Malliavin derivative of Lévy driven BSDEs and applications2015... · Application: a BSDE with a bounded solution 3. Application: a BSDE with a bounded solution conjectures: thegeneratorcanbequadratic(inz),

Malliavin derivative of Lévy driven BSDEs andapplications

Christel Geiss

University of Jyväskylä (Finland)

Workshop Stochastic Analysis, Controlled Dynamical Systemsand Applications

Jena, March 9 -13, 2015

C. Geiss Malliavin derivative of BSDEs 1 / 26

Malliavin derivative (Lévy setting)

Outline

1 Malliavin derivative (Lévy setting)

2 Malliavin derivative of BSDEs

3 Application: a BSDE with a bounded solution



1. Malliavin derivative (Lévy setting)

Itô’s chaos expansion:Lt = σWt +

∫(0,t]×|x |≤1 xN(ds, dx) +

∫(0,t]×|x |>1 xN(ds, dx),

for any ξ ∈ L2 := L2(Ω,FLT ,P) exists the chaos expansion

ξ =∞∑n=0

In (fn) , a.s.

In (fn) multiple integrals w.r.t. M(ds, dx) = σdWsδ0(dx) +N(ds, dx)

for example:

I0 (f0) = Eξ, I1 (f1) =

∫[0,T ]×R

f1(s, x)M(ds, dx)



I2 (f2) = 2∫[0,T ]×R

(∫[0,s)×R f2((u, y), (s, x))M(du, dy)

)M(ds, dx)

if f2((u, y), (s, x)) = f2((s, x), (u, y)).

Let Ln2 := L2(([0,T ]×R)n,B(([0,T ]×R)n),m⊗n)

with m(ds, dx) = ds(σ2δ0(dx) + ν(dx))

For any symmetric fn ∈ Ln2, gm ∈ Lm2

EIn(fn)Im(gm) =

n!〈fn, gn〉Ln2 , m = n

0, else.



Malliavin Derivative via chaos expansion (Solé, Utzet, Vives, 2007)Let

D1,2 :=

ξ =

∞∑n=0

In(fn) ∈ L2 :∞∑n=0

(n + 1)!‖fn‖2Ln2 <∞

For ξ ∈ D1,2 we define Dξ ∈ L2(P⊗m) by

Dt,xξ =∞∑n=1

nIn−1(fn((t, x), · )).



’Malliavin derivative in direction of the Brownian part’

D01,2 :=

ξ =

∞∑n=0

In(fn) ∈ L2 : fn ∈ Ln2, n ∈ N,

∞∑n=1

(n + 1)!

∫ T

0‖fn((t, 0), ·)‖2Ln−1

2dt <∞

(1)

’Malliavin derivative for the jump part’

DR01,2 :=

ξ =

∞∑n=0

In(fn) ∈ L2 : fn ∈ Ln2, n ∈ N,

∞∑n=1

(n + 1)!

∫[0,T ]×R0

‖fn((t, x), ·)‖2Ln−12

dtν(dx) <∞.

For σ > 0 and ν 6= 0 : D1,2 = D01,2 ∩ DR0

1,2. (2)



A characterization for DR01,2

Malliavin derivative related to Picard’s difference operator approachFor any ξ ∈ L2 we have by the factorization Lemma:∃G : D([0,T ])→ R :

ξ(ω) = G(

(Ls(ω))0≤s≤T

)=: G (L(ω)) a.a. ω ∈ Ω,

where D([0,T ]) denotes the Skorohod space of càdlàg functions.



A characterization for DR01,2

Lemma 1 (Alos et al. 2008, G. and Steinicke)

1 If ξ = G (L) ∈ L2 then G (L) ∈ DR01,2 ⇐⇒

G (L + x1I[t,T ])− G (L) ∈ L2(P⊗m)

and it holds then for x 6= 0 P⊗m-a.e.

Dt,xξ = G (L + x1I[t,T ])− G (L)

2 Let Λ ∈ B(D[0,T ]) be a set with P (L ∈ Λ) = 0. Then

P⊗m(

(ω, t, x) ∈ Ω× [0,T ]×R0 : L(ω) + x1I[t,T ] ∈ Λ)

= 0.


Malliavin derivative of BSDEs

Outline






2. Malliavin derivative of BSDEs

Yt = ξ +

∫ T

tf (., s,Ys ,Zs ,Us)ds −

∫ T

tZsdWs

−∫(t,T ]×R

Us(x)N(ds, dx), 0 ≤ t ≤ T

Brownian motion case, deterministic generator: Pardoux & Peng(1992)Idea of the proof: Picard iteration,Y n+1t = ξ +

∫ Tt f (s,Y n

s ,Zns )ds −

∫ Tt Zn+1

s dWs

assume Y n,Zn have a Malliavin derivative, conclude this forY n+1,Zn+1

f (ω, s, y , z), El Karoui, Peng, & Quenez, (1997)Delong & Imkeller (2010) delay equation, with jumps



Which conditions are needed such that f (·, s,Y ns ,Z

ns ) ∈ D1,2?

F : Ω×Rd → R measurable and G1, ...,Gd ∈ D1,2,when does it follow that F (·,G1, ...,Gd) ∈ D1,2?

(Ω,F ,P) := (ΩW × ΩJ , FW ⊗FJ ,PW ⊗ PJ).

If we set E : = L2(ΩJ ,FJ ,PJ) it holds

ξ ∈ DR01,2 and ξ ∈ DW

1,2(E ) ⇐⇒ ξ ∈ D1,2.

Regard ωW 7→ ξ(ωW, ωJ) ∈ E .



generalized chain rule

Theorem 2 (G. and Steinicke)

Suppose G1, ...,Gd ∈ DW1,q(E ) for some q ≥ 1 and let p > 1,

1 F (ω, ·) ∈ C1(Rd) for a.a. ω,2 F (·, y) ∈ DW

1,p(E ) for all y ,

3 (DWF )(·, y) |y=G ∈ Lp(PW ;HS(L2[0,T ],E )) and∑dk=1 ∂ykF (·,G )DWGk ∈ Lp(PW ;HS(L2[0,T ],E )),

4 ’locally Lipschitz’ ∀N ∈ N ∃KN ∈⋃

r>1 Lr (P) and for a.a. ω

‖(DWF (·, y))(ω)− (DWF (·, y))(ω)‖L2[0,T ]≤ KN(ω)|y − y |,|y |, |y | ≤ N



Theorem 2 (G. and Steinicke)

ThenF (·,G ) ∈ DW

1,p(E )

and

DWF (·,G ) = (DWF )(·,G ) +d∑

k=1

∂ykF (·,G )DWGk

∈ Lp(PW ;HS(L2[0,T ],E )).



idea of the proofSobolev classes w.r.t. the Gaussian measureDW

1,p(E ) = ’Kusuoka-Stroock Sobolev class’ p ∈ (1,∞) : Sugita(1985), p = 1 : Bogachev (1998)Kusuoka-Stroock Sobolev class is defined as

KW1,p(E ) := ξ ∈ Lp :ξ r .a.c. and s.G .d .,

Dξ ∈ Lp(PW ;HS(L2[0,T ],E ))r.a.c. =’ray absolutely continuous’,s.G.d. = ’stochastically Gateâux differentiable’by a characterization of Janson (1997) one may describe r.a.c. usingthe Gateâux differential

KW1,p(E ) = ξ ∈ Lp : ∃Dξ ∈ Lp(PW ;HS(L2[0,T ],E ))

∀h ∈ L2[0,T ] : ρuhξ(ω)− ξ(ω) =

∫ u

0ρsh〈Dξ, h〉ds a.s.

with absolutely integrable integrand



ρuh Cameron-Martin shift on the Wiener space:

ρuhξ(ω) := ξ(ω + u

∫ ·0h(s)ds)

ξ is s.G.d. ⇐⇒ ∃L2[0,T ]-valued r .v .Dξ : ∀h ∈ L2[0,T ]

limu→0

ρuhξ(ω)− ξ(ω)

u= 〈Dξ, h〉, a.s.

convergence in probability !In general, the Malliavin derivative and s.G.d. are not equal:If A ∈ F , then 1IA is s.G.d. and D1IA = 0,but 6 ∃D1IA for 0 < P(A) < 1.



Yt = ξ +

∫ T

tf

s,Ys ,Zs ,

∫R0

g(Us(x))g1(x)ν(dx)︸︷︷︸[g(Us)]ν

ds

−∫ T

tZsdWs −

∫(t,T ]×R0

Us(x)N(ds, dx), 0 ≤ t ≤ T



Theorem 3 (G. & Steinicke)

Under the assumptions given below it holds1 For the solution (Y ,Z ,U) it holds

Y ,Z ∈ L2([0,T ];D1,2), U ∈ L2([0,T ]×R0;D1,2),

and t 7→ Dr ,yYt admits a càdlàg version for m- a.e. (r , y).

2p(

(Dr ,0Yr )r∈[0,T ]

)is a version of (Zr )r∈[0,T ]

p(

(Dr ,xYr )r∈[0,T ],x∈R0

)is a version of (Ur (x))r∈[0,T ],x∈R0



Theorem 3 (G. & Steinicke)3 For m- a.e. (r , v) the process (Dr ,vY ,Dr ,vZ ,Dr ,vU) is the unique

solution of

Dr ,vYt = Dr ,vξ +

∫ T

tFr ,v (s,Dr ,vYs ,Dr ,vZs ,Dr ,vUs) ds

−∫ T

tZsdWs −

∫(t,T ]×R0

Us(x)N(ds, dx)

where Fr ,v (s, y , z , u) for is given by

(Dr ,0fg ) (s,Ys ,Zs ,Us)

+ 〈∇f (s,Ys ,Zs , [g(Us)]ν) , (y , z , [g ′(Us)u]ν )〉 v = 0

fg((L + v1I[r ,T ]), s,Ys + y ,Zs + z ,Us + u

)−fg (L, s,Ys ,Zs ,Us) v 6= 0,



assumptions (A1):1 ξ ∈ D1,2.

2 E∫ T0 |f (., t, 0, 0, 0)|2 dt <∞.

3 ∀t ∈ [0,T ] : R3 3 y 7→ ∂yi f (., t, y), i = 1, 2, 3 is P-a.s. boundedand continuous.

4 ∀(t, y) ∈ [0,T ]×R3 : f (., t, y) ∈ D1,2

5 ∀t ∈ [0,T ], ∀N ∈ N ∃ K tN ∈

⋃p>1 Lp :

for a.a. ω and ∀|y |, |y | ≤ N

‖ (D.,0f ) (., t, y)(ω)− (D.,0f ) (., t, y)(ω)‖L2[0,T ] < K tN(ω)|y − y |.

6 ∀G ∈ (L2)3 : ∃Γ ∈ L2(P⊗m), such that for a.e. t it holds

|(Ds,x f ) (., t,G )| ≤ Γs,x P⊗m− a.e.

7 g ∈ C 1(R) with bounded derivative and g1 ∈ L2(R0,B(R0), ν).


Application: a BSDE with a bounded solution

Outline






3. Application: a BSDE with a bounded solution

Yt = ξ +

∫ T

tf (s,Ys ,Zs) +

∫R0

gα(Us(x))g1(x)ν(dx)ds

−∫ T

tZsdWs −

∫(t,T ]×R0

Us(x)N(ds, dx),

gα(u) :=eαu − αu − 1

αfor some α > 0

assumptions (A2):1 ν(R0) <∞2 |ξ| ≤ B a.s.3 k1, k2 ≥ 0 such that for all z , u and for P⊗ λ -a.e. ω, s it holds|f (ω, s, y , z , u)| ≤ k1 + k2|y |,



3. Application: a BSDE with a bounded solutionassumptions (A3):|DW· ξ| ≤ A P⊗ λ-a.e.

|DW· f (t, y , z)| ≤ q(t) P⊗ λ-a.e.

If (A1) and (A2) hold then (Becherer 2006):|Us(x)| ≤ 2‖supt |Yt |‖L∞ <∞ for P⊗ λ⊗ ν- a.e. (ω, s, x)

Corollary 41 If (A1) and (A2) hold the solution (Y ,Z ,U) is Malliavin differentiable.2 If additionally (A3) holds, then P⊗ λ-a.e.

|Zt | ≤ eB(T−t)A + eB(T−t)∫ T

tBe−B(T−s)q(s)ds

The second assertion uses a comparison theorem from Royer (2006) andthe ideas of Cheridito & Nam (2014).



3. Application: a BSDE with a bounded solution

conjectures:

the generator can be quadratic (in z) ,due to Becherer, Bütter & Kentia (2015)ν(R0) <∞ might be not needed



Hans-Jürgen Engelbert,

thank youandmany happy returns!


References

E. Alós, J. A. León and J. Vives, An anticipating Itô formula for Lévyprocesses ALEA Lat. Am. J. Probab. Math. Stat. 4 (2008), 285-305.

D. Becherer, Bounded solutions to backward SDEs with jumps for utilityoptimization and indifference hedging, Ann. Appl. Probab. Vol.16, No. 4(2006),pp.1733-2275.

V. Bogachev, Gaussian measures, AMS, 1998.

Ł. Delong, P. Imkeller, On Malliavin’s differentiability of BSDEs with timedelayed generators driven by Brownian motions and Poisson randommeasures, Stochastic Process. Appl. 120 (2010), 1748-1775.

N. El Karoui, S. Peng, and M.C. Quenez, Backward Stochastic DifferentialEquations in Finance, Math. Finance 7 (1997), 1-71.

C. Geiss, A. Steinicke, Malliavin derivative of random functions andapplications to Lévy driven BSDEs, arXiv1404.4477


References

S. Geiss, J. Ylinen, Decoupling on the Wiener space and applications toBSDEs, arXiv1409.5322, 2014.

K. Itô, Spectral type of the shift transformation of differential process withstationary increments, Trans. Amer. Math. Soc. 81:253-263, 1956.

S. Janson, Gaussian Hilbert Spaces, Cambridge, 1997.

É. Pardoux, S. Peng, Backward Stochastic Differential Equations andQuasilinear Parabolic Partial Differential Equations, Stochastic partialdifferential equations and their applications (Charlotte, NC, 1991), 200-217,Lecture Notes in Control and Inform. Sci., 176, Springer, Berlin, 1992.

J. Solé, F. Utzet, J Vives, Chaos expansions and Malliavin calculus for Lévyprocesses, Stochastic analysis and applications, 595-612, Abel Symp., 2,Springer, Berlin, 2007.

H. Sugita, On a characterization of the Sobolev spaces over an abstractWiener space J. Math. Kyoto Univ. 25-4(1985) 717-725.


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Malliavin derivative of Lévy driven BSDEs and applications2015... · Application: a BSDE with a bounded solution 3. Application: a BSDE with a bounded solution conjectures: thegeneratorcanbequadratic(inz),