Functional calculus on integer-valued measures and ... · Overview 1 Introduction 2 Functional calculus for integer-valued measures 3 Martingale representation formula, purely discontinuous

Functional calculus on integer-valued measuresand martingale representation formulas for jump

processes

P. Blacque-Florentin

(joint with R. Cont)Imperial College London

AMaMeF & Swissquote Conference,8 Sep. 2015

P. Blacque-Florentin (Imperial) Functional calculus on N-valued measures Lausanne 2015 1 / 26

Overview

1 Introduction

2 Functional calculus for integer-valued measures

3 Martingale representation formula, purely discontinuous case

4 Including a continuous component

5 Example: Supremum of a Levy process


Martingale representation theorem for random measures

Let J(dtdy) be an integer-valued random measure on [0,T ]× Rd\0with compensator µ(dtdy) on a probability space (Ω,F ,P).The filtration (Ft) generated by J is said to have the predictablerepresentation property if any Ft-adapted square-integrable martingale issuch that

Y (t) = Y (0) +

∫ t

0

∫Rd\0

ψ(s, y)(J − µ)(ds dy)

with ψ : [0,T ]× Rd\0 × Ω→ Rd\0, Ft-predictable.Example: The predictable representation property for J holds if J is aPoisson random measure (Ito, Ikeda-Watanabe). Not true in general formeasures with non-deterministic compensators (Cohen 2013).


Martingale representation formulas

Problem of finding an explicit representation appears in manyapplications like hedging, control of jump processes or BSDEs withjumps.

Has been approached through Malliavin calculus for jump processes(Bismut 73, Lokka 05, Sole-Utzet-Vives 05) and Markoviantechniques (Jacod-Meleard-Protter 00).

In these results, ψ is represented in the form: ψ = pE [Dt,zY |Ft ],where D is an appropriate “Malliavin” derivative operator, for whichmany constructions have been proposed.


Outline

We introduce a pathwise calculus for functionals of integer-valuedmeasures.

Use it to provide an explicit version of the martingale representationformula for functionals of integer-valued measures.

These results extend the Functional Ito calculus to integer-valuedrandom measures.


Canonical space of integer-valued random measures

M =M([0,T ]× Rd\0) space of σ-finite integer-valued measures on[0,T ]× Rd\0:

j : B([0,T ]× Rd\0)→ N

such that j is σ-finite and there exists a sequence of(ti , zi ) ∈ [0,T ]× Rd\0 with elements neither necessarily distinct norordered such that

j(.) =∞∑i=0

δ(ti ,zi )(.)


Stopped measure

For any t ∈ [0,T ] and j ∈M([0,T ]× Rd\0), define

jt(.) := (. ∩ ([0, t]× (Rd\0))).

Similarlyjt−(.) := (. ∩ ([0, t)× (Rd\0))).

Non-anticipative functionals on measures

A map F : [0,T ]×M([0,T ]× (Rd\0)→ R is called

a non-anticipative functional if F (t, j) = F (t, jt).

a predictable functional if F (t, j) = F (t, jt−).


Definition

For z ∈ Rd , define the pathwise finite difference operator ∇j ,z

∇j ,zF (t, j) = F (t, jt− + δ(t,z))− F (t, jt−) (FD)

Denote

∇jF : [0,T ]×M× (Rd − 0) 7→ R(t, j , z) → ∇j ,zF (t, j)

Then the operator ∇j : F 7→ ∇jF maps non-anticipative functionals intopredictable functionals.


Integral functionals of integere-valued measures

Proposition: integral functionals

Let ψ : [0,T ]×Rd\0 → R be a kernel with support bounded away from0, and

F (t, j) =

∫ t

0

∫Rd\0

ψ(s, y)(j − µ)(ds dy),

with µ : B([0,T ]× Rd0 )×M([0,T ]× Rd\0)→ R+ predictable in j and

σ-finite. Then F is non-anticipative and ∇jF = ψ, i.e.

∀(t, z) ∈ [0,T ]× Rd , ∇j ,zF (t, j) = ψ(t, z).


Constructing the probability space

We now consider a integer-valued random measure with law P on (Ω,F),with compensator µ(dt dz), where

Ω =M([0,T ]× Rd\0); for ω ∈ Ω,

ω(t, .) = jt(.)

F a σ-algebra making the J(A), A ∈ B([0,T ]× Rd0 ) measurable.

The filtration F generated by J.

We will now show that the pathwise operator ∇j admits a unique closureon the space of square-integrable F-martingales, which is the adjoint ofthe stochastic integral with respect to the J − µ.


Space of square-integrable integrands

Assume that the compensator µ of J satisfies

µ(dsdy) = ν(s × dy)ds and

∫ T

0

∫Rd\0

|z |2 ∨ 1µ(dsdz) <∞,P-a.s.

defineL2P(µ): space of predictable random fields ψ : [0,T ]×Rd → R such that

‖ψ‖2L2P(µ) := E [

∫[0,T ]×Rd\0

|ψ(s, y)|2µ(ds dy)] <∞

and

I2P(µ) :=

Y : [0,T ]× Ω→ R|Y (t) =

∫[0,t]×Rd\0

ψ(s, y)(J − µ)(dsdy), ψ ∈ L2P(µ)

‖Y ‖2I2P(µ) := E [|Y (T )|2]


Set S of regular simple predictable fields

ψ : [0,T ]× Rd ×M([0,T ]× Rd\0)→ Rd belongs to S if

ψ is predictable: ψ(t, z , j) = ψ(t, z , jt−)

and

ψ(t, z , jt) =

I ,K∑i=0k=1

ψik(jti )1(ti ,ti+1](t)1Ak(z)

with Ak ∈ B(Rd\0),0 6∈ Ak and

ψik = gik(S1nτ1 , . . . ,S

1nτn),

gik bounded and 0 ≤ τ1 ≤ τn ≤ ti ,

Sεt := j([0, t]× (ε,∞)d)


Stochastic operators

The operator of compensated stochastic integration w.r.t J is defined as

I : L2P(µ)→ I2P(µ)

ψ 7→∫ .

0

∫Rd\0

ψ(s, y)(J − µ)(dsdy)

The operator ∇J is defined on I (S) as

∇J : I (S)→ L2P(µ)

F (t, Jt) =

∫ .

0

∫Rd\0

ψ(s, y)(J − µ)(dsdy)

7→ ∇j ,zF (t, Jt)

= F (t, Jt− + δt,z)− F (t, Jt−)

= ψ(t, z)


Density of regular integral functionals

Proposition

The set I (S) of processes Y that have the following functionalrepresentation

Y (.) = F (., J) =

∫ .

0

∫Rd\0

ψ(s, y)(J − µ)(dsdy) (1)

with ψ ∈ S, is dense in I2P(µ).

In other words, integral processes having a regular functionalrepresentation are dense in I2P(µ).


∇J as the adjoint of the stochastic integral

Theorem

The operator ∇J : I (S)→ L2P(µ) is closable in I2P(µ), and is the adjoint ofthe stochastic integral in the sense of the following integration by parts.

< Y , I (φ) >I2P(µ):= E

[Y (T )

∫ T

0

∫Rd\0

φ(s, y)(J − µ)(dsdy)

]

= E

[∫ T

0

∫Rd\0

∇JY (s, y)φ(s, y)µ(dsdy)

]=:< ∇JY , φ >L2P(µ)


Representation theorem for square-integrable martingales

Martingale representation formula

If the filtration F generated by J has the martingale representationproperty, and if a process Y (.) = F (., J) –with F an non-anticipativefunctional– is a square integrable martingale, then

Y (t) = Y (0) +

∫ t

0

∫Rd\0

∇JY (s, y)(J − µ)(dsdy)


Including the continuous component

On a filtered probability space (Ω,F ,F,P) constructed similarly as before,and F generated by an integer valued random measure J with compensator

µ(dsdy) = ν(s × dy)ds and

∫ T

0

∫Rd\0

|z |2µ(dsdz) <∞,P-a.s.,

and a continuous martingale X , any square-integrable martingale writes,P-a.s.

Y (T ) = Y (0) +

∫ T

0∇XY (s)dX (s) +

∫ T

0

∫Rd0

∇JY (s, z)J(ds dz),

with ∇XY defined as follows.


Defining Sc as:

Set Sc of regular simple predictable processes

ψ : [0,T ]××D([0,T ])×M([0,T ]× Rd\0)→ Rd belongs to Sc if

φ is predictable: ψ(t, z , x , j) = ψ(t, z , xt−, jt−)

and

φ(t, xt , jt) =I∑

i=0k=1

φi (xti , jti )1(ti ,ti+1](t)

with

φi = gi (x(τ1), · · · , x(τn),S1nτ1 , . . . ,S

1nτn),

gik ∈ C∞c (R2n,Rn) and 0 ≤ τ1 ≤ τn ≤ ti ,

Sεt := j([0, t]× (ε,∞)d)


Define L2P([X ]) := space of predictable processes ψ : [0,T ]× Ω→ Rsuch that

‖ψ‖2L2P([X ]) := E [

∫[0,T ]×Rd\0

|ψ(s, y)|2[X ](ds dy)] <∞

and

I2P([X ]) :=

Y : [0,T ]× Ω→ R|Y (t) =

∫[0,t]×Rd\0

φ(s)dX (s), ψ ∈ L2P([X ])

‖Y ‖2I2P([X ]) := E [|Y (T )|2]


Defining:

IX :L2P([X ])→ I2P([X ])

φ 7→∫ .

0φ(s)dX (s),

The operator

∇x :IX (Sc)→ I2P([X ])

F (t, xt , jt) 7→ limh→0

F (t, xt + h1[t,∞), jt)− F (t, xt , jt)

h

= φ(t)

can be closed in I2P([X ]) in the same fashion as in the jump case, and theclosure is I2P([X ]) itself.


Continuous- and jump-part comparison

Creating the martingale-generating measure

M(ds dz) := 1z=0dX (s) + zJ(ds dz),

the martingale representation formula rewrites as:

Y (t) = Y (0) +

∫ t

0

∫Rd

∇Y (s, z)M(ds dz) P-a.s.

where

∇Y (s, z) :=

∇XY (s, y) if z = 0∇JY (s,z)

z otherwise.

The continuous component is the limit operator of the operator appearingin the jump case.


Example: representation of the supremum of a Levyprocess

The representation formula for the supremum X of a Levy process X

1 was proved by Shiryaev and Yor (2004) using Ito’s formula.

2 was reproved more recently by Remillard-Renaud(2011) usingMalliavin calculus.

Main challenges in the functional Ito case:

1 Infinite variation: infinite variation, induced by a continuouscomponent and/or an infinite jump activity destroys the pathwisecharacterisation of the quantities.

2 In case the Levy process has a continuous component: the supremumis not a vertically differentiable functional.

→ we need to truncate the jumps and smoothen the functional.


Functional approximation

Define the Levy process

X (t) = X (0) + µt + σW (t) +

∫ t

0

∫|z|<1

zJ(dsdz) +

∫ t

0

∫|z|≥1

zJ(dsdz)

and its approximation

X n(t) = X (0)+µt+σW (t)+

∫ t

0

∫(−1,− 1

n)∪( 1

n,1)

zJ(dsdz)+

∫ t

0

∫|z|≥1

zJ(dsdz)

It can be shown that

E [X (T )|Ft ] = X (t) +

∫ ∞X (t)−X (t)

FT−t(u)du,

with FT−t(u) = P(X (T − t) ≤ u).


Furthermore, consider the approximation of the supremum functional,

La(f , t) =1

alog(

∫ t

0eaf (s)ds).

Define the approximation:

Y a,n(t) = La(X n, t) +

∫ ∞La(X n,t)−X n(t)

FT−t(u)du

Since X nL2

−→ Xn→∞

and La(f ,T ) −→a→0

sups∈[0,T ] f (s), one can show:

limn→∞

lima→∞

E [|Y a,n(T )− X (T )|2] = 0


Operator computations

We can now compute

∇JYa,n(t, z) =

∫ La(X n,t)−X n(t)

La(X n,t)−X n(t)−zFT−t(u)du

−→a→∞n→∞

∫ X (t)−X (t)

X (t)−X (t)−zFT−t(u)du = ∇JX (t, z)

and

∇WY a,n(t) = limh→0

1

h

∫ La(X n,t)−X n(t)

La(X n,t)−X n(t)−σhFT−t(u)du

= FT−t(La(X n, t)− X n(t))

−→a→∞n→∞

σFT−t(X (T )− X (t)) = ∇WX (t)


References

R. Cont and D-A. Fournie (2013)Functional Ito calculus and stochastic integral representation ofmartingalesAnn. Probab. 41 (2013), no. 1, 109–133

P. B-F and R. Cont (2015):Functional Ito calculus and martingale representation formula forinteger-valued random measureshttp://arxiv.org/abs/1508.00048


http://arxiv.org/abs/1508.00048

Documents

Functional calculus on integer-valued measures and ... · Overview 1 Introduction 2 Functional calculus for integer-valued measures 3 Martingale representation formula, purely discontinuous