Contents Introduction The convex frame The multidimensional frame Solving Quadratic BSDEs Hélène HIBON 29/06/16
Contents Introduction The convex frame The multidimensional frame Introduction 1 The convex frame 2 Motivation The Legendre-Fenchel transformation The result The multidimensional frame 3 Local → Global solution Stability results
Contents Introduction The convex frame The multidimensional frame Introduction 1 The convex frame 2 Motivation The Legendre-Fenchel transformation The result The multidimensional frame 3 Local → Global solution Stability results
Contents Introduction The convex frame The multidimensional frame Introduction 1 The convex frame 2 Motivation The Legendre-Fenchel transformation The result The multidimensional frame 3 Local → Global solution Stability results
Contents Introduction The convex frame The multidimensional frame What is a solution to a BSDE ? For BSDE ( ξ, g ) : a pair ( Y , Z ) of predictable processes such that a.s t �→ Y t is continuous , t �→ Z t belongs to L 2 ( 0 , T ) , t �→ g ( t , Y t , Z t ) belongs to L 1 ( 0 , T ) and � T � T Y t = ξ + g ( s , Y s , Z s ) ds − Z s dWs t t where { W t := ( W 1 t , ..., W d t ) ∗ , 0 ≤ t ≤ T } is a d -dimensional standard Brownian motion defined on some probability space (Ω , F , P )
Contents Introduction The convex frame The multidimensional frame Notations and requirements n denotes the dimension for Y ⋄ { F t , 0 ≤ t ≤ T } : augmented natural filtration of the Brownian motion W ⋄ S ∞ ( R n ) : set of R n -valued F t -adapted essentially bounded continuous processes. Banach space when provided with the essential sup norm || . || ∞ ⋄ M 2 ( R n × d ) : set of predictable R n × d -valued processes { Z t } t ∈ [ 0 , T ] such that � 1 / 2 �� T 0 | Z s | 2 ds || Z || M 2 := E < ∞ • Comparison theorem and Kobylanski’s monotone convergence theorem ⋄ E ( M ) : the stochastic exponential of a one-dimensional local martingale M ⋄ β · M : the stochastic integral of an adapted process β with respect to a local continuous martingale M
Contents Introduction The convex frame The multidimensional frame Notations and requirements n denotes the dimension for Y ⋄ { F t , 0 ≤ t ≤ T } : augmented natural filtration of the Brownian motion W ⋄ S ∞ ( R n ) : set of R n -valued F t -adapted essentially bounded continuous processes. Banach space when provided with the essential sup norm || . || ∞ ⋄ M 2 ( R n × d ) : set of predictable R n × d -valued processes { Z t } t ∈ [ 0 , T ] such that � 1 / 2 �� T 0 | Z s | 2 ds || Z || M 2 := E < ∞ • Comparison theorem and Kobylanski’s monotone convergence theorem ⋄ E ( M ) : the stochastic exponential of a one-dimensional local martingale M ⋄ β · M : the stochastic integral of an adapted process β with respect to a local continuous martingale M
Contents Introduction The convex frame The multidimensional frame Notations and requirements n denotes the dimension for Y ⋄ { F t , 0 ≤ t ≤ T } : augmented natural filtration of the Brownian motion W ⋄ S ∞ ( R n ) : set of R n -valued F t -adapted essentially bounded continuous processes. Banach space when provided with the essential sup norm || . || ∞ ⋄ M 2 ( R n × d ) : set of predictable R n × d -valued processes { Z t } t ∈ [ 0 , T ] such that � 1 / 2 �� T 0 | Z s | 2 ds || Z || M 2 := E < ∞ • Comparison theorem and Kobylanski’s monotone convergence theorem ⋄ E ( M ) : the stochastic exponential of a one-dimensional local martingale M ⋄ β · M : the stochastic integral of an adapted process β with respect to a local continuous martingale M
Contents Introduction The convex frame The multidimensional frame Notations and requirements n denotes the dimension for Y ⋄ { F t , 0 ≤ t ≤ T } : augmented natural filtration of the Brownian motion W ⋄ S ∞ ( R n ) : set of R n -valued F t -adapted essentially bounded continuous processes. Banach space when provided with the essential sup norm || . || ∞ ⋄ M 2 ( R n × d ) : set of predictable R n × d -valued processes { Z t } t ∈ [ 0 , T ] such that � 1 / 2 �� T 0 | Z s | 2 ds || Z || M 2 := E < ∞ • Comparison theorem and Kobylanski’s monotone convergence theorem ⋄ E ( M ) : the stochastic exponential of a one-dimensional local martingale M ⋄ β · M : the stochastic integral of an adapted process β with respect to a local continuous martingale M
Contents Introduction The convex frame The multidimensional frame • Girsanov’s theorem and BMO martingales Definition : M = ( M t , F t ) uniformly integrable martingale with M 0 = 0 is BMO 2 if there exists c > 0 so that E [ < M > ∞ τ | F τ ] ≤ c for all bounded s.t τ . || M || 2 BMO 2 := the smallest such constant. Lemma (Kazamaki) : For K > 0 , there are constants c 1 ( K ) > 0 and c 2 ( K ) > 0 such that for any one-dimensional BMO 2 martingale N such that || N || BMO 2 ( P ) ≤ K and any BMO 2 martingale M , c 1 || M || BMO 2 ( P ) ≤ || ˜ M || BMO 2 (˜ P ) ≤ c 2 || M || BMO 2 ( P ) where ˜ M := M − � M , N � and d ˜ P := E ( N ) ∞ 0 dP .
Contents Introduction The convex frame The multidimensional frame • Girsanov’s theorem and BMO martingales Definition : M = ( M t , F t ) uniformly integrable martingale with M 0 = 0 is BMO 2 if there exists c > 0 so that E [ < M > ∞ τ | F τ ] ≤ c for all bounded s.t τ . || M || 2 BMO 2 := the smallest such constant. Lemma (Kazamaki) : For K > 0 , there are constants c 1 ( K ) > 0 and c 2 ( K ) > 0 such that for any one-dimensional BMO 2 martingale N such that || N || BMO 2 ( P ) ≤ K and any BMO 2 martingale M , c 1 || M || BMO 2 ( P ) ≤ || ˜ M || BMO 2 (˜ P ) ≤ c 2 || M || BMO 2 ( P ) where ˜ M := M − � M , N � and d ˜ P := E ( N ) ∞ 0 dP .
Contents Introduction The convex frame The multidimensional frame • Existence and uniqueness of solution to BSDE with quadratic generator Theorem (Briand, Hu - 2006) : 2 | z | 2 with ( α t ) t ∈ [ 0 , T ] progressively measurable If | g ( t , y , z ) | ≤ α t + β | y | + γ and ∃ λ > γ e β T s.t E [ exp ( λ | ξ | + λ � T 0 α t dt )] < ∞ then their exists a solution with Z ∈ M 2 and − 1 γ ln E [ φ t ( − ξ ) |F t ] ≤ Y t ≤ 1 γ ln E [ φ t ( ξ ) |F t ] � d φ t = − H ( t , φ t ) dt with H ( t , p ) := α t γ 1 p ∈ ] −∞ , 1 [ where φ . ( z ) solves φ T = e γ z + p ( α t γ + β ln ( p )) 1 p ≥ 1 Theorem (Delbaen, Hu, Richou - 2011) : Suppose g convex with respect to z , K − Lip with respect to y and α t + ¯ β | y | + ¯ γ 2 | z | 2 . g ( t , y , z ) ≤ ¯ 1 ) Suppose g ( t , y , z ) ≥ − α t − r ( | y | + | z | ) . If exponential moment of order ε for ξ − + � T C e − CT ( Y − ) ∗ ] < ∞ ε 0 α t dt then a solution and a cste C s.t E [ e γ for ξ + + � T ¯ β T , p > ¯ If exponential moment of order pe 0 ¯ α t dt then a solution t + � T s.t E [ e pA ∗ ] < ∞ with A t := Y + 0 ¯ α t dt γ, ∃ ε > 0 s.t E [ e pA ∗ + e ε ( Y − ) ∗ ] < ∞ 2 ) If there exists a solution verifying ∃ p > ¯ then it is unique (among such solutions).
Contents Introduction The convex frame The multidimensional frame • Existence and uniqueness of solution to BSDE with quadratic generator Theorem (Briand, Hu - 2006) : 2 | z | 2 with ( α t ) t ∈ [ 0 , T ] progressively measurable If | g ( t , y , z ) | ≤ α t + β | y | + γ and ∃ λ > γ e β T s.t E [ exp ( λ | ξ | + λ � T 0 α t dt )] < ∞ then their exists a solution with Z ∈ M 2 and − 1 γ ln E [ φ t ( − ξ ) |F t ] ≤ Y t ≤ 1 γ ln E [ φ t ( ξ ) |F t ] � d φ t = − H ( t , φ t ) dt with H ( t , p ) := α t γ 1 p ∈ ] −∞ , 1 [ where φ . ( z ) solves φ T = e γ z + p ( α t γ + β ln ( p )) 1 p ≥ 1 Theorem (Delbaen, Hu, Richou - 2011) : Suppose g convex with respect to z , K − Lip with respect to y and α t + ¯ β | y | + ¯ γ 2 | z | 2 . g ( t , y , z ) ≤ ¯ 1 ) Suppose g ( t , y , z ) ≥ − α t − r ( | y | + | z | ) . If exponential moment of order ε for ξ − + � T C e − CT ( Y − ) ∗ ] < ∞ ε 0 α t dt then a solution and a cste C s.t E [ e γ for ξ + + � T ¯ β T , p > ¯ If exponential moment of order pe 0 ¯ α t dt then a solution t + � T s.t E [ e pA ∗ ] < ∞ with A t := Y + 0 ¯ α t dt γ, ∃ ε > 0 s.t E [ e pA ∗ + e ε ( Y − ) ∗ ] < ∞ 2 ) If there exists a solution verifying ∃ p > ¯ then it is unique (among such solutions).
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