From Stochastic Calculus of Variations on Wiener space to Stochastic - - PowerPoint PPT Presentation

From “Stochastic Calculus of Variations on Wiener space” to “Stochastic Calculus of Variations on Poisson space”.

Maurizio Pratelli Department of Mathematics, University of Pisa pratelli@dm.unipi.it Brixen, July 16, 2007

Malliavin’s derivative: “Calculus of Variations approach”

Ω = C0(0, T) F ∈ L2(Ω) (a functional on Wiener space) Cameron-Martin space CM: h ∈CM if h(t) = t

0 ˙

h(s) ds , ˙ h ∈ L2(0, T) and hCM = ˙ hL2 .

Suppose that ∃ Zs ∈ L2(Ω × [0, T]) such that lim

ε→0

F(ω + ǫh) − F(ω) ǫ =

T

Zs(ω)˙ h(s) ds then F is derivable (in Malliavin’s sense) and Zs = DsF (more gen- erally DhF = T

0 DsF ˙

h(s) ds ). With this definition, D is like a Fr´ echet derivative, but only along the directions in CM. Why?

Girsanov’s theorem: if dP∗ dP = exp

T

˙ h(s)dWs − 1 2

T

˙ h2(s) ds

• = LT

law of W(.) + h(.) under P = law of W(.) under P∗ (recall that on the canonical space Wt(ω) = ω(t)).

Introducing dPǫ dP = exp

• ǫ

T

˙ h(s)dWs − ǫ2 2

T

˙ h2(s) ds

• = Lǫ

T

we have I E

F(ω + ǫh) − F(ω)

ǫ

• = I

E

• F(ω) Lǫ

T − 1

ǫ

• since limǫ→0

Lǫ

T −1

ǫ

= T

0 ˙

h(s) dWs

we obtain the integration by parts formula I E

T

DsF ˙ h(s) ds

• = I

E

• F

T

˙ h(s) dWs

• (˙

h(s) can be replaced by Hs ∈ L2 Ω × [0, T] adapted) Intuitively: Malliavin’s calculs is the analysis of the variations of the paths along the directions supported by Girsanov’s theorem.

More generally: for k ∈ L2 0, T

• , define W(k) = T

0 k(s) dWs (Wiener’s

integral) and define smooth functional F = φ W(k1), . . . , W(kn) (φ smooth). We obtain easily DsF =

n

• i=1

∂φ ∂xi

• W(k1), . . . , W(kn)

ki(s)

The operator D : S ⊂ L2 Ω

• → L2

Ω × [0, T]

• (S space of smooth

functionals) is closable (by the integration by parts formula) (from now on we consider the closure). The adjoint operator D∗ = δ : L2 Ω × [0, T]

• is called divergence
• r Skorohod integral and D∗ restricted to the adapted processes

coincides with Ito’s integral. This is equivalent to the Clark-Ocone-Karatzas formula: if F is derivable F = I E[F] +

T

I E

• DsF
• Fs
• dWs

Very important is the so-called Chain rule: Ds φ F 1, . . . , F n =

n

• i=1

∂φ ∂xi

• · · ·

Ds F i (if φ : I Rn → I R is derivable in the classic sense and F 1, . . . , F n in the Malliavin’s sense).

Summing up:

• integration by parts formula
• D∗ restricted to adapted processes coincides with Ito’s integral
• Clark-Ocone-Karatzas formula
• chain rule

A remark: Skorohod (anticipating) integral is not an integral (limit

• f Riemann’s sums).

Intuitively

T

Hs dXs = lim

• i

Hti

• Xti+1 − Xti
• Formula: if Hs is adapted and F derivable

T

• FHs
• δWs = F

T

Hs dWs −

T

DsF Hs ds

Malliavin derivative in Chaos Expansion.

An introductory example: alternative description on the space H1,2 0, 2π

• .

f ∈ L2 0, 2π can be written f = a0 +

• k≥1
• ak cos kx + bk sin kx
• k

|ak|2 + |bk|2 < +∞ If there is a finite number of terms f′ =

• k
• k bk cos kx − k ak sin kx

Therefore f is derivable (in weak sense) and f′ ∈ L2 0, 2π if

• k

k2 |ak|2 + |bk|2 < +∞ and f′ =

• k

k

• bk cos kx − ak sin kx
• short and easy definition of (weak) derivative and of the space

H1,2 0, 2π ;

• the meaning of derivative is hidden.

Wiener Chaos Expansion Sn =

• 0 < t1 < · · · < tn < T
• , given f ∈ Sn

Jn(f) =

• ]0,T]

dWtn

• ]0,tn]

dWtn−1 · · ·

• ]0,t2]

f(t1, . . . , tn) dWt1 I E

• Jn(f)2

=

• f
• 2

L2(Sn)

If Cn = image of L2(Sn) by Jn , we have L2 Ω = C0 ⊕ C1 ⊕ C2 . . .

If L2 [0, T]n is the subspace of symmetric functions of L2 [0, T]n , define: In(f) = n!

• · · ·
• Sn

f(· · · ) dWtn · · · dWt1 we have I E

• In(f)2

= n!

• f
• 2

L2([0,T]n) .

F ∈ L2 Ω can be written F =

n≥0 In

• fn
• with

n≥0 n! fn2 L2 < +∞

By direct calculus Dt In

• fn(t1, . . . , t)
• = n In−1
• fn(t1, . . . , tn−1, t)
• We can define

Dt F =

n≥1 n In−1

• . . .

provided that

• n≥1 n n! fn2

L2 < +∞

A similar characterization can be given for Skorohod integral.

With this approach:

• concise and more elementary definitions of Malliavin’s derivative

and divergence

• some proof are easier, some more complicated (e.g. “chain rule”)
• the idea of derivative is hidden

A good result with this approach: Energy identity for Skorohod integral (Nualart, Pardoux, Shigekawa) I E

T

Zs δWs

2

= I E

T

Z2

s ds +

T T

(DtZs + DsZt) ds dt

• Other approaches: discretization (Ocone, Mallavin–Thalmaier), weak

derivation ...

Main applications of Malliavin calculus:

• Clark–Ocone–Karatzas formula (explicit characterization of the

integrand)

• Regularity of the law of some r.v. (solutions of S.D.E.)
• Sensitivity analysis in Mathematical Finance (Monte Carlo weights

for the Greek’s)

An idea of “sensitivity analysis” (Fourni´ e, Lasry, Lebuchoux, Lions, Touzi [99], and F.L.L.L. [01]):

∂ ∂ζ I

E

• f
• F ζ

= I E

• f′

F ζ ∂ζF ζ = = I E

Dw

• f
• F ζ

Dw F ζ

∂ζF ζ = I E

• f
• F ζ

D∗

w

∂ζF ζ

DwF ζ

• .

The “weight” W = D∗

w

∂ζF ζ

DwF ζ

• is independent of f (and not unique).

In order to extend to more general situations (from diffusion models to jump–diffusion models), we need:

• an integration by parts formula
• chain rule.

Plain Poisson process

Let Pt be a Poisson process with jump times τ1 < τ2 < . . . (σi = τi − τi−1 are independent exponential density) and Nt = (Pt − t) the compensated Poisson.

Point of view of Chaos Expansion:

Starting from

• Jn(f) =
• ]0,T]

dNtn

• ]0,tn[

dNtn−1 · · ·

• ]0,t2[

f(t1, . . . , tn) dNt1

A similar theory, based on chaotic representation, can be developed w.r.t. Nt (Lokka, Oksendal and ...)

• similar definition of derivative Dc and Skorohod integral
• (Dc)∗ coincides with ordinary stochastic integrals on predictable

processes

• Clark–Ocone–Karatzas formula

A serious drawback: the chain rule is not satisfied. In fact, the “chaotic” derivative satisfies the formula Dc

t

• FG
• = F Dc

tG + G Dc tF + Dc tF Dc tG

(Chain rule is (morally) equivalent to the formula Dt(FG) = FDtG + GDtF ).

An alternative point of view: Variations on the paths

(via Girsanov theorem) Given h(t) = t

0 ˙

h(s) ds , ˙ h ∈ L2(0, T) and ˙ h uniformly bounded from below, consider a perturbed probability dPǫ dP = Lǫ

T = exp

• − ǫ

T

˙ h(s)ds

s≤T

• 1 + ǫ˙

h(s)∆Ps

• Let αǫ(t) = t
• 1 + ǫ ˙

h(r) dr (a variation on time): law of Pαǫ(.) under P = law of P(.) under Pǫ

Similar definition for derivative of a Poisson functional: lim

ε→0

F(Pαǫ.) − F(P.) ǫ Since limǫ→0

Lǫ

T −1

ǫ

= T

0 ˙

h(s) dNs we obtain the integration by parts formula. Some differences with Gaussian case: only a deterministic perturba- tion is allowed, (the integration by parts formula is less immediate).

On smooth functionals of the form F = φ τ1, . . . , τn

• we obtain by a direct calculus

Dv

t φ

τ1, . . . , τn

• = −

n

• i=1

∂φ ∂xi

• . . .

I[0,τi](t) Good properties: (Dv)∗ concides with stochastic integrals for pre- dictable processes (Clark–Ocone–Karatzas), the chain rule is sat- isfied.

A drawback: the analysis of divergence is more complicated (w.r.t. chaotic point of view) A serious drawback: PT is not derivable (not in the domain of the

• perator Dv)!

PT =

• i≥1

I[0,T](τi) is not a smooth function of the jump times.

A remark: the domains of the operators Dc and Dv are completely

• different. Typical derivable functionals are:
• stochastic integrals T

0 h(s) dNs

(or iterated stoch. int.) for the

• perator Dc ;
• smooth functions φ
• τ1, . . . , τn
• for the operator Dv .

The “variations” point of view was investigated in some papers by Privault with a different approach (Bouleau–Hirsch, who started by proving Clark–Ocone-Karatzas formula). A similar approach is in Elliot–Tsoi ”93. Privault obtained sensitivity results for models of the kind dSt = St−

• m(t) dt +

n

• j=1

αj dP j

t

• (P 1, . . . , P n) independent Poisson processes, for Asian options of the

form T

0 f(t, St) dt .

Compound Poisson processes

Xt =

• j≤Pt

Uj − λ t I E

• Uj
• =

[0,t]×I R

x d

• µ − ν
• Pt Poisson process with intensity λ , U1, U2, . . . i.i.d.

µ =

• n

ǫ(τn,Un) ; ν(ω, dt, dx) = λ dt dF(x)

Chaotic expansion approach developed by L´ eon et coll. (2002), Oksendal and coll. (many papers) with attention to anticipative calculus, anticipative Ito’s formulae ...

Variations on the paths

Two possibilities: variations on jump times and on jump amplitude (supported by Girsanov’s theorem)

Variations on jump times.

Integration by parts formula I E

T

Dt

sF Hs ds

• = I

E

• F

T

Hs dNs

• (No hope for a Clark-Ocone-Karatzas formula)

Variations on jump amplitude.

This is the good point of view, and it was investigated by Bismut, Bass–Cranston, Jacod–Bichteler–Pellaumail under a restriction: dF(x) is the Lebesgue measure under a suitable open interval E . Their results can be extended to the case dF(x) = f(x) dx (where the “density” f is continuous and strictly positive on an open interval E =]a, b[ ).

Methods:

• look at the process Xt in the form

[0,t]×I R x d

µ − ν

• use Girsanov theorem for random measures
• consider s.d.e. with respect to random measures.

Integration by parts formula I E

[0,T]×E

Dj

(s,x)F H(s, x) dsdF(x)

• = I

E

• F

[0,T]×E

H d(µ−ν)

A remark: some papers extend sensitivity analysis to jump-diffusion models by using the chaotic approach. How is it possible? Idea: if F(ω, ω′) (ω in the Wiener space, ω′ in Poisson space), we have Dωφ

• F(ω, ω′)
• = φ′

F) DωF(ω, ω′) (where Dω is the derivative w.r.t. Wiener component, ω′ is only a parameter). Davis–Johannson (2006) under a separability assumption, Teichmann– Forster–Lutkebohmert (2007) under more general hypothesis.

Separability assumption: St = f Xc

t , Xd t

• where Xc satisfies an equation

dXc

t = Xc t

• m(t) dt + σc

t dWt

• and Xd

t satisfies a similar equation on the Poisson space.

Bavouzet–Messaoud uses integration by parts w.r.t. jump ampli- tude, but only after discretization. These methods seems not convenient for more general L´ evy pro- cesses.

Happy Belated Birthday Wolfgang !