Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade -...

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Stochastic target game problems B. Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech Université Pierre et Marie Curie, Mai 2012 Joint work with L. Moreau (Paris-Dauphine) and M. Nutz (Columbia)

Transcript of Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade -...

Page 1: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Stochastic target game problems

B. Bouchard

Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech

Université Pierre et Marie Curie, Mai 2012

Joint work with L. Moreau (Paris-Dauphine) and M. Nutz (Columbia)

Page 2: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Problem formulation and Motivations

Page 3: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Problem formulation

Determine the viability sets

Λ(t) := (z , p) : ∃ ν ∈ U s. t. E[`(Z ν[ϑ],ϑ

t,z (T ))]≥ p ∀ ϑ ∈ V

In which :

• V is a set of admissible adverse controls• U is a set of admissible strategies

• Z ν[ϑ],ϑt,z is an adapted Rd -valued process s.t. Z ν[ϑ],ϑ

t,z (t) = z• ` is a given loss/utility function• p a threshold.

Page 4: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Examples

2 Z ν[ϑ],ϑt,z = (Xϑ

t,x ,Yν[ϑ],ϑt,x ,y ) where

• X ν[ϑ],ϑt,x models financial assets or factors with dynamics

depending on ϑ

• Y ν[ϑ],ϑt,x ,y models a wealth process

• ϑ is the control of the market : parameter uncertainty (e.g.volatility), adverse players, etc...

• ν[ϑ] is the financial strategy given the past observations of ϑ.

2 Flexible enough to embed constraints, transaction costs, marketimpact, etc...

Page 5: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Examples

Λ(t) := (z , p) : ∃ ν ∈ U s.t. E[`(Z ν[ϑ],ϑ

t,z (T ))]≥ p ∀ ϑ ∈ V

2 Almost sure constraint :

Λ(t) := z : ∃ ν ∈ U s.t. Z ν[ϑ],ϑt,z (T ) ∈ O P− a.s. ∀ ϑ ∈ V

for `(z) = 1z∈O, p = 1.⇒ Super-hedging in finance for O := y ≥ g(x).

2 Compare with Peng (G -expectations) and Soner, Touzi andZhang (2BSDE).

Page 6: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Examples

Λ(t) := (z , p) : ∃ ν ∈ U s.t. E[`(Z ν[ϑ],ϑ

t,z (T ))]≥ p ∀ ϑ ∈ V

2 Almost sure constraint :

Λ(t) := z : ∃ ν ∈ U s.t. Z ν[ϑ],ϑt,z (T ) ∈ O P− a.s. ∀ ϑ ∈ V

for `(z) = 1z∈O, p = 1.⇒ Super-hedging in finance for O := y ≥ g(x).

2 Compare with Peng (G -expectations) and Soner, Touzi andZhang (2BSDE).

Page 7: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Examples

Λ(t) := (z , p) : ∃ ν ∈ U s.t. E[`(Z ν[ϑ],ϑ

t,z (T ))]≥ p ∀ ϑ ∈ V

2 Almost sure constraint :

Λ(t) := z : ∃ ν ∈ U s.t. Z ν[ϑ],ϑt,z (T ) ∈ O P− a.s. ∀ ϑ ∈ V

for `(z) = 1z∈O, p = 1.⇒ Super-hedging in finance for O := y ≥ g(x).

2 Compare with Peng (G -expectations) and Soner, Touzi andZhang (2BSDE).

Page 8: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Examples

2 Constraint in probability :

Λ(t) := (z , p) : ∃ ν ∈ U s.t. P[Z ν[ϑ],ϑ

t,z (T ) ∈ O]≥ p ∀ ϑ ∈ V

for `(z) = 1z∈O, p ∈ (0, 1).⇒ Quantile-hedging in finance for O := y ≥ g(x).

Page 9: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Examples

Λ(t) := (z , p) : ∃ ν ∈ U s.t. E[`(Z ν[ϑ],ϑ

t,z (T ))]≥ p ∀ ϑ ∈ V

2 Expected loss control for `(z) = −[y − g(x)]−

2 Can impose several constraint : B. and Thanh Nam (discreteP&L constraints).

2 Give sense to problems that would be degenerate under P− a.s.constraints : B. and Dang (guaranteed VWAP pricing).

Page 10: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Examples

Λ(t) := (z , p) : ∃ ν ∈ U s.t. E[`(Z ν[ϑ],ϑ

t,z (T ))]≥ p ∀ ϑ ∈ V

2 Expected loss control for `(z) = −[y − g(x)]−

2 Can impose several constraint : B. and Thanh Nam (discreteP&L constraints).

2 Give sense to problems that would be degenerate under P− a.s.constraints : B. and Dang (guaranteed VWAP pricing).

Page 11: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Examples

Λ(t) := (z , p) : ∃ ν ∈ U s.t. E[`(Z ν[ϑ],ϑ

t,z (T ))]≥ p ∀ ϑ ∈ V

2 Expected loss control for `(z) = −[y − g(x)]−

2 Can impose several constraint : B. and Thanh Nam (discreteP&L constraints).

2 Give sense to problems that would be degenerate under P− a.s.constraints : B. and Dang (guaranteed VWAP pricing).

Page 12: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Examples

Λ(t) := (z , p) : ∃ ν ∈ U s.t. E[`(Z ν[ϑ],ϑ

t,z (T ))]≥ p ∀ ϑ ∈ V

2 Expected loss control for `(z) = −[y − g(x)]−

2 Can impose several constraint : B. and Thanh Nam (discreteP&L constraints).

2 Give sense to problems that would be degenerate under P− a.s.constraints : B. and Dang (guaranteed VWAP pricing).

Page 13: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Aim

Provide a direct PDE characterization Λ

:this requires a Dynamic Programming Principle.

Page 14: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Aim

Provide a direct PDE characterization Λ :this requires a Dynamic Programming Principle.

Page 15: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Geometric Dynamic Programming Principle I

The case without adverse control

Page 16: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

The GDPP for P− a.s. criteria

2 P− a.s. criteria (Soner and Touzi)

Λ(t) := z : ∃ ν ∈ U s.t. Z νt,z(T ) ∈ O

2 GDPP : for all stopping time τ with values in [t,T ]

Λ(t) = z : ∃ ν ∈ U s.t. Z νt,z(τ) ∈ Λ(τ)

2 Extended by B. and Thanh Nam to constraints on the wholepath (obstacle version).2 Relies on a measurable selection argument : not very flexible,requires topological properties.2 Is enough to provide a PDE characterization for (t, z) 7→ 1z∈Λ(t).

Page 17: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

The GDPP for P− a.s. criteria

2 P− a.s. criteria (Soner and Touzi)

Λ(t) := z : ∃ ν ∈ U s.t. Z νt,z(T ) ∈ O

2 GDPP : for all stopping time τ with values in [t,T ]

Λ(t) = z : ∃ ν ∈ U s.t. Z νt,z(τ) ∈ Λ(τ)

2 Extended by B. and Thanh Nam to constraints on the wholepath (obstacle version).2 Relies on a measurable selection argument : not very flexible,requires topological properties.2 Is enough to provide a PDE characterization for (t, z) 7→ 1z∈Λ(t).

Page 18: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

The GDPP for P− a.s. criteria

2 P− a.s. criteria (Soner and Touzi)

Λ(t) := z : ∃ ν ∈ U s.t. Z νt,z(T ) ∈ O

2 GDPP : for all stopping time τ with values in [t,T ]

Λ(t) = z : ∃ ν ∈ U s.t. Z νt,z(τ) ∈ Λ(τ)

2 Extended by B. and Thanh Nam to constraints on the wholepath (obstacle version).

2 Relies on a measurable selection argument : not very flexible,requires topological properties.2 Is enough to provide a PDE characterization for (t, z) 7→ 1z∈Λ(t).

Page 19: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

The GDPP for P− a.s. criteria

2 P− a.s. criteria (Soner and Touzi)

Λ(t) := z : ∃ ν ∈ U s.t. Z νt,z(T ) ∈ O

2 GDPP : for all stopping time τ with values in [t,T ]

Λ(t) = z : ∃ ν ∈ U s.t. Z νt,z(τ) ∈ Λ(τ)

2 Extended by B. and Thanh Nam to constraints on the wholepath (obstacle version).2 Relies on a measurable selection argument : not very flexible,requires topological properties.

2 Is enough to provide a PDE characterization for (t, z) 7→ 1z∈Λ(t).

Page 20: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

The GDPP for P− a.s. criteria

2 P− a.s. criteria (Soner and Touzi)

Λ(t) := z : ∃ ν ∈ U s.t. Z νt,z(T ) ∈ O

2 GDPP : for all stopping time τ with values in [t,T ]

Λ(t) = z : ∃ ν ∈ U s.t. Z νt,z(τ) ∈ Λ(τ)

2 Extended by B. and Thanh Nam to constraints on the wholepath (obstacle version).2 Relies on a measurable selection argument : not very flexible,requires topological properties.2 Is enough to provide a PDE characterization for (t, z) 7→ 1z∈Λ(t).

Page 21: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

The GDPP for criteria in expectation2 Controlled loss :

Λ(t) := (z , p) : ∃ ν ∈ U s.t. E[`(Z ν

t,z(T ))]≥ p

2 GDPP : for all stopping time τ with values in [t,T ]

Λ(t)6=(z , p) : ∃ ν ∈ U s.t. (Z νt,z(τ), p) ∈ Λ(τ)

since

(Z νt,z(τ), p) ∈ Λ(τ)“⇒′′ Mν(τ) := E

[`(Z ν

t,z(T ))|Fτ]≥ p.

But

Λ(t) = (z , p) : ∃ (ν,M) ∈ U×Mt,p s.t. (Z νt,z(τ),M(τ)) ∈ Λ(τ)

Page 22: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

The GDPP for criteria in expectation2 Controlled loss :

Λ(t) := (z , p) : ∃ ν ∈ U s.t. E[`(Z ν

t,z(T ))]≥ p

2 GDPP : for all stopping time τ with values in [t,T ]

Λ(t)=(z , p) : ∃ ν ∈ U s.t. (Z νt,z(τ), p) ∈ Λ(τ)

since

Z νt,z(τ) ∈ V (τ, p)“⇒′′ Mν(τ) := E

[`(Z ν

t,z(T ))|Fτ]≥ p.

But

Λ(t) = (z , p) : ∃ (ν,M) ∈ U×Mt,p s.t. (Z νt,z(τ),M(τ)) ∈ Λ(τ)

Page 23: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

The GDPP for criteria in expectation2 Controlled loss :

Λ(t) := (z , p) : ∃ ν ∈ U s.t. E[`(Z ν

t,z(T ))]≥ p

2 GDPP : for all stopping time τ with values in [t,T ]

Λ(t)6=(z , p) : ∃ ν ∈ U s.t. (Z νt,z(τ), p) ∈ Λ(τ)

since

(Z νt,z(τ), p) ∈ Λ(τ)“⇒′′ Mν(τ) := E

[`(Z ν

t,z(T ))|Fτ]≥ p.

But

Λ(t) = (z , p) : ∃ (ν,M) ∈ U×Mt,p s.t. (Z νt,z(τ),M(τ)) ∈ Λ(τ)

Page 24: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

The GDPP for criteria in expectation2 Controlled loss :

Λ(t) := (z , p) : ∃ ν ∈ U s.t. E[`(Z ν

t,z(T ))]≥ p

2 GDPP : for all stopping time τ with values in [t,T ]

Λ(t)6=(z , p) : ∃ ν ∈ U s.t. (Z νt,z(τ), p) ∈ Λ(τ)

since

(Z νt,z(τ), p) ∈ Λ(τ)“⇒′′ Mν(τ) := E

[`(Z ν

t,z(T ))|Fτ]≥ p.

But

Λ(t) = (z , p) : ∃ ν ∈ U s.t. (Z νt,z(τ),Mν(τ)) ∈ Λ(τ)

Page 25: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

The GDPP for criteria in expectation2 Controlled loss :

Λ(t) := (z , p) : ∃ ν ∈ U s.t. E[`(Z ν

t,z(T ))]≥ p

2 GDPP : for all stopping time τ with values in [t,T ]

Λ(t)6=(z , p) : ∃ ν ∈ U s.t. (Z νt,z(τ), p) ∈ Λ(τ)

since

(Z νt,z(τ), p) ∈ Λ(τ)“⇒′′ Mν(τ) := E

[`(Z ν

t,z(T ))|Fτ]≥ p.

But

Λ(t) = (z , p) : ∃ (ν,M) ∈ U×Mt,p s.t. (Z νt,z(τ),M(τ)) ∈ Λ(τ)

Page 26: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

The GDPP for criteria in expectation

Λ(t) := (z , p) : ∃ ν ∈ U s.t. E[`(Z ν

t,z(T ))]≥ p

2 Alternative formulation in terms of the setMt,p of martingalesstarting from p at t :

Λ(t) := (z , p) : ∃ (ν,M) ∈ U×Mt,p s.t. `(Z νt,z(T )) ≥ M(T )

2 In Brownian diffusion settings :

Λ(t) := (z , p) : ∃ (ν, α) ∈ U×A s.t. `(Z νt,z(T )) ≥ Mα

t,p(T )

whereMα

t,p := p +

∫ ·tαsdWs .

2 Back to P− a.s. criteria up to an increase of the controls and thesystem : B., Elie and Touzi.

Page 27: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

The GDPP for criteria in expectation

Λ(t) := (z , p) : ∃ ν ∈ U s.t. E[`(Z ν

t,z(T ))]≥ p

2 Alternative formulation in terms of the setMt,p of martingalesstarting from p at t :

Λ(t) := (z , p) : ∃ (ν,M) ∈ U×Mt,p s.t. `(Z νt,z(T )) ≥ M(T )

2 In Brownian diffusion settings :

Λ(t) := (z , p) : ∃ (ν, α) ∈ U×A s.t. `(Z νt,z(T )) ≥ Mα

t,p(T )

whereMα

t,p := p +

∫ ·tαsdWs .

2 Back to P− a.s. criteria up to an increase of the controls and thesystem : B., Elie and Touzi.

Page 28: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

The GDPP for criteria in expectation

Λ(t) := (z , p) : ∃ ν ∈ U s.t. E[`(Z ν

t,z(T ))]≥ p

2 Alternative formulation in terms of the setMt,p of martingalesstarting from p at t :

Λ(t) := (z , p) : ∃ (ν,M) ∈ U×Mt,p s.t. `(Z νt,z(T )) ≥ M(T )

2 In Brownian diffusion settings :

Λ(t) := (z , p) : ∃ (ν, α) ∈ U×A s.t. `(Z νt,z(T )) ≥ Mα

t,p(T )

whereMα

t,p := p +

∫ ·tαsdWs .

2 Back to P− a.s. criteria up to an increase of the controls and thesystem : B., Elie and Touzi.

Page 29: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Geometric Dynamic Programming Principle II

The game formulation

Page 30: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

The GDPP for games2 Back to the original problem :

Λ(t) := (z , p) : ∃ ν ∈ U s.t. E[`(Z ν[ϑ],ϑ

t,z (T ))]≥ p ∀ ϑ ∈ V

2 Submartingale property : for ν fixed

Sν,ϑs := ess infϑ∈V

E[`(Z ν[ϑ⊕s ϑ],ϑ⊕s ϑ

t,z (T ))|Fs

]defines a family of submartingales parameterized by ϑ.

2 Doob-Meyer decomposition : ∃ a family of martingales Mν,ϑ, ϑsuch that Sν,ϑ ≥ Mν,ϑ with Mν,ϑ(t) = p.

2 (z , p) ∈ Λ(t) if and only if there exists ν ∈ U and a family ofmartingales Mϑ, ϑ with Mϑ(t) = p such that

`(Z ν[ϑ],ϑt,z (T )) ≥ Mϑ(T ) P− a.s. ∀ ϑ ∈ V.

Page 31: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

The GDPP for games2 Back to the original problem :

Λ(t) := (z , p) : ∃ ν ∈ U s.t. E[`(Z ν[ϑ],ϑ

t,z (T ))]≥ p ∀ ϑ ∈ V

2 Submartingale property : for ν fixed

Sν,ϑs := ess infϑ∈V

E[`(Z ν[ϑ⊕s ϑ],ϑ⊕s ϑ

t,z (T ))|Fs

]defines a family of submartingales parameterized by ϑ.

2 Doob-Meyer decomposition : ∃ a family of martingales Mν,ϑ, ϑsuch that Sν,ϑ ≥ Mν,ϑ with Mν,ϑ(t) = p.

2 (z , p) ∈ Λ(t) if and only if there exists ν ∈ U and a family ofmartingales Mϑ, ϑ with Mϑ(t) = p such that

`(Z ν[ϑ],ϑt,z (T )) ≥ Mϑ(T ) P− a.s. ∀ ϑ ∈ V.

Page 32: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

The GDPP for games2 Back to the original problem :

Λ(t) := (z , p) : ∃ ν ∈ U s.t. E[`(Z ν[ϑ],ϑ

t,z (T ))]≥ p ∀ ϑ ∈ V

2 Submartingale property : for ν fixed

Sν,ϑs := ess infϑ∈V

E[`(Z ν[ϑ⊕s ϑ],ϑ⊕s ϑ

t,z (T ))|Fs

]defines a family of submartingales parameterized by ϑ.

2 Doob-Meyer decomposition : ∃ a family of martingales Mν,ϑ, ϑsuch that Sν,ϑ ≥ Mν,ϑ with Mν,ϑ(t) = p.

2 (z , p) ∈ Λ(t) if and only if there exists ν ∈ U and a family ofmartingales Mϑ, ϑ with Mϑ(t) = p such that

`(Z ν[ϑ],ϑt,z (T )) ≥ Mϑ(T ) P− a.s. ∀ ϑ ∈ V.

Page 33: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

The GDPP for games2 Back to the original problem :

Λ(t) := (z , p) : ∃ ν ∈ U s.t. E[`(Z ν[ϑ],ϑ

t,z (T ))]≥ p ∀ ϑ ∈ V

2 Submartingale property : for ν fixed

Sν,ϑs := ess infϑ∈V

E[`(Z ν[ϑ⊕s ϑ],ϑ⊕s ϑ

t,z (T ))|Fs

]defines a family of submartingales parameterized by ϑ.

2 Doob-Meyer decomposition : ∃ a family of martingales Mν,ϑ, ϑsuch that Sν,ϑ ≥ Mν,ϑ with Mν,ϑ(t) = p.

2 (z , p) ∈ Λ(t) if and only if there exists ν ∈ U and a family ofmartingales Mϑ, ϑ with Mϑ(t) = p such that

`(Z ν[ϑ],ϑt,z (T )) ≥ Mϑ(T ) P− a.s. ∀ ϑ ∈ V.

Page 34: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

The GDPP for games

Λ(t) := (z , p) : ∃ ν ∈ U s.t. E[`(Z ν[ϑ],ϑ

t,z (T ))]≥ p ∀ ϑ ∈ V

2 (z , p) ∈ Λ(t) if and only if there exists ν ∈ U and a family ofmartingales Mϑ, ϑ with Mϑ(t) = p such that

(Z ν[ϑ],ϑt,z (τ),Mϑ(τ)) ∈ Λ(τ) P− a.s. ∀ ϑ ∈ V.

2 Main difficulty : how to rely on a measurable selection argumentas in the previous cases ?2 Solution : provide a weak formulation, under a suitable continuityassumption.

Page 35: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

The GDPP for games

Λ(t) := (z , p) : ∃ ν ∈ U s.t. E[`(Z ν[ϑ],ϑ

t,z (T ))]≥ p ∀ ϑ ∈ V

2 (z , p) ∈ Λ(t) if and only if there exists ν ∈ U and a family ofmartingales Mϑ, ϑ with Mϑ(t) = p such that

(Z ν[ϑ],ϑt,z (τ),Mϑ(τ)) ∈ Λ(τ) P− a.s. ∀ ϑ ∈ V.

2 Main difficulty : how to rely on a measurable selection argumentas in the previous cases ?

2 Solution : provide a weak formulation, under a suitable continuityassumption.

Page 36: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

The GDPP for games

Λ(t) := (z , p) : ∃ ν ∈ U s.t. E[`(Z ν[ϑ],ϑ

t,z (T ))]≥ p ∀ ϑ ∈ V

2 (z , p) ∈ Λ(t) if and only if there exists ν ∈ U and a family ofmartingales Mϑ, ϑ with Mϑ(t) = p such that

(Z ν[ϑ],ϑt,z (τ),Mϑ(τ)) ∈ Λ(τ) P− a.s. ∀ ϑ ∈ V.

2 Main difficulty : how to rely on a measurable selection argumentas in the previous cases ?2 Solution : provide a weak formulation, under a suitable continuityassumption.

Page 37: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

A Weak version of the Geometric DynamicProgramming Principle for game formulations

Page 38: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Abstract setting

2 To (ν, ϑ) ∈ U× V associate a Rd -valued càdlàg processZ ν,ϑ

t,z (·) = Z ν[ϑ],ϑt,z (·) with values in Rd such that Z ν,ϑ

t,z (t) = z .

2 Set

I (t, z , ν, ϑ) := E[`(Z ν,ϑ

t,z (T ))|Ft

]and J(t, z , ν) := ess inf

ϑ∈VI (t, z , ν, ϑ).

2 We consider

Λ(t) :=

(z , p) ∈ Rd × R : ∃ ν ∈ U s.t. J(t, z , ν) ≥ p P− a.s..

Page 39: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Abstract setting

2 To (ν, ϑ) ∈ U× V associate a Rd -valued càdlàg processZ ν,ϑ

t,z (·) = Z ν[ϑ],ϑt,z (·) with values in Rd such that Z ν,ϑ

t,z (t) = z .

2 Set

I (t, z , ν, ϑ) := E[`(Z ν,ϑ

t,z (T ))|Ft

]and J(t, z , ν) := ess inf

ϑ∈VI (t, z , ν, ϑ).

2 We consider

Λ(t) :=

(z , p) ∈ Rd × R : ∃ ν ∈ U s.t. J(t, z , ν) ≥ p P− a.s..

Page 40: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Abstract setting

2 To (ν, ϑ) ∈ U× V associate a Rd -valued càdlàg processZ ν,ϑ

t,z (·) = Z ν[ϑ],ϑt,z (·) with values in Rd such that Z ν,ϑ

t,z (t) = z .

2 Set

I (t, z , ν, ϑ) := E[`(Z ν,ϑ

t,z (T ))|Ft

]and J(t, z , ν) := ess inf

ϑ∈VI (t, z , ν, ϑ).

2 We consider

Λ(t) :=

(z , p) ∈ Rd × R : ∃ ν ∈ U s.t. J(t, z , ν) ≥ p P− a.s..

Page 41: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Concatenation possibilities

2 Adverse controls : ϑ := ϑ0 ⊕t (ϑ11A + ϑ21Ac ) ∈ V,for ϑ0, ϑ1, ϑ2 ∈ V, and A ∈ Ft

2 Non-anticipating strategies :ν : ϑ ∈ V 7→ ν[ϑ] := ν0[ϑ]⊕t

∑j≥1 νj [ϑ]1Aϑ

j∈ U,

for Aϑj , ϑ ∈ Vj≥1 ⊂ Ft such that Aϑj , j ≥ 1 forms a partition ofΩ for each ϑ ∈ V.

2 Non-anticipating event sets : Aϑ, ϑ ∈ V ∈ Ft ifAϑ, ϑ ∈ V ⊂ Ft is such thatAϑ1 ∩ ϑ1 =(0,t] ϑ2 = Aϑ2 ∩ ϑ1 =(0,t] ϑ2 for ϑ1, ϑ2 ∈ V.

2 Non-anticipating stopping times : τϑ ∈ (s1, s2], ϑ ∈ V andτϑ /∈ (s1, s2], ϑ ∈ V belong to Fs2 for τϑ, ϑ ∈ V ⊂ Tt .

Page 42: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Concatenation possibilities

2 Adverse controls : ϑ := ϑ0 ⊕t (ϑ11A + ϑ21Ac ) ∈ V,for ϑ0, ϑ1, ϑ2 ∈ V, and A ∈ Ft

2 Non-anticipating strategies :ν : ϑ ∈ V 7→ ν[ϑ] := ν0[ϑ]⊕t

∑j≥1 νj [ϑ]1Aϑ

j∈ U,

for Aϑj , ϑ ∈ Vj≥1 ⊂ Ft such that Aϑj , j ≥ 1 forms a partition ofΩ for each ϑ ∈ V.

2 Non-anticipating event sets : Aϑ, ϑ ∈ V ∈ Ft ifAϑ, ϑ ∈ V ⊂ Ft is such thatAϑ1 ∩ ϑ1 =(0,t] ϑ2 = Aϑ2 ∩ ϑ1 =(0,t] ϑ2 for ϑ1, ϑ2 ∈ V.

2 Non-anticipating stopping times : τϑ ∈ (s1, s2], ϑ ∈ V andτϑ /∈ (s1, s2], ϑ ∈ V belong to Fs2 for τϑ, ϑ ∈ V ⊂ Tt .

Page 43: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Concatenation possibilities

2 Adverse controls : ϑ := ϑ0 ⊕t (ϑ11A + ϑ21Ac ) ∈ V,for ϑ0, ϑ1, ϑ2 ∈ V, and A ∈ Ft

2 Non-anticipating strategies :ν : ϑ ∈ V 7→ ν[ϑ] := ν0[ϑ]⊕t

∑j≥1 νj [ϑ]1Aϑ

j∈ U,

for Aϑj , ϑ ∈ Vj≥1 ⊂ Ft such that Aϑj , j ≥ 1 forms a partition ofΩ for each ϑ ∈ V.

2 Non-anticipating event sets : Aϑ, ϑ ∈ V ∈ Ft ifAϑ, ϑ ∈ V ⊂ Ft is such thatAϑ1 ∩ ϑ1 =(0,t] ϑ2 = Aϑ2 ∩ ϑ1 =(0,t] ϑ2 for ϑ1, ϑ2 ∈ V.

2 Non-anticipating stopping times : τϑ ∈ (s1, s2], ϑ ∈ V andτϑ /∈ (s1, s2], ϑ ∈ V belong to Fs2 for τϑ, ϑ ∈ V ⊂ Tt .

Page 44: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Concatenation possibilities

2 Adverse controls : ϑ := ϑ0 ⊕t (ϑ11A + ϑ21Ac ) ∈ V,for ϑ0, ϑ1, ϑ2 ∈ V, and A ∈ Ft

2 Non-anticipating strategies :ν : ϑ ∈ V 7→ ν[ϑ] := ν0[ϑ]⊕t

∑j≥1 νj [ϑ]1Aϑ

j∈ U,

for Aϑj , ϑ ∈ Vj≥1 ⊂ Ft such that Aϑj , j ≥ 1 forms a partition ofΩ for each ϑ ∈ V.

2 Non-anticipating event sets : Aϑ, ϑ ∈ V ∈ Ft ifAϑ, ϑ ∈ V ⊂ Ft is such thatAϑ1 ∩ ϑ1 =(0,t] ϑ2 = Aϑ2 ∩ ϑ1 =(0,t] ϑ2 for ϑ1, ϑ2 ∈ V.

2 Non-anticipating stopping times : τϑ ∈ (s1, s2], ϑ ∈ V andτϑ /∈ (s1, s2], ϑ ∈ V belong to Fs2 for τϑ, ϑ ∈ V ⊂ Tt .

Page 45: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Path dependency

2 Z ν,ϑ1t,z (s)(ω) = Z ν,ϑ2

t,z (s)(ω) for P-a.e. ω ∈ ϑ1 =(0,s] ϑ2, for allν ∈ U and ϑ1, ϑ2 ∈ V.

2 Z ν1,ϑt,z (s)(ω) = Z ν2,ϑ

t,z (s)(ω) for P-a.e. ω ∈ ν1[ϑ] =(t,s] ν2[ϑ],for all ϑ ∈ V and ν1, ν2 ∈ U.

2 There exists a constant K (t, z) ∈ R such that

esssupν∈U

ess infϑ∈V

E[`(Z ν,ϑ

t,z (T ))|Ft

]= K (t, z) P− a.s.

Page 46: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Path dependency

2 Z ν,ϑ1t,z (s)(ω) = Z ν,ϑ2

t,z (s)(ω) for P-a.e. ω ∈ ϑ1 =(0,s] ϑ2, for allν ∈ U and ϑ1, ϑ2 ∈ V.

2 Z ν1,ϑt,z (s)(ω) = Z ν2,ϑ

t,z (s)(ω) for P-a.e. ω ∈ ν1[ϑ] =(t,s] ν2[ϑ],for all ϑ ∈ V and ν1, ν2 ∈ U.

2 There exists a constant K (t, z) ∈ R such that

esssupν∈U

ess infϑ∈V

E[`(Z ν,ϑ

t,z (T ))|Ft

]= K (t, z) P− a.s.

Page 47: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Path dependency

2 Z ν,ϑ1t,z (s)(ω) = Z ν,ϑ2

t,z (s)(ω) for P-a.e. ω ∈ ϑ1 =(0,s] ϑ2, for allν ∈ U and ϑ1, ϑ2 ∈ V.

2 Z ν1,ϑt,z (s)(ω) = Z ν2,ϑ

t,z (s)(ω) for P-a.e. ω ∈ ν1[ϑ] =(t,s] ν2[ϑ],for all ϑ ∈ V and ν1, ν2 ∈ U.

2 There exists a constant K (t, z) ∈ R such that

esssupν∈U

ess infϑ∈V

E[`(Z ν,ϑ

t,z (T ))|Ft

]= K (t, z) P− a.s.

Page 48: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Path regularity and growth conditions

2 There exists C > 0, q ≥ 0, ε > 0 and a continuous map % suchthat

|`(z)| ≤ C (1 + |z |q) , ess sup(ν,ϑ)∈U×V

E[|Z ν,ϑ

t,z (T )|(q∨1)+ε|Ft

]≤ %(z)

and

ess sup(ν,ϑ)∈U×V

E[|Z ν⊕s ν,ϑ⊕s ϑ

t,z (T )− Z ν,ϑ⊕s ϑs,z ′ (T )| |Fs

]≤ C |Z ν,ϑ

t,z (s)− z ′|.

2 As δ → 0

supϑ∈V,τ∈Tt

P

[sup

0≤h≤δ

∣∣∣Z ν,ϑt,z (τ + h)− Z ν,ϑ

t,z (τ)∣∣∣ ≥ ι]→ 0.

Page 49: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Path regularity and growth conditions

2 There exists C > 0, q ≥ 0, ε > 0 and a continuous map % suchthat

|`(z)| ≤ C (1 + |z |q) , ess sup(ν,ϑ)∈U×V

E[|Z ν,ϑ

t,z (T )|(q∨1)+ε|Ft

]≤ %(z)

and

ess sup(ν,ϑ)∈U×V

E[|Z ν⊕s ν,ϑ⊕s ϑ

t,z (T )− Z ν,ϑ⊕s ϑs,z ′ (T )| |Fs

]≤ C |Z ν,ϑ

t,z (s)− z ′|.

2 As δ → 0

supϑ∈V,τ∈Tt

P

[sup

0≤h≤δ

∣∣∣Z ν,ϑt,z (τ + h)− Z ν,ϑ

t,z (τ)∣∣∣ ≥ ι]→ 0.

Page 50: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Non-anticipating admissible martingale strategies

2 Mϑ1(s)(ω) = Mϑ2(s)(ω) for P-a.e. ω ∈ ϑ1 =(0,s] ϑ2, for allMϑ, ϑ ∈ V ∈Mt,p and ϑ1, ϑ2 ∈ V.

Page 51: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Weak formulation

2 We consider relaxed versions of Λ

Λ(t) :=

(z , p) ∈ Rd × R : there exist (tn, zn, pn)→ (t, z , p)

such that (zn, pn) ∈ Λ(tn) and tn ≥ t for all n ≥ 1

and

Λι(t) :=

(z , p) ∈ Rd × R : (t ′, z ′, p′) ∈ Bι(t, z , p)implies (z ′, p′) ∈ Λ(t ′)

Page 52: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Weak formulation

Theorem(GDP1) : If (z , p + ε) ∈ Λ(t) for some ε > 0, then ∃ ν ∈ U andMϑ, ϑ ∈ V ⊂Mt,p s.t.(

Z ν,ϑt,z (τ) ,Mϑ (τ)

)∈ Λ (τ) P− a.s. ∀ ϑ ∈ V, τ ∈ Tt .

(GDP2) : Let ι > 0, ν ∈ U, Mϑ, ϑ ∈ V ∈Mt,p andτϑ, ϑ ∈ V ∈ Tt be s.t.(

Z ν,ϑt,z (τϑ),Mϑ(τϑ)

)∈ Λι(τ

ϑ) P− a.s. for all ϑ ∈ V.

Assume further that Z ν,ϑt,z(τϑ), ϑ ∈ V is uniformly bounded in

L∞. Then (z , p − ε) ∈ Λ(t) for all ε > 0.

Page 53: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Weak formulation

Theorem(GDP1) : If (z , p + ε) ∈ Λ(t) for some ε > 0, then ∃ ν ∈ U andMϑ, ϑ ∈ V ⊂Mt,p s.t.(

Z ν,ϑt,z (τ) ,Mϑ (τ)

)∈ Λ (τ) P− a.s. ∀ ϑ ∈ V, τ ∈ Tt .

(GDP2) : Let ι > 0, ν ∈ U, Mϑ, ϑ ∈ V ∈Mt,p andτϑ, ϑ ∈ V ∈ Tt be s.t.(

Z ν,ϑt,z (τϑ),Mϑ(τϑ)

)∈ Λι(τ

ϑ) P− a.s. for all ϑ ∈ V.

Assume further that Z ν,ϑt,z(τϑ), ϑ ∈ V is uniformly bounded in

L∞. Then (z , p − ε) ∈ Λ(t) for all ε > 0.

Page 54: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

More abstract setting

2 Rd replaced by a metric space (Z, dZ)

2 Lipschitz continuity, and growth estimates on Z and ` replacedby :For all s ∈ [t,T ], ε > 0, there exists a partition (Bj)j of Z intoBorel sets and a sequence (zj)j≥1 ⊂ Z such that, for all ν ∈ U,ϑ ∈ V and j ≥ 1 :

• E[`(Z ν,ϑ

t,z (T ))|Fs

]≥ I (s, zj , ν, ϑ)− ε on C ν,ϑ

j := Z ν,ϑt,z (s) ∈

Bj.

• ess infϑ∈V

E[`(Z ν,ϑ⊕s ϑ

t,z (T ))|Fs

]≤ J(s, zj , ν(ϑ⊕s ·)) + ε on C ν,ϑ

j .

• |K (s, zj)− K (s,Z ν,ϑt,z (s))| ≤ ε on C ν,ϑ

j .

2 Plus some additional control on the negative part of I .

Page 55: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

More abstract setting

2 Rd replaced by a metric space (Z, dZ)2 Lipschitz continuity, and growth estimates on Z and ` replacedby :For all s ∈ [t,T ], ε > 0, there exists a partition (Bj)j of Z intoBorel sets and a sequence (zj)j≥1 ⊂ Z such that, for all ν ∈ U,ϑ ∈ V and j ≥ 1 :

• E[`(Z ν,ϑ

t,z (T ))|Fs

]≥ I (s, zj , ν, ϑ)− ε on C ν,ϑ

j := Z ν,ϑt,z (s) ∈

Bj.

• ess infϑ∈V

E[`(Z ν,ϑ⊕s ϑ

t,z (T ))|Fs

]≤ J(s, zj , ν(ϑ⊕s ·)) + ε on C ν,ϑ

j .

• |K (s, zj)− K (s,Z ν,ϑt,z (s))| ≤ ε on C ν,ϑ

j .2 Plus some additional control on the negative part of I .

Page 56: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

More abstract setting

2 Rd replaced by a metric space (Z, dZ)2 Lipschitz continuity, and growth estimates on Z and ` replacedby :For all s ∈ [t,T ], ε > 0, there exists a partition (Bj)j of Z intoBorel sets and a sequence (zj)j≥1 ⊂ Z such that, for all ν ∈ U,ϑ ∈ V and j ≥ 1 :

• E[`(Z ν,ϑ

t,z (T ))|Fs

]≥ I (s, zj , ν, ϑ)− ε on C ν,ϑ

j := Z ν,ϑt,z (s) ∈

Bj.

• ess infϑ∈V

E[`(Z ν,ϑ⊕s ϑ

t,z (T ))|Fs

]≤ J(s, zj , ν(ϑ⊕s ·)) + ε on C ν,ϑ

j .

• |K (s, zj)− K (s,Z ν,ϑt,z (s))| ≤ ε on C ν,ϑ

j .2 Plus some additional control on the negative part of I .

Page 57: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

More abstract setting

2 Rd replaced by a metric space (Z, dZ)2 Lipschitz continuity, and growth estimates on Z and ` replacedby :For all s ∈ [t,T ], ε > 0, there exists a partition (Bj)j of Z intoBorel sets and a sequence (zj)j≥1 ⊂ Z such that, for all ν ∈ U,ϑ ∈ V and j ≥ 1 :

• E[`(Z ν,ϑ

t,z (T ))|Fs

]≥ I (s, zj , ν, ϑ)− ε on C ν,ϑ

j := Z ν,ϑt,z (s) ∈

Bj.

• ess infϑ∈V

E[`(Z ν,ϑ⊕s ϑ

t,z (T ))|Fs

]≤ J(s, zj , ν(ϑ⊕s ·)) + ε on C ν,ϑ

j .

• |K (s, zj)− K (s,Z ν,ϑt,z (s))| ≤ ε on C ν,ϑ

j .2 Plus some additional control on the negative part of I .

Page 58: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

More abstract setting

2 Rd replaced by a metric space (Z, dZ)2 Lipschitz continuity, and growth estimates on Z and ` replacedby :For all s ∈ [t,T ], ε > 0, there exists a partition (Bj)j of Z intoBorel sets and a sequence (zj)j≥1 ⊂ Z such that, for all ν ∈ U,ϑ ∈ V and j ≥ 1 :

• E[`(Z ν,ϑ

t,z (T ))|Fs

]≥ I (s, zj , ν, ϑ)− ε on C ν,ϑ

j := Z ν,ϑt,z (s) ∈

Bj.

• ess infϑ∈V

E[`(Z ν,ϑ⊕s ϑ

t,z (T ))|Fs

]≤ J(s, zj , ν(ϑ⊕s ·)) + ε on C ν,ϑ

j .

• |K (s, zj)− K (s,Z ν,ϑt,z (s))| ≤ ε on C ν,ϑ

j .

2 Plus some additional control on the negative part of I .

Page 59: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

More abstract setting

2 Rd replaced by a metric space (Z, dZ)2 Lipschitz continuity, and growth estimates on Z and ` replacedby :For all s ∈ [t,T ], ε > 0, there exists a partition (Bj)j of Z intoBorel sets and a sequence (zj)j≥1 ⊂ Z such that, for all ν ∈ U,ϑ ∈ V and j ≥ 1 :

• E[`(Z ν,ϑ

t,z (T ))|Fs

]≥ I (s, zj , ν, ϑ)− ε on C ν,ϑ

j := Z ν,ϑt,z (s) ∈

Bj.

• ess infϑ∈V

E[`(Z ν,ϑ⊕s ϑ

t,z (T ))|Fs

]≤ J(s, zj , ν(ϑ⊕s ·)) + ε on C ν,ϑ

j .

• |K (s, zj)− K (s,Z ν,ϑt,z (s))| ≤ ε on C ν,ϑ

j .2 Plus some additional control on the negative part of I .

Page 60: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Sketch of proof for GDP12 Assume

Sν,ϑs := ess infϑ∈Vs

E[`(Z ν[ϑ⊕s ϑ],ϑ⊕s ϑ

t,z (T ))|Fs

]is such that Sν,ϑt ≥ p.

2 It admits a càlàg decomposition (up to a modification),+Doob-Meyer-type decomposition : Sν,ϑ ≥ Mν,ϑ a càdlàg martingale.

2

K (τ,Z ν,ϑt,z (τ)) ≥ Mν,ϑ (τ)

which leads to(Z ν,ϑ

t,z (τ) ,Mν,ϑ (τ)− ε)∈ Λ (τ) P− a.s.

2 Ok for stopping times τ with values in a countable set. Pass tothe limit.

Page 61: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Sketch of proof for GDP12 Assume

Sν,ϑs := ess infϑ∈Vs

E[`(Z ν[ϑ⊕s ϑ],ϑ⊕s ϑ

t,z (T ))|Fs

]is such that Sν,ϑt ≥ p.

2 It admits a càlàg decomposition (up to a modification),+Doob-Meyer-type decomposition : Sν,ϑ ≥ Mν,ϑ a càdlàg martingale.

2

K (τ,Z ν,ϑt,z (τ)) ≥ Mν,ϑ (τ)

which leads to(Z ν,ϑ

t,z (τ) ,Mν,ϑ (τ)− ε)∈ Λ (τ) P− a.s.

2 Ok for stopping times τ with values in a countable set. Pass tothe limit.

Page 62: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Sketch of proof for GDP12 Assume

Sν,ϑs := ess infϑ∈Vs

E[`(Z ν[ϑ⊕s ϑ],ϑ⊕s ϑ

t,z (T ))|Fs

]is such that Sν,ϑt ≥ p.

2 It admits a càlàg decomposition (up to a modification),+Doob-Meyer-type decomposition : Sν,ϑ ≥ Mν,ϑ a càdlàg martingale.

2

K (τ,Z ν,ϑt,z (τ)) ≥ Mν,ϑ (τ)

which leads to(Z ν,ϑ

t,z (τ) ,Mν,ϑ (τ)− ε)∈ Λ (τ) P− a.s.

2 Ok for stopping times τ with values in a countable set. Pass tothe limit.

Page 63: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Sketch of proof for GDP12 Assume

Sν,ϑs := ess infϑ∈Vs

E[`(Z ν[ϑ⊕s ϑ],ϑ⊕s ϑ

t,z (T ))|Fs

]is such that Sν,ϑt ≥ p.

2 It admits a càlàg decomposition (up to a modification),+Doob-Meyer-type decomposition : Sν,ϑ ≥ Mν,ϑ a càdlàg martingale.

2

K (τ,Z ν,ϑt,z (τ)) ≥ Mν,ϑ (τ)

which leads to(Z ν,ϑ

t,z (τ) ,Mν,ϑ (τ)− ε)∈ Λ (τ) P− a.s.

2 Ok for stopping times τ with values in a countable set. Pass tothe limit.

Page 64: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Sketch of proof for GDP12 Assume

Sν,ϑs := ess infϑ∈Vs

E[`(Z ν[ϑ⊕s ϑ],ϑ⊕s ϑ

t,z (T ))|Fs

]is such that Sν,ϑt ≥ p.

2 It admits a càlàg decomposition (up to a modification),+Doob-Meyer-type decomposition : Sν,ϑ ≥ Mν,ϑ a càdlàg martingale.

2

K (τ,Z ν,ϑt,z (τ)) ≥ Mν,ϑ (τ)

which leads to(Z ν,ϑ

t,z (τ) ,Mν,ϑ (τ)− ε)∈ Λ (τ) P− a.s.

2 Ok for stopping times τ with values in a countable set. Pass tothe limit.

Page 65: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Sketch of proof for GDP22 τϑn an approximation of τϑ on a sequence of finite grids πn.

2 If (Z ν,ϑ

t,z

(τϑ),Mϑ

(τϑ))∈ Λι

(τϑ)

P− a.s.

then (Z ν,ϑ

t,z

(τϑn

),Mϑ

(τϑn

))∈ Λ

(τϑn

)on Eϑn for all ϑ ∈ V

with P[Eϑn]→ 1 as n→∞ (uniformly in ϑ).

2 Hence,K (τϑn ,Z

ν,ϑt,z (τϑn )) ≥ Mϑ(τϑn ).

2 Regularity + covering : there exists νε ∈ U such that

E[`(Z νε,ϑ

t,z (T ))|Fτϑn]≥ Mϑ(τϑn )− ε.

which impliesE[`(Z νε,ϑ

t,z (T ))|Ft

]≥ p − ε.

Page 66: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Sketch of proof for GDP22 τϑn an approximation of τϑ on a sequence of finite grids πn.

2 If (Z ν,ϑ

t,z

(τϑ),Mϑ

(τϑ))∈ Λι

(τϑ)

P− a.s.

then (Z ν,ϑ

t,z

(τϑn

),Mϑ

(τϑn

))∈ Λ

(τϑn

)on Eϑn for all ϑ ∈ V

with P[Eϑn]→ 1 as n→∞ (uniformly in ϑ).

2 Hence,K (τϑn ,Z

ν,ϑt,z (τϑn )) ≥ Mϑ(τϑn ).

2 Regularity + covering : there exists νε ∈ U such that

E[`(Z νε,ϑ

t,z (T ))|Fτϑn]≥ Mϑ(τϑn )− ε.

which impliesE[`(Z νε,ϑ

t,z (T ))|Ft

]≥ p − ε.

Page 67: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Sketch of proof for GDP22 τϑn an approximation of τϑ on a sequence of finite grids πn.

2 If (Z ν,ϑ

t,z

(τϑ),Mϑ

(τϑ))∈ Λι

(τϑ)

P− a.s.

then (Z ν,ϑ

t,z

(τϑn

),Mϑ

(τϑn

))∈ Λ

(τϑn

)on Eϑn for all ϑ ∈ V

with P[Eϑn]→ 1 as n→∞ (uniformly in ϑ).

2 Hence,K (τϑn ,Z

ν,ϑt,z (τϑn )) ≥ Mϑ(τϑn ).

2 Regularity + covering : there exists νε ∈ U such that

E[`(Z νε,ϑ

t,z (T ))|Fτϑn]≥ Mϑ(τϑn )− ε.

which impliesE[`(Z νε,ϑ

t,z (T ))|Ft

]≥ p − ε.

Page 68: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Sketch of proof for GDP22 τϑn an approximation of τϑ on a sequence of finite grids πn.

2 If (Z ν,ϑ

t,z

(τϑ),Mϑ

(τϑ))∈ Λι

(τϑ)

P− a.s.

then (Z ν,ϑ

t,z

(τϑn

),Mϑ

(τϑn

))∈ Λ

(τϑn

)on Eϑn for all ϑ ∈ V

with P[Eϑn]→ 1 as n→∞ (uniformly in ϑ).

2 Hence,K (τϑn ,Z

ν,ϑt,z (τϑn )) ≥ Mϑ(τϑn ).

2 Regularity + covering : there exists νε ∈ U such that

E[`(Z νε,ϑ

t,z (T ))|Fτϑn]≥ Mϑ(τϑn )− ε.

which impliesE[`(Z νε,ϑ

t,z (T ))|Ft

]≥ p − ε.

Page 69: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Application to Brownian controlled SDEs

Page 70: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Framework

2 U is the set of maps ν : V → U such thatϑ1 =(0,τ ] ϑ2 ⊂ ν[ϑ1] =(0,τ ] ν[ϑ2] for all ϑ1, ϑ2 ∈ V and τ ∈ T .

2 Z ν,ϑt,z = (X ν,ϑ

t,x ,Yν,ϑt,x ,y ) is the strong solution of

Z (s) = z +

∫ s

tµ(Z (r), ν[ϑ]r , ϑr )dr +

∫ s

tσ(Z (r), ν[ϑ]r , ϑr )dWr .

2Mt,p is identified to Mαt,p, α ∈ A with

Mαt,p := p +

∫ ·tαsdWs

2 X the set of maps α[·] : V 7→ A such thatϑ1 =(0,τ ] ϑ2 ⊂ α[ϑ1] =(0,τ ] α[ϑ2] for ϑ1, ϑ2 ∈ V and τ ∈ T .

Page 71: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Framework

2 U is the set of maps ν : V → U such thatϑ1 =(0,τ ] ϑ2 ⊂ ν[ϑ1] =(0,τ ] ν[ϑ2] for all ϑ1, ϑ2 ∈ V and τ ∈ T .

2 Z ν,ϑt,z = (X ν,ϑ

t,x ,Yν,ϑt,x ,y ) is the strong solution of

Z (s) = z +

∫ s

tµ(Z (r), ν[ϑ]r , ϑr )dr +

∫ s

tσ(Z (r), ν[ϑ]r , ϑr )dWr .

2Mt,p is identified to Mαt,p, α ∈ A with

Mαt,p := p +

∫ ·tαsdWs

2 X the set of maps α[·] : V 7→ A such thatϑ1 =(0,τ ] ϑ2 ⊂ α[ϑ1] =(0,τ ] α[ϑ2] for ϑ1, ϑ2 ∈ V and τ ∈ T .

Page 72: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Framework

2 U is the set of maps ν : V → U such thatϑ1 =(0,τ ] ϑ2 ⊂ ν[ϑ1] =(0,τ ] ν[ϑ2] for all ϑ1, ϑ2 ∈ V and τ ∈ T .

2 Z ν,ϑt,z = (X ν,ϑ

t,x ,Yν,ϑt,x ,y ) is the strong solution of

Z (s) = z +

∫ s

tµ(Z (r), ν[ϑ]r , ϑr )dr +

∫ s

tσ(Z (r), ν[ϑ]r , ϑr )dWr .

2Mt,p is identified to Mαt,p, α ∈ A with

Mαt,p := p +

∫ ·tαsdWs

2 X the set of maps α[·] : V 7→ A such thatϑ1 =(0,τ ] ϑ2 ⊂ α[ϑ1] =(0,τ ] α[ϑ2] for ϑ1, ϑ2 ∈ V and τ ∈ T .

Page 73: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Framework

2 U is the set of maps ν : V → U such thatϑ1 =(0,τ ] ϑ2 ⊂ ν[ϑ1] =(0,τ ] ν[ϑ2] for all ϑ1, ϑ2 ∈ V and τ ∈ T .

2 Z ν,ϑt,z = (X ν,ϑ

t,x ,Yν,ϑt,x ,y ) is the strong solution of

Z (s) = z +

∫ s

tµ(Z (r), ν[ϑ]r , ϑr )dr +

∫ s

tσ(Z (r), ν[ϑ]r , ϑr )dWr .

2Mt,p is identified to Mαt,p, α ∈ A with

Mαt,p := p +

∫ ·tαsdWs

2 X the set of maps α[·] : V 7→ A such thatϑ1 =(0,τ ] ϑ2 ⊂ α[ϑ1] =(0,τ ] α[ϑ2] for ϑ1, ϑ2 ∈ V and τ ∈ T .

Page 74: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Framework

2 Assume ` : (x , y) ∈ Rd−1 × R 7→ `(x , y) ∈ R is non-decreasingin its y -variable.

2 This allows to associate to Λ the real valued function :

y(t, x) := infy : (x , y) ∈ Λ(t).

Page 75: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

Framework

2 Assume ` : (x , y) ∈ Rd−1 × R 7→ `(x , y) ∈ R is non-decreasingin its y -variable.

2 This allows to associate to Λ the real valued function :

y(t, x) := infy : (x , y) ∈ Λ(t).

Page 76: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

GDPCorollary(GDP1)y If y > y(t, x , p + ε) with ε > 0, then ∃ ν ∈ U andαϑ, ϑ ∈ V ⊂ A s.t.

Y ν,ϑt,x ,y (τ) ≥ y∗

(τ,X ν,ϑ

t,x (τ),Mαϑ

t,p (τ))

P-a.s.

for all ϑ ∈ V, τ ∈ Tt .

(GDP2)y Fix a bounded open set O 3 (t, x , y , p), (ν, α) ∈ U× X

and let τϑ denote the first exit time of (·,X ν,ϑt,x ,Y

ν,ϑt,x ,y ,M

α[ϑ]t,p ),

ϑ ∈ V. Assume that there exists η > 0 and a continuous functionϕ ≥ y such that

Y ν,ϑt,x ,y (τϑ) ≥ ϕ

(τ,X ν,ϑ

t,x (τϑ),Mα[ϑ]t,p (τϑ)

)+ η P− a.s. for all ϑ ∈ V.

Then, y ≥ y(t, x , p − ε) for all ε > 0.

Page 77: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

GDPCorollary(GDP1)y If y > y(t, x , p + ε) with ε > 0, then ∃ ν ∈ U andαϑ, ϑ ∈ V ⊂ A s.t.

Y ν,ϑt,x ,y (τ) ≥ y∗

(τ,X ν,ϑ

t,x (τ),Mαϑ

t,p (τ))

P-a.s.

for all ϑ ∈ V, τ ∈ Tt .

(GDP2)y Fix a bounded open set O 3 (t, x , y , p), (ν, α) ∈ U× X

and let τϑ denote the first exit time of (·,X ν,ϑt,x ,Y

ν,ϑt,x ,y ,M

α[ϑ]t,p ),

ϑ ∈ V. Assume that there exists η > 0 and a continuous functionϕ ≥ y such that

Y ν,ϑt,x ,y (τϑ) ≥ ϕ

(τ,X ν,ϑ

t,x (τϑ),Mα[ϑ]t,p (τϑ)

)+ η P− a.s. for all ϑ ∈ V.

Then, y ≥ y(t, x , p − ε) for all ε > 0.

Page 78: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

PDE characterization - “waving hands” version

2 Assuming smoothness, existence of optimal strategies...

2 y = y(t, x , p) impliesY ν[ϑ],ϑ(t+) ≥ y(t+,X ν[ϑ],ϑ(t+),Mα[ϑ](t+)) for all ϑ.

2 This implies dY ν[ϑ],ϑ(t) ≥ dy(t,X ν[ϑ],ϑ(t),Mα[ϑ](t)) for all ϑ

2 Hence, for all ϑ,

µY (x , y , ν[ϑ]t , ϑt) ≥ Lν[ϑ]t ,ϑt ,α[ϑ]tX ,M y(t, x , p)

σY (x , y , ν[ϑ]t , ϑt) = σX (x , ν[ϑ]t , ϑt)Dxy(t, x , p) + α[ϑ]tDpy(t, x , p)

with y = y(t, x , p)

Page 79: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

PDE characterization - “waving hands” version

2 Assuming smoothness, existence of optimal strategies...

2 y = y(t, x , p) impliesY ν[ϑ],ϑ(t+) ≥ y(t+,X ν[ϑ],ϑ(t+),Mα[ϑ](t+)) for all ϑ.

2 This implies dY ν[ϑ],ϑ(t) ≥ dy(t,X ν[ϑ],ϑ(t),Mα[ϑ](t)) for all ϑ

2 Hence, for all ϑ,

µY (x , y , ν[ϑ]t , ϑt) ≥ Lν[ϑ]t ,ϑt ,α[ϑ]tX ,M y(t, x , p)

σY (x , y , ν[ϑ]t , ϑt) = σX (x , ν[ϑ]t , ϑt)Dxy(t, x , p) + α[ϑ]tDpy(t, x , p)

with y = y(t, x , p)

Page 80: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

PDE characterization - “waving hands” version

2 Assuming smoothness, existence of optimal strategies...

2 y = y(t, x , p) impliesY ν[ϑ],ϑ(t+) ≥ y(t+,X ν[ϑ],ϑ(t+),Mα[ϑ](t+)) for all ϑ.

2 This implies dY ν[ϑ],ϑ(t) ≥ dy(t,X ν[ϑ],ϑ(t),Mα[ϑ](t)) for all ϑ

2 Hence, for all ϑ,

µY (x , y , ν[ϑ]t , ϑt) ≥ Lν[ϑ]t ,ϑt ,α[ϑ]tX ,M y(t, x , p)

σY (x , y , ν[ϑ]t , ϑt) = σX (x , ν[ϑ]t , ϑt)Dxy(t, x , p) + α[ϑ]tDpy(t, x , p)

with y = y(t, x , p)

Page 81: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

PDE characterization - “waving hands” version

2 PDE characterization

infv∈V

sup(u,a)∈N vy

(µY (·, y, u, v)− Lu,v ,a

X ,M y)

= 0

where

N vy := (u, a) ∈ U × Rd : σY (·, y, u, v) = σX (·, u, v)Dxy + aDpy.

2 Need to be relaxed...

Page 82: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

PDE characterization - “waving hands” version

2 PDE characterization

infv∈V

sup(u,a)∈N vy

(µY (·, y, u, v)− Lu,v ,a

X ,M y)

= 0

where

N vy := (u, a) ∈ U × Rd : σY (·, y, u, v) = σX (·, u, v)Dxy + aDpy.

2 Need to be relaxed...

Page 83: Stochastic target game problems - CEREMADEStochastic target game problems B.Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech UniversitéPierreetMarieCurie,Mai2012

References

2 H.M. Soner and N. Touzi (2002). Dynamic programming forstochastic target problems and geometric flows, JEMS, 4, 201-236.2 B. , R. Elie and N. Touzi (2009). Stochastic Target problems withcontrolled loss, SIAM SICON, 48 (5), 3123-3150.2 L. Moreau (2011), Stochastic target problems with controlled loss in ajump diffusion model, SIAM SICON, 49, 2577-2607, 2011.2 B. and N. M. Dang (2010). Generalized stochastic target problems forpricing and partial hedging under loss constraints - Application in optimalbook liquidation, to appear in F&S.2 B. and V. Thanh Nam (2011). A stochastic target approach for P&Lmatching problems, to appear in MOR.2 R. Buckdahn and J. Li. Stochastic differential games and viscositysolutions of Hamilton-Jacobi-Bellman-Isaacs equations. SIAM SICON,47(1) :444-475, 2008.