

∫ t

0

∂2f∂x2 (s,Xs)σ2 (ω, s) ds a.s.

In other words,

df (t ,X (t)) =∂f∂t

dt +∂f∂x

dX (t) +12∂2f∂x2 σ

2(t)dt

=

(∂f∂t

+∂f∂x

b(t) +12∂2f∂x2 σ

2(t))

dt +∂f∂xσ(t)dB(t)

VUB Stochastic calculus : Introduction (II)

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

General unidimensional Ito formulaThis suggests the following calculation rules

dt .dt = 0dB(t).dB(t) = dt

dB(t).dt = 0

We recover the Ito formula by applying a Taylor development untilorder 2 combined with those calculation rules :1) Case Y (t) = f (t ,B(t)) :

dYt =∂f∂t

(t ,Bt ) dt +∂f∂x

(t ,Bt ) dBt +12∂2f∂x2 (t ,Bt ) dBt · dBt

+12∂2f∂t2 (t ,Bt ) dt .dt +

∂2f∂x∂t

(t ,Bt ) dt .dB(t)

=∂f∂t

(t ,Bt ) dt +∂f∂x

(t ,Bt ) dBt +12∂2f∂x2 (t ,Bt ) dt

VUB Stochastic calculus : Introduction (II)

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

2) Case Y (t) = f (t ,X (t)), where dX (t) = b(t)dt + σ(t)dB(t) :

dYt =∂f∂t

(t ,Xt ) dt +∂f∂x

(t ,Xt ) dXt +12∂2f∂x2 (t ,Xt ) dXt · dXt

+12∂2f∂t2 (t ,Xt ) dt .dt +

∂2f∂x∂t

(t ,Xt ) dt .dX (t)

By applying the formal calculation rules :

dXt · dXt =

= (b (ω, t) dt + σ (ω, t) dBt ) · (b (ω, t) dt + σ (ω, t) dBt )

= b (ω, t)2 dt · dt + 2b (ω, t)σ (ω, t) dBt · dt + σ (ω, t)2 dBt · dBt

= σ (ω, t)2 dt

anddXt .dt = b(ω, t)dt .dt + σ(ω, t)dBt .dt = 0

We hence get :

dYt =∂f∂t

(t ,Xt ) dt +∂f∂x

(t ,Xt ) dXt +12∂2f∂x2 (t ,Xt )σ

2dt

VUB Stochastic calculus : Introduction (II)

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

General multidimensional Ito formula

Theorem (Ito formula for 2 Ito processes)

Let f ∈ C1,2,2 (R+ × R× R)

and Xt , t ∈ [0,T ] and Yt , t ∈ [0,T ] be Itoprocesses with the following representation

Xt =∫ t

0 a (ω, s) ds +∫ t

0 b (ω, s) dBs , Yt =∫ t

0 α (ω, s) ds +∫ t

0 β (ω, s) dBs

Then

Zt = f (t ,Xt,Yt ) = f (0, 0, 0) +

∫ t

0

∂f∂t

(s,Xs,Ys) ds

+

∫ t

0

∂f∂x

(s,Xs,Ys) dXs +

∫ t

0

∂f∂y

(s,Xs,Ys) dYs

+12

∫ t

0

∂2f∂x2

(s,Xs,Ys) b2 (ω, s) ds

+

∫ t

0

∂2f∂x∂y

(s,Xs,Ys) b (ω, s)β (ω, s) ds

+12

∫ t

0

∂2f∂y2

(s,Xs,Ys)β2 (ω, s) ds a.s..

VUB Stochastic calculus : Introduction (II)

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

General multidimensional Ito formulaWe can get the result by using the calculation rules extended to thecase of several Brownian motions :

(B1

t)

and(B2

t)

withcorr

(dB1

t ,dB2t)

= ρtdt :

· dt dB1t dB2

tdt 0 0 0dB1

t 0 dt ρtdtdB2

t 0 ρtdt dt

By applying a Taylor development until order 2(and applying dt .dt = 0,dYt .dt = dXt .dt = 0) :

dZt = ft (t ,Xt ,Yt ) dt + fx (t ,Xt ,Yt ) dXt ++fy (t ,Xt ,Yt ) dYt

+ 12 fxx (t ,Xt ,Yt ) dXt · dXt

+fxy (t ,Xt ,Yt ) dXt · dYt

+ 12 fyy (t ,Xt ,Yt ) dYt · dYt .

VUB Stochastic calculus : Introduction (II)

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

General multidimensional Ito formulaMultidimensional process

Let S∗ be an Ito process of dimension d :

dS∗ (t) = µ (t) dt + σ (t)dB (t)

avec S∗ =(S1, ...,Sd

)T, B =

(B1, ...,Bk

)T,

µ =(µ1, ..., µd

)Tand σ =

(σi,j)

of dimension d × k .

Hence for 0 ≤ i ≤ d :

dSi (t) = µi (t) dt + σi (t) dB (t)

Where

σi =(σi,1, ..., σi,k

)and where

σi (s) dBs =k∑

j=1σi,j (s) dBj (s) .

VUB Stochastic calculus : Introduction (II)

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General multidimensional Ito formula

We suppose that ∀ (i , j) µi and σi,j are adapted processes such that∫ T

0| µi (t) | dt <∞ a.s. ;

∫ T

0| σi,j (t) |2 dt <∞ a.s. (3)

Multidimensional Ito lemmaLet X =

(X 1, ...,X d

)Tbe an Ito process with multiple dimension :

dXt = µtdt + σtdBt

whereµt is a Ft -adapted process, µt : Ω −→ Rd

σt is a random adapted matrix of dimension (d × k) ,Bt a Brownian motion of dimension k ,µ, σ satisfy (3).

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Multidimensional Ito formula

Lemma

Let f ∈ C1,2 ([0,T ]× Rd ;R). We denote by :

fx (t , x) =

(∂f∂x1

(t , x) , ...,∂f∂xd

(t , x)

)row vector

fxx (t , x) the matrix of second derivatives(

∂2

∂xi , ∂xjf (t , x)

)i,j

ft (t , x) =∂f∂t

(t , x) .

Let Yt = f (t ,Xt ). Then

dYt =

ft (t ,Xt )+fx (t ,Xt )µt +

12

[tr(σtσ

Tt fxx (t ,Xt )

)]dt

+ fx (t ,Xt )σtdBt

where, for a square matrix A = (Ai,j ), the trace of A, denoted by trA, is thesum of diagonal elements.

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Stochastic differential equations (SDE) : Introduction -geometric Brownian motion

Let (W (t)) be a standard BM. Then a geometric Brownian motionis defined by

X (t) = X (0)eµt+σW (t) = f (t ,W (t)),

where f (t , x) = X (0) exp(µt + σx

).

By using the Ito lemma :

(i)∂f∂t

= µf , (ii)∂f∂x

= σf , (iii)∂2f∂x2 = σ2f

this gives :

dX = df = µf︸︷︷︸(i)

dt + σf︸︷︷︸(ii)

dW +12σ2f︸︷︷︸(iii)

dt = (µ+12σ2)Xdt + σXdW .

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Stochastic differential equations (SDE) : Introduction -Geometric Brownian motion

By denoting :

δ = µ+σ2

2we get to the following result :

LemmaThe process (geometric Brownian motion ) :

X (t) = X (0) exp(

(δ − σ2

2)t + σW (t)

)is solution to the stochastic differential equation :

dX (t) = δX (t)dt + σX (t)dW (t)

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Stochastic differential equations (SDE) : Introduction -Geometric Brownian motion

Properties of geometric Brownian motion

1 E[X (t)] = X (0)eδt

2 Var [X (t)] = X 2(0)e2δt (eσ2t − 1)

Proof :It suffices to remark that X (t)/X (0) ∼ LN((δ − σ2

2 )t , σ√

t) and to usethe expressions for the moments of a log-normal distribution :

If Y ∼ LN(µ, σ), then :

E[Y ] = eµ+σ2/2, Var [Y ] = (eσ2− 1)e2µ+σ2

.

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Stochastic differential equations (SDE) : Introduction -Geometric Brownian motion

The SDE followed by process X (t) can be interpreted as follows :

X (t + dt)− X (t)X (t)

= δ dt + σ dW (t).

In other words, the relative return (or arithmetic return) of the asset on[t , t + dt ] can be decomposed in a linear deterministic trend perturbedby a noise term ;

δ is called instantaneous return and σ the volatility (measure of theuncertainty / risk of the asset).

The GBM process can be seen as a perturbed exponential function (ifσ = 0, Xt = X0 exp(δt)).

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Stochastic differential equations (SDE) : Introduction -Geometric Brownian motion

Financial application of Geometric Brownian motion :Modelling the dynamics of a stock price (Samuelson’s model, 1973)

The first model for the dynamics of a stock : Bachelier (1900),additive Brownian motion :

In that model, the price of a stock is supposed to follow acontinuous random walk, i.e. a Brownian motion (absoluteBrownian motion) :

S(t , ω) = S(0) + σW (t , ω)

where W is a standard BM, and σ is a volatility parameter.

Main drawback of this model : stock prices can becomenegative...

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Stochastic differential equations (SDE) : Introduction -Geometric Brownian motion

Financial application of Geometric Brownian motion :

Samuelson (1973) proposes to model the price of a stock by ageometric Brownian motion (relative Brownian motion)

S(t) = S(0)exp((

δ − σ2

2

)t + σW (t)

)where δ represents the expected return of the stock and σ itsvolatility.

Advantage : the process is always positive, and generalizes thedeterministic formula with compounded interests (see before)

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Stochastic differential equations (SDE)

Ordinary differential equationsLet us consider the ODE :

Y ′(t) = f (t ,Y (t)), Y (0) = Y0

(or, in differential notations : dY (t) = f (t ,Y (t))dt)

We have existence results required that f satisfies some Lipschitzcondition : there exists a constant K such that

|f (t , x)− f (t , y)| ≤ K |x − y | ∀x , y

(...)

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Stochastic differential equations (SDE)

Ordinary differential equationsMoreover, we can obtain numerically the solutions by using the Picarditeration method :

Y (0)(t) = Y0

Y (n)(t) = Y0 +

∫ t

0f (s,Y (n−1))ds

We get a very similar result for SDEs

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Stochastic differential equations (SDE)Theorem (Existence and uniqueness)

Suppose that the coefficients of the following SDE :

dXt = µ(t ,Xt )dt + σ(t ,Xt )dBt t ∈ [0,T ], X0 = x0 (4)

satisfy the two following conditions :1 Lipschitz condition w.r.t the space variable : there exists a

constant K s.t. :

|µ(t , x)− µ(t , y)|2 + |σ(t , x)− σ(t , y)|2 ≤ K |x − y |2 (5)

∀x , y ∈ R,∀t ∈ [0,T ]

2 Growth condition w.r.t the space variable :

|µ(t , x)|2 + |σ(t , x)|2 ≤ K (1 + |x |2), (6)

∀x ∈ R,∀t ∈ [0,T ]

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Stochastic differential equations (SDE)

Theorem (Existence and uniqueness - cont’d)

then there exists an adapted continuous solution Xt of equation (4)uniformly bounded in L2(dP) :

sup0≤t≤T

IE[X 2

t]<∞.

Moreover, if Xt and Yt are two bounded continuous solutions in L2 of(4), then :

P [Xt = Yt for all t ∈ [0,T ]] = 1.

The process (Xt ) solution of such an SDE is called a diffusionprocess.

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Stochastic differential equations (SDE)

This solution can be approached by par iterations following a Picardmethodology :

X (n) = X0 +

∫ t

0µ(s,X (n−1)(s))ds +

∫ t

0σ(s,X (n−1)(s))dB(s)

Case of the geometric Brownian motionIn that case, we are well under the assumptions of this theorem asµ(x , t) = δx and σ(t , x) = σx are linear (hence condition (6) isverified), and Lipschitz w.r.t. variable x (as in C1).

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Stochastic differential equations (SDE)

Important processes in financial modelling

We have seen the geometric Brownian motion as a solution of anSDE. This is convenient to model market quantities that must remainpositive (ex : stocks).

Other important process : Ornstein-Uhlenbeck process :

dX (t) = a(b − X (t))dt + σdB(t)

Corresponding deterministic equation :

Y ′(t) = a(b − Y (t)) (Y (0) = Y0)

Solution :Y (t) = Y0e−at + b

(1− e−at)

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Stochastic differential equations (SDE)Illustration of the solution of that EDO forY0 = 0.07,a = 0.02,b = 0.03

0 100 200 300 400 500 600 700 800 900 10000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

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Stochastic differential equations (SDE)

Let the SDE of the OU process : dX (t) = a(b − X (t))dt + σdB(t)

We will search a solution of the form X (t) = Y (t) + Z (t)e−at whereY (t) is a (deterministic) solution of dY (t) = a(b − Y (t))dt

We have :

dX (t) = dY (t) + e−atdZ (t)− ae−atZ (t)dt= a(b − Y (t))dt + e−atdZ (t)− ae−atZ (t)dt= a(b − X (t))dt + σdB(t)= a(b − Y (t))dt − ae−atZ (t)dt + σdB(t)

It directly comes :

dZ (t) = eatσdB(t)⇔ Z (t) = σ

∫ t

0easdB(s)

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Stochastic differential equations (SDE)

The Ornstein-Uhlenbeck process hence becomes :

X (t) = X0e−at + b(1− e−at )︸ ︷︷ ︸deterministic trend

+σe−at∫ t

0easdB(s)︸ ︷︷ ︸

random noise

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Stochastic differential equations (SDE)

Property : Mean of an Ornstein-Uhlenbeck process :

IE [X (t)] = X0e−at + b(1− e−at )

→ weighted average of the initial position and the asymptotic positionb.

Weights are negative exponential functions governed by the restoringforce a.

This is also the solution of the corresponding ODE (which correspondto the SDE in the case σ = 0).

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Stochastic differential equations (SDE)Illustration of a simulated trajectory of the SDE forX0 = 0.07,a = 0.02,b = 0.03, σ = 0.008

0 10 20 30 40 50 60 70 80 90 100−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

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Stochastic differential equations (SDE)Illustration of simulated trajectories of the SDE forX0 = 0.07,a = 0.02,b = 0.03, σ = 0.008

0 10 20 30 40 50 60 70 80 90 100−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

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Stochastic differential equations (SDE)Illustration of simulated trajectories of the SDE forX0 = 0.07,a = 0.02,b = 0.03, σ = 0.0008

0 10 20 30 40 50 60 70 80 90 1000.02

0.03

0.04

0.05

0.06

0.07

0.08

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Stochastic differential equations (SDE)

Financial application of the Ornstein-Uhlenbeck process : interestrates modelling (Vasicek model, 1977)

dr(t) = a(b − r(t))dt + σdW (t)

(SDR for the short rate r(t))a = restoring force (mean reversion speed) on the interest ratesb = mean reversion target, i.e. asymptotic short rateσ = volatility on interest rates

We can show (see later) that this model can be used to model thedynamics of the entire yield curve (not only the short rate).

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Girsanov theorem

Change of measure in discrete universe

We have seen in the framework of a binomial model that, under someconditions, prices of options can be obtained as expectations ofdiscounted payoffs, in which the expectation is taken w.r.t. anotherprobability measure, called the risk-neutral measure, and equivalentto the initial measure, called the historical measure

What means “equivalent measure” ?

Two (probability) measures are equivalent if they have the same setsof zero (probability) measure :

P ∼ Q

P(A) = 0⇔ Q(A) = 0 ∀A ∈ F .

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Girsanov theorem

Change of measure in discrete universe (cont’d)

Let us consider an example in a discrete world, where the set ofpossible scenarios constitutes a finite or countable set :

Ω = ω1, ...ωN

P and Q will be equivalent probability measures if for instance :

P(ωi) = pi > 0

Q(ωi) = qi > 0

→ coincidence on the set of possible scenarios, but not on theiroccurrence probability

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Girsanov theorem

Change of measure in discrete universe (cont’d)

When we pass to an equivalent measure, the way we can pass fromthe one to the other can be captured by the so called Radon -Nikodym derivative.

The R-N derivative is a random variable, also called density ofmeasure Q w.r.t. measure P, given in the discrete case by :

Z (ωi ) =Q(ωi)P(ωi)

=qi

pi

By definition, this random variable has a mean 1 under the initialmeasure P :

IEP[Z ] =N∑

i=1

piqi

pi=

N∑i=1

qi = 1

(like the integral of the density of a continuous random variable...)

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Girsanov theorem

Change of measure in discrete universe (cont’d)

R-N derivative allows to easily compute expectations under the newmeasure Q in function of expectations under the old measure P : if Yis a random variable, then

IEQ[Y ] = IEP[Z .Y ]

Proof

IEQ[Y ] =N∑

i=1

Y (ωi )qi =N∑

i=1

(Y (ωi )

qi

pi

)pi =

N∑i=1

(Y (ωi )Z (ωi )) pi = IEP[Y .Z ]

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Girsanov theorem - Example of the binomial model

We consider a market in which risky assets can increase or decreaseon a given period (from t=0 to t=1), with the possibility to borrow orinvest with a risk free rate.

The set of possible scenarios is hence

Ω = up, down = u,d

Initial historical measure : historical /real world measure.

• For instance :pu = pd =

12

Risk free rate : r = 3%

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Girsanov theorem - Example of the binomial modelWe suppose that there is a risky asset (stock), whose initial price isS(0) = 1, and which can evolve until t=1 by an increase or adecrease :

S(1) = Ywhere Y is a binomial random variable such that

Y (u) = 1.09, Y (d) = 1.01

We have seen that it is possible to introduce an equivalent measure,called the risk-neutral measure, depending only on the volatility of thestock and the risk-free rate (and not on the expected return of thestock) :

qu =1 + r − Y (d)/S(0)

Y (u)/S(0)− Y (d)/S(0), qd = 1− qu

which gives here :

qu =1.03− 1.011.09− 1.01

=14, qd =

34

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Girsanov theorem - Example of the binomial model

Let us compute the density of Q w.r.t. P :

Z (u) =qu

pu=

1/41/2

=12

Z (d) =qd

pd=

3/41/2

=32

Let us compute the expectations of Y under P and Q :

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Girsanov theorem - Example of the binomial model

Let us compute the expectations of Y under P and Q :1) Real world :

EP[Y ] = (1.09 + 1.01).12

= 1.05

2) Risk-neutral world :

Direct calculation :

IEQ[Y ] = 1.09× 14

+ 1.01× 34

= 1.03

Computation by using the density of Q w.r.t P

IEQ[Y ] = IEP[Y .Z ] = (1.09× 12

)× 12

+ (1.01× 32

)× 12

= 1.03

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Girsanov theorem : en temps continu

Similarly to the discrete binomial model, when using a geometricBrownian motion, we will consider a new measure, called risk –neutral, under which all financial assets have an expected returnequal to the risk free rate r .

This implies a change of drift of the process :

dS(t) = δS(t)dt + σS(t)dW (t)= (r + δ − r)S(t)dt + σS(t)dW (t)= rS(t)dt + σS(t)dW (t) + (δ − r)S(t)dt= rS(t)dt + σS(t)dW ∗(t)

where we denote by W ∗(t) = W (t) +δ − rσ

t (Brownian motion withdrift).

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Girsanov theorem : in continuous time

The process W ∗ is not a standard Brownian motion under the realworld probability (presence of a drift).

Girsanov answers to the following questions :How to change measure s.t. W ∗ is again a standard Brownianmotion ?More generally, what becomes the process W when we changethe measure ?

For that purpose we introduce the concept of equivalent measure andRadon-Nikodym derivative in the general case.

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Girsanov theorem

Let (Ω,F) be a probability space with 2 measures P1 and P2.

Definition

the measure P2 is called absolutely continuous w.r.t. measure P1 if :

P2(B) = 0 ∀B ∈ F suchthat P1(B) = 0

Definition

P1 and P2 are equivalent if P2 is absolutely continuous with respect to P1 andvice versa.

→ P1 and P2 are equivalent if they have the same sets of zero measure.

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→ P1 and P2 are equivalent if they have same sets of zero measure.

• In terms of stochastic process, this means that the set of possibletrajectories of a process under P1 is identical to the set of possibletrajectories under P2.• For instance, if X is a stochastic process with jumps under P1, it willremain a jump process under P2.

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Girsanov theorem

Theorem (Radon- Nikodym theorem)

Measure P2 is absolutely continuous w.r.t. measure P1 iff there existsa random variable Z ≥ 0 such that

P2(A) =

∫A

Z (ω)dP1(ω) = IEP1 [IAZ ]

Variable Z is called the Radon-Nikodym derivative of P2 withrespect to P1, or the density of P2 with respect to P1.

Proof : see course of analysis (measure theory).

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Girsanov theorem

Remark

The density Z is a random variable with mean 1 (under P1) :to see that, it suffices to take A = Ω above :

P2(Ω) = 1 =

∫Ω

Z (ω)dP1(ω) = IEP1 [Z ].

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Stochastic calculus - II

Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

Girsanov theorem : exponential martingale

Let :• (Ω,F ,P) be a probability space•W be a standard Brownian motion under P• (Ft ) be associated filtration

DefinitionThe process Y defined by :

Y (t) = eθW (t)− 12 θ

2t

is called exponential martingale

We have already seen that Y is a martingale.

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Girsanov theorem : exponential martingale

This family of processes Y has the following properties :The random variables Y (t) have a log-normal distribution, withparameters

m = −12θ2t , σ2 = θ2t

IE [Y (t)] = 1Y (t) ≥ 0 for all t

−→ Y (t) appears as a natural candidate of Radon-Nikodymderivative to use in measure changes

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Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

Girsanov theorem

Theorem

Let W (y , ω); t ∈ [0,T ], ω ∈ Ω be a Brownian motion defined on aprobability space (Ω,F ,P).Let W ∗ be a process defined by :

W ∗(t) = W (t)− θt (θ ∈ R)

Then the process W ∗ is a Brownian motion under measure Qequivalent to P whose density is defined by :

ρT = Y (T ) = exp(θW (T )− θ2

2T)

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Girsanov theoremProof : We use the characteristic function of increments :

IEQ[exp(iu(W ∗(t)−W ∗(s)))|Fs]??= exp(−u2

2(t − s))︸ ︷︷ ︸

fn caract. d’une N(0,t-s)

??

or, equivalently, do we have that V (t) = exp(iuW ∗(t) + u2

2 t) is a martingaleunder Q?

IEQ[V (t)] =

∫Ω

V (t , ω)dQ(ω) =

∫Ω

V (t , ω)Y (T , ω)dP(ω)

=

(∫Ω

V (t , ω)Y (t , ω)dP(ω)

)(∫Ω

Y (T , ω)

Y (t , ω)dP(ω)

)(BM has independent increments)

=

∫Ω

V (t , ω)Y (t , ω)dP(ω)

since Y is an exponential martingale. Indeed :VUB Stochastic calculus : Introduction (II)

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Girsanov theoremIndeed :∫

Ω

Y (T , ω)

Y (t , ω)dP(ω) = IEP

[Y (T )

Y (t)

]= IEP

[IEP

[Y (T )

Y (t)|Ft

]]= IEP

[1

Y (t)IEP[Y (T )|Ft ]

]= IE

[1

Y (t).Y (t)

]= IE [1] = 1

Showing that V (t) is a martingale under Q is hence equivalent to showing thatV (t)Y (t) is a martingale under P.

V (t)Y (t) = exp(

iuW∗(t) +u2

2t)

exp(θW (t)−

12θ2t)

= exp(

iu(W (t)− θt) +u2

2t)

exp(θW (t)−

12θ2t)

= exp ((θ + iu)W (t)) exp(−

12

t(θ2 − u2 + iuθ)

)= exp ((θ + iu)W (t)) exp

(−

12

t(θ + iu)2)

︸ ︷︷ ︸martingale exponentielle

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Girsanov theorem – Generalization

Generalization of the drift of the Brownian motion :

W ∗(t) = W (t)− θt

↓W ∗(t) = W (t)−

∫ t

0b(s, ω)ds

where b is a stochastic process.

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Ito formulaStochastic differential equationsGirsanov theoremFeynman – Kac Lemma

Girsanov theorem – Generalization

Hypotheses on process b :

i. b is an process adapted to the filtrationii. b satisfies the Novikov condition :

IE

[12

∫ T

0b2(s, ω)ds

]<∞

iii. We define the random variable :

ρT = exp

(∫ T

0b(s, ω)dW (s)− 1

2

∫ T

0b2(s, ω)ds

)

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Girsanov theorem – Generalization

TheoremIf b satisfies conditions (i,ii,iii), then the process

W ∗(t) = W (t)−∫ t

0b(s, ω)ds

is a standard Brownian motion under Q whose density w.r.t. P is givenby :

dQdP

= ρT

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Feynman – Kac Lemma

Link between :

stochastic processes (and SDE)and

partial differential equations (PDE)

Objective : To represent the solution of a (parabolic) PDE under theform of an expectation• In practice, the solution of the parabolic PDE will represent here theprice of a derivative product (option) on the underlying asset whoserisk-neutral dynamics is described by an SDE.• This allows to solve the PDE by using probabilistic methods, and inparticular to approach numerically the solution by Monte-Carlomethods (instead of using numerical methods for PDEs consisting todiscretize the equation itself).

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Theorem (Feynman – Kac Lemma)

Let S = S(t , ω) be a solution of the SDE :

dS(t) = µ(t ,S(t))dt + σ(t ,S(t))dW (t) 0 ≤ t ≤ T

where W is a standard Brownian motion.

Let us define the function of 2 real variables (t , x) :

F (t , x) = IE [H(S(T ))|S(t) = x ]

where H is a function of one real variable. If∫ T

0IE

[(σ(t ,S(t))

∂F∂x

(t ,S(t))

)2]

dt <∞ (H1)

then function F is solution of the PDE with the terminal condition :∂F∂t

(t , x) + µ(t , x)∂F∂x

(t , x) +12σ2(t , x)

∂2F∂x2 (t , x) = 0

F (T , x) = H(x)

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Feynman – Kac Lemma : Remark

In fact, we can define the infinitesimal generator of diffusion S by thefollowing differential operator G :

G =12σ2(x)

∂2

∂x2 + µ(x)∂

∂x

F-K lemma tels us that F (t , x) = IE [H(ST )|St = x ] is solution of

∂F∂t

+ G(F ) = 0

with terminal conditions F (T , x) = H(x).

We can also see that the converse is also true : every solution of thePDE has the integral representation.

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Feynman-Kac Lemma

Proof : We apply the Ito formula to the process Y (t) = F (t ,S(t)) :

F (T ,S(T )) = F (t ,S(t)) +

∫ T

t

(∂F∂t

(s,S(s)) + µ(s,S(s))∂F∂x

(s,S(s))

)ds

+

∫ T

t

12σ2(s,S(s))

∂2F∂x2 (s,S(s))ds +

∫ T

tσ(s,S(s))

∂F∂x

(s,S(s))dW (s)

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Feynman – Kac Lemma

We take the conditional expectation of both members by conditioningw.r.t. event [S(t) = x ] :

Left hand side :

IE [F (T ,S(T ))|S(t) = x ] = IE [IE [H(S(T ))|S(T ) = S(T )]|S(t) =x ] = IE [H(S(T ))|S(t) = x ] = F (t , x)

Right hand side : first term :

IE [F (t ,S(t))|S(t) = x ] = F (t , x) = left hand sideRight hand side : last term :

IE[∫ T

t σ(s,S(s))∂F∂x (s,S(s))dW (s)

]= 0 since this is a property

of the stochastic integral and we have condition (H1)

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Feynman – Kac Lemma

It remains hence :

IE

[∫ T

t

(∂F∂s

(s,S(s)) + µ(s,S(s))∂F∂x

(s,S(s)) +12σ

2(s,S(s))∂2F∂x2

(s,S(s))

)ds|S(t) = x

]= 0

for any x , t and T .One can see that this implies that the integrand is zero, and thatthe EDP is satisfied.Terminal condition :

F (T , x) = IE [H(S(T ))|S(T ) = x ] = H(x)

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Feynman – Kac Lemma

One can also show the converse result. For that purpose, we depart from F solution ofthe backward PDE, and we apply the Ito formula, which leads as previously to :

F (T ,S(T )) = F (t ,S(t)) +

∫ T

t

(∂F∂t

(s,S(s)) + µ(s,S(s))∂F∂x

(s,S(s))

)ds

+

∫ T

t

12σ2(s,S(s))

∂2F∂x2

(s,S(s))ds +

∫ T

tσ(s,S(s))

∂F∂x

(s,S(s))dW (s)

We take the conditional expectation of both members w.r.t. [S(t) = x ].

Left hand side :IE [F (T ,S(T ))|S(t) = x ] = IE [H(S(T ))|S(t) = x ] since F satisfies the terminalcondition ;

First term right hand side :IE [F (t ,S(t))|S(t) = x ] = F (t , x) ;

The last term of the right hand side is = 0 as in the previous result ;

What remains is zero since F satisfies the PDE

We hence arrive to the conclusion that IE [H(S(T ))|S(t) = x ] = F (t , x).

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Feynman – Kac Lemma

Example : solving the equation

∂F∂t

+12∂2F∂x2 = 0,

with terminal condition F (T , x) = H(x).

The corresponding SDE (µ = 0, σ = 1) becomes :

dS(t) = dW (t)

In other words :S(t) = S(0) + W (t)

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Feynman – Kac Lemma

Feynman-Kac’s representation :

F (t , x) = IE [H(W (T ))|W (t) = x ]

Now we have already seen that

W (T )|W (t) = x ∼ N(x ,T − t)

, hence this expectation is equal to :

F (t , x) =

∫ +∞

−∞

1(T − t)

√2π

e−(y−x)2

2(T−t) H(y)dy

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Feynman – Kac Lemma : Variant

Variant : Introduction of a discounting within the expectation(→ expectation of the discounted payoff) :

F (t , x) = IE [e−r(T−t)H(S(T ))|S(t) = x ]

is solution of the backward PDE : ∂F∂t

(t , x) + µ(t , x)∂F∂x

(t , x) +12σ2(t , x)

∂2F∂x2 (t , x) = rF (t , x)

F (T , x) = H(x)

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Feynman – Kac Lemma : Financial application

Application to Black-Scholes model

Let us consider a call option on asset X , with maturity date T andstrike K, where the dynamics of the price of the asset X is modelledby a geometric Brownian motion.

We will see later that this price can be obtained as an expectation ofa discounted payoff, the one of the call :

H(X (T )) = (X (T )− K )+

We will see that the price of the call at instant t ≤ T , if the underlyingasset X is worth x at t , can be obtained as :

C(t , x) = IEQ

[e−r(T−t)(X (T )− K )+|X (t) = x

]

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Feynman – Kac Lemma : Financial application

where Q ∼ P is such that dX (t) = rX (t)dt + σX (t)dW (t) with W astandard B.M. under Q.

This price will satisfy the Black-Scholes PDE :

∂C∂t

+ rx∂C∂x

+12σ2x2 ∂

2C∂x2 = rC

with terminal condition C(T , x) = (x − K )+

→ application of this variant of Feynman-Kac lemma with µ(t , x) = rxand σ(t , x) = σx

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Lemme de Feynman - Kac : Financial application

Generalization in the case where the risk free rate is stochastic :

The function F defined by :

F (t , x) = IE[e−

∫ Tt r(s,X(s))dsH(X (T ))|X (t) = x

]is solution of :

∂F∂t

(t , x) + µ(t , x)∂F∂x

(t , x) +12σ2(t , x)

∂2F∂x2 (t , x) = r(t , x)F (t , x)

with terminal condition F (T , x) = H(x)

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