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From “Stochastic Calculus of Variations on
Wiener space” to “Stochastic Calculus of
Variations on Poisson space”.
Maurizio Pratelli
Department of Mathematics, University of Pisa
Brixen, July 16, 2007
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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) =∫ t0 h(s) ds , h ∈ L2(0, T )
and ‖h‖CM = ‖h‖L2 .
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Suppose that ∃Zs ∈ L2(Ω× [0, T ]) such that
limε→0
F (ω + εh)− F (ω)
ε=
∫ T
0Zs(ω)h(s) ds
then F is derivable (in Malliavin’s sense) and Zs = DsF (more gen-
erally DhF =∫ T0 DsF h(s) ds ).
With this definition, D is like a Frechet derivative, but only along the
directions in CM. Why?
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Girsanov’s theorem: if
dP∗
dP= exp
( ∫ T
0h(s)dWs − 1
2
∫ T
0h2(s) ds
)= LT
law of(W(.) + h(.)
)under P = law of W(.) under P∗
(recall that on the canonical space Wt(ω) = ω(t)).
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Introducing
dPε
dP= exp
(ε
∫ T
0h(s)dWs − ε2
2
∫ T
0h2(s) ds
)= Lε
T
we have
IE[F (ω + εh)− F (ω)
ε
]= IE
[F (ω)
LεT − 1
ε
]
since limε→0Lε
T−1ε =
∫ T0 h(s) dWs
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we obtain the integration by parts formula
IE[ ∫ T
0DsF h(s) ds
]= IE
[F
∫ T
0h(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.
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More generally: for k ∈ L2(0, T
), define W (k) =
∫ T0 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)
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The operator D : S ⊂ L2(Ω
) → L2(Ω × [0, T ]
)(S space of smooth
functionals) is closable (by the integration by parts formula) (fromnow on we consider the closure).
The adjoint operator D∗ = δ : L2(Ω × [0, T ]
)is called divergence
or Skorohod integral and D∗ restricted to the adapted processescoincides with Ito’s integral.
This is equivalent to the Clark-Ocone-Karatzas formula: if F isderivable
F = IE[F ] +
∫ T
0IE
[DsF
∣∣Fs]dWs
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Very important is the so-called Chain rule:
Ds φ(F1, . . . , Fn
)=
n∑
i=1
∂φ
∂xi
( · · · )Ds F i
(if φ : IRn → IR is derivable in the classic sense and F1, . . . , Fn in the
Malliavin’s sense).
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Summing up:
• integration by parts formula
• D∗ restricted to adapted processes coincides with Ito’s integral
• Clark-Ocone-Karatzas formula
• chain rule
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A remark: Skorohod (anticipating) integral is not an integral (limitof Riemann’s sums).
Intuitively
∫ T
0Hs dXs = lim
∑
i
Hti
(Xti+1 −Xti
)
Formula: if Hs is adapted and F derivable
∫ T
0
(FHs
)δWs = F
∫ T
0Hs dWs −
∫ T
0DsF Hs ds
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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
)
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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.
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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
IE[Jn(f)2
]=
∥∥f∥∥2
L2(Sn)
If Cn = image of L2(Sn) by Jn , we have L2(Ω
)= C0 ⊕ C1 ⊕ C2 . . .
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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 IE[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! ‖fn‖2L2 < +∞
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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! ‖fn‖2L2 < +∞
A similar characterization can be given for Skorohod integral.
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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
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A good result with this approach: Energy identity for Skorohod
integral (Nualart, Pardoux, Shigekawa)
IE[( ∫ T
0Zs δWs
)2]= IE
[ ∫ T
0Z2
s ds +
∫ T
0
∫ T
0(DtZs + DsZt) dsdt
]
Other approaches: discretization (Ocone, Mallavin–Thalmaier), weak
derivation ...
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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)
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An idea of “sensitivity analysis” (Fournie, Lasry, Lebuchoux, Lions,
Touzi [99], and F.L.L.L. [01]):
∂∂ζ IE
[f(F ζ
)]= IE
[f ′
(F ζ
)∂ζF
ζ]=
= IE[Dw
[f(F ζ
)]Dw F ζ ∂ζF
ζ]= IE
[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).
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In order to extend to more general situations (from diffusion models
to jump–diffusion models), we need:
• an integration by parts formula
• chain rule.
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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
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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
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A serious drawback: the chain rule is not satisfied.
In fact, the “chaotic” derivative satisfies the formula
Dct
(FG
)= F Dc
tG + G DctF + Dc
tF DctG
(Chain rule is (morally) equivalent to the formula
Dt(FG) = FDtG + GDtF ).
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An alternative point of view: Variations on the paths
(via Girsanov theorem)
Given h(t) =∫ t0 h(s) ds , h ∈ L2(0, T ) and h uniformly bounded from
below, consider a perturbed probability
dPε
dP= Lε
T = exp(− ε
∫ T
0h(s)ds
) ∏
s≤T
(1 + εh(s)∆Ps
)
Let αε(t) =∫ t0
(1 + ε h(r)
)dr (a variation on time):
law of Pαε(.) under P = law of P(.) under Pε
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Similar definition for derivative of a Poisson functional:
limε→0
F (Pαε.)− F (P.)
ε
Since limε→0Lε
T−1ε =
∫ T0 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).
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On smooth functionals of the form
F = φ(τ1, . . . , τn
)
we obtain by a direct calculus
Dvt 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.
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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
operator Dv)!
PT =∑
i≥1
I[0,T ](τi)
is not a smooth function of the jump times.
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A remark: the domains of the operators Dc and Dv are completely
different. Typical derivable functionals are:
• stochastic integrals∫ T0 h(s) dNs (or iterated stoch. int.) for the
operator Dc ;
• smooth functions φ(τ1, . . . , τn
)for the operator Dv .
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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 dPjt
)
(P1, . . . , Pn) independent Poisson processes, for Asian options of the
form∫ T0 f(t, St) dt .
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Compound Poisson processes
Xt =∑
j≤Pt
Uj − λ t IE[Uj
]=
∫ ∫
[0,t]×IRxd
(µ− ν
)
Pt Poisson process with intensity λ , U1, U2, . . . i.i.d.
µ =∑n
ε(τn,Un) ; ν(ω,dt,dx) = λdtdF (x)
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Chaotic expansion approach developed by Leon 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)
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Variations on jump times.
Integration by parts formula
IE[ ∫ T
0Dt
sF Hs ds]= IE
[F
( ∫ T
0Hs dNs
)]
(No hope for a Clark-Ocone-Karatzas formula)
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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[ ).
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Methods:
• look at the process Xt in the form∫ ∫
[0,t]×IR xd(µ− ν
)
• use Girsanov theorem for random measures
• consider s.d.e. with respect to random measures.
Integration by parts formula
IE[ ∫ ∫
[0,T ]×ED
j(s,x)
F H(s, x) dsdF (x)]= IE
[F
( ∫ ∫
[0,T ]×EH d(µ−ν)
)]
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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.
![Page 37: Stochastic Calculus of Variations on Poisson space.brixen07/slides/pratelli.pdf · Similar deflnition for derivative of a Poisson functional: lim "!0 F(Pfi†:) ¡ F(P:) † Since](https://reader033.vdocuments.mx/reader033/viewer/2022053008/5f0b809e7e708231d430d612/html5/thumbnails/37.jpg)
Separability assumption:
St = f(Xc
t , Xdt
)
where Xc satisfies an equation
dXct = Xc
t
(m(t) dt + σc
t dWt
)
and Xdt satisfies a similar equation on the Poisson space.
![Page 38: Stochastic Calculus of Variations on Poisson space.brixen07/slides/pratelli.pdf · Similar deflnition for derivative of a Poisson functional: lim "!0 F(Pfi†:) ¡ F(P:) † Since](https://reader033.vdocuments.mx/reader033/viewer/2022053008/5f0b809e7e708231d430d612/html5/thumbnails/38.jpg)
Bavouzet–Messaoud uses integration by parts w.r.t. jump ampli-
tude, but only after discretization.
These methods seems not convenient for more general Levy pro-
cesses.
![Page 39: Stochastic Calculus of Variations on Poisson space.brixen07/slides/pratelli.pdf · Similar deflnition for derivative of a Poisson functional: lim "!0 F(Pfi†:) ¡ F(P:) † Since](https://reader033.vdocuments.mx/reader033/viewer/2022053008/5f0b809e7e708231d430d612/html5/thumbnails/39.jpg)
Happy Belated Birthday
Wolfgang !