BSDEs Reflected BSDEs Differentiability of Reflected BSDEs Differentiability of reflected BSDEs with quadratic growth joint work with S. Ankirchner and P. Imkeller IRTG Stochastic Models of Complex Processes Disentis, July 2008 Anja Richter, richtera@math.hu-berlin.de Differentiability of reflected BSDEs with quadratic growth
BSDEs Reflected BSDEs Differentiability of Reflected BSDEs Outline BSDEs Definition Application in Finance Reflected BSDEs Definition Utility maximization Differentiability of Reflected BSDEs Setting Tools Results Anja Richter, richtera@math.hu-berlin.de Differentiability of reflected BSDEs with quadratic growth
BSDEs Definition Reflected BSDEs Application in Finance Differentiability of Reflected BSDEs What is a BSDE? Parameters: ◮ ξ r.v. F T -measurable ◮ f : Ω × [0 , T ] × R × R d → R predictable mapping A BSDE with terminal condition ξ and generator/driver f is an equation of the type � T � T Y t = ξ − Z s dW s + f ( s , Y s , Z s ) ds . (1) t t A solution is a pair of adapted processes ( Y , Z ) such that (1) makes sense. Anja Richter, richtera@math.hu-berlin.de Differentiability of reflected BSDEs with quadratic growth
BSDEs Definition Reflected BSDEs Application in Finance Differentiability of Reflected BSDEs Utility maximization ◮ incomplete financial market, i.e. d < m stocks dS i t = S i t ( b i t dt + σ i t dW t ) , i = 1 , . . . d , where b ∈ R d and σ ∈ R d , m . ◮ small investor: wealth process ( p s := π s σ s , θ s := σ − 1 s b s ) � t � t dS s V p t = v + π s = v + p s ( dW s + θ s ds ) S s 0 0 ◮ utility function U ( x ) = − exp − α x ( α > 0 risk aversion) ◮ Optimization problem under constraint C U ( V p � � Val ( v ) = sup E T ) p ∈ C Anja Richter, richtera@math.hu-berlin.de Differentiability of reflected BSDEs with quadratic growth
BSDEs Definition Reflected BSDEs Application in Finance Differentiability of Reflected BSDEs Utility maximization ◮ incomplete financial market, i.e. d < m stocks dS i t = S i t ( b i t dt + σ i t dW t ) , i = 1 , . . . d , where b ∈ R d and σ ∈ R d , m . ◮ small investor: wealth process � t � t dS s V p t = v + = v + p s ( dW s + θ s ds ) π s S s 0 0 ◮ utility function U ( x ) = − exp − α x ( α > 0 risk aversion) ◮ ξ European Option ◮ Optimization problem under constraint C U ( V p � � Val ( v ) = sup E T + ξ ) p ∈ C Anja Richter, richtera@math.hu-berlin.de Differentiability of reflected BSDEs with quadratic growth
BSDEs Definition Reflected BSDEs Application in Finance Differentiability of Reflected BSDEs Utility maximization U ( V p � � Optimization problem: Val ( v ) = sup p ∈ C E T + ξ ) Idea: Find a process Y with terminal condition Y T = ξ such that ◮ U ( V p t + Y t ) is a supermartingale for all p ◮ U ( V p opt + Y t ) is a martingale for one p opt t → BSDE with terminal condition ξ � T � T Y t = ξ − Z s dW s + f ( s , Z s ) ds . t t Anja Richter, richtera@math.hu-berlin.de Differentiability of reflected BSDEs with quadratic growth
BSDEs Definition Reflected BSDEs Application in Finance Differentiability of Reflected BSDEs Utility maximization U ( V p � � Optimization problem: Val ( v ) = sup p ∈ C E T + ξ ) Theorem (Hu, Imkeller, M¨ uller 2005) Val ( v ) = U ( v + Y 0 ) where ( Y , Z ) is the unique solution of � T � T Y t = ξ − Z s dW s + f ( s , Z s ) ds t t 2 dist 2 ( 1 αθ − z , C ) − z θ + 1 and f ( · , z ) = − α 2 α | θ | 2 . ! f grows quadratically in z ! Anja Richter, richtera@math.hu-berlin.de Differentiability of reflected BSDEs with quadratic growth
BSDEs Definition Reflected BSDEs Utility maximization Differentiability of Reflected BSDEs What is a RBSDE? Parameters: ◮ ( ξ t ) t ∈ [0 , T ] continuous on [0 , T [ and lim t → T ξ t ≤ ξ T ◮ f : Ω × [0 , T ] × R × R d → R predictable mapping A RBSDE with barrier ξ and generator/driver f is an equation of the type � T � T Y t = ξ T − Z s dW s + f ( s , Y s , Z s ) ds + K T − K t , (2) t t � T Y t ≥ ξ t , ( Y t − ξ t ) dK t = 0 , 0 where K is a continuous nondecreasing process. A solution is a triple of adapted processes ( Y , Z , K ) such that (2) makes sense. Anja Richter, richtera@math.hu-berlin.de Differentiability of reflected BSDEs with quadratic growth
BSDEs Definition Reflected BSDEs Utility maximization Differentiability of Reflected BSDEs Utility maximization Same setting as before: � t ◮ wealth process V p t = v + 0 p s ( dW s + θ s ds ) ◮ utility function U ( x ) = − e − α x ( α > 0 risk aversion) Question: What happens if the investor holds an American option with payoff function ( ξ t ) t ∈ [0 , T ] ? Optimization problem: U ( V p � � Val ( v ) = sup ν, p E T + ξ ν ) Anja Richter, richtera@math.hu-berlin.de Differentiability of reflected BSDEs with quadratic growth
BSDEs Definition Reflected BSDEs Utility maximization Differentiability of Reflected BSDEs Utility maximization Optimization problem: Val ( v ) = sup ν, p E [ U ( V p T + ξ ν )] Theorem (A.R.) Val ( v ) = U ( v + Y 0 ) where ( Y , Z , K ) is the unique solution of � T � T Y t = ξ T − Z s dW s + f ( s , Z s ) ds + K T − K t , t t � T Y t ≥ ξ t , 0 ( Y t − ξ t ) dK t , with K continuous, nondecreasing and 2 dist 2 ( 1 αθ − z , C ) − z θ + 1 f ( · , z ) = − α 2 α | θ | 2 . ! f grows quadratically in z ! Anja Richter, richtera@math.hu-berlin.de Differentiability of reflected BSDEs with quadratic growth
BSDEs Setting Reflected BSDEs Tools Differentiability of Reflected BSDEs Results Parameterized RBSDE Parameter dependence on x ∈ R � T � T Y x Z x f ( s , Z x s ) ds + K x T − K x t = ξ T ( x ) − s dW s + t . t t � T Y x ( Y x t − ξ t ( x )) dK x t ≥ ξ t ( x ) , t = 0 , 0 Question: Are the solution processes Y x , Z x and K x continuous or even differentiable with respect to x ? Anja Richter, richtera@math.hu-berlin.de Differentiability of reflected BSDEs with quadratic growth
BSDEs Setting Reflected BSDEs Tools Differentiability of Reflected BSDEs Results Our setting: Quadratic RBSDEs ◮ Consider RBSDE � T � T Y t = ξ T − Z s dW s + f ( s , Z s ) ds + K T − K t , t t � T Y t ≥ ξ t , ( Y t − ξ t ) dK t = 0 , 0 with ◮ ξ bounded adapted process, continuous on [0 , T [ and lim t → T ξ t ≤ ξ T ◮ f s.t. ∀ ( t , z ): | f ( t , z ) | ≤ M (1 + | z | 2 ), and continuous in z ◮ Kobylanski (02) proved solution processes are sup t | Y t | < ∞ Z 2 � and E [ s ds ] < ∞ Anja Richter, richtera@math.hu-berlin.de Differentiability of reflected BSDEs with quadratic growth
BSDEs Setting Reflected BSDEs Tools Differentiability of Reflected BSDEs Results BMO Martingales Definition (BMO) Uniformly integrable martingales M with M 0 = 0 and 1 2 � ∞ < ∞ � M � BMO = sup � E [ � M � T − � M � τ |F τ ] τ E ( M ) := exp { M − 1 2 � M �} Theorem (Kazamaki 1994) ◮ M BMO = ⇒ dQ = E ( M ) T dP is a probability measure ◮ M BMO = ⇒ ∃ p > 1 such that E ( M ) ∈ L p Theorem (A.R.) � ( Y , Z , K ) solution of the above RBSDE = ⇒ ZdW is BMO Anja Richter, richtera@math.hu-berlin.de Differentiability of reflected BSDEs with quadratic growth
BSDEs Setting Reflected BSDEs Tools Differentiability of Reflected BSDEs Results Moment estimates � Using Itˆ o formula, the BMO property of ZdW and inequalities of H¨ older, BDG, Doob,Young, for p > 1: Theorem (A.R.) �� � T � � � + E P � � � p | Y t | 2 p | Z s | 2 ds K 2 p E P + E P sup T t ∈ [0 , T ] 0 � 1 � � T � q 2 | ξ t | 2 pq 2 + � 2 pq 2 ξ 2 pq 2 ≤ CE P + sup f ( s , 0) ds . T t ∈ [0 , T ] 0 With similar methods we can estimate the variation in the solution induced by a variation in the data! Anja Richter, richtera@math.hu-berlin.de Differentiability of reflected BSDEs with quadratic growth
BSDEs Setting Reflected BSDEs Tools Differentiability of Reflected BSDEs Results Results Theorem (A.R.) Let ξ be differentiable in x, lipschitz in norm, f be differentiable in z, ∇ z f of linear growth in z, Then for p > 1 and | x − x ′ | < 1 � � t − Y x ′ | Y x t | 2 p ≤ C | x − x ′ | p E sup t ∈ [0 , T ] ��� T � p � t − Z x ′ t | 2 ds | Z x ≤ C | x − x ′ | p E 0 � � | K x t − K x ′ t | 2 p ≤ C | x − x ′ | p . E sup t ∈ [0 , T ] Anja Richter, richtera@math.hu-berlin.de Differentiability of reflected BSDEs with quadratic growth
BSDEs Setting Reflected BSDEs Tools Differentiability of Reflected BSDEs Results Spaces: ◮ S p space of predictable processes X such that � 1 � p t | X t | p � X � S p = E sup < ∞ ◮ H p space of predictable processes X such that 2 � 1 � p ��� T p | X t | 2 dt � X � H p = E < ∞ 0 Corollary (A.R.) ◮ ( Y x t ) and ( K x t ) are continuous in t and x. ◮ R → H 2 p : x �→ Z x is H¨ older continuous with α = 1 2 . ◮ R → S 2 p : x �→ Y x is H¨ older continuous with α = 1 2 . Anja Richter, richtera@math.hu-berlin.de Differentiability of reflected BSDEs with quadratic growth
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