Post on 23-Jun-2020
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) =∫ t0 h(s) ds , h ∈ L2(0, T )
and ‖h‖CM = ‖h‖L2 .
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?
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)).
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
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.
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)
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
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).
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 (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
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
IE[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 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 < +∞
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.
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)
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 ...
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” (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).
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
Dct
(FG
)= F Dc
tG + G DctF + Dc
tF DctG
(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) =∫ 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
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).
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.
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.
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 .
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 .
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)
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)
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)
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]×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)
)]
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 , 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.
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.
Happy Belated Birthday
Wolfgang !