1/34 Existence and uniqueness of solution for multidimensional BSDE with local conditions on the coefficient EL HASSAN ESSAKY Cadi Ayyad University Multidisciplinary Faculty Safi, Morocco ITN—Roscof, March 18-23, 2010 El Hassan Essaky Multidisciplinary Faculty ( Cadi Ayyad University Multidisciplinary Faculty Safi, Morocco ITN—Roscof, Existence-uniqueness of solution for BSDE 1 / 34
2/34 1. BSDEs–Introduction (Ω , F , ( F t ) t ≤ 1 , P ) be a complete probability space F t = σ ( B s , 0 ≤ s ≤ t ) ∨ N be a filtration Consider the following ODE � dY t = 0 , t ∈ [ 0 , T ] , (1) Y T = ξ ∈ I R . El Hassan Essaky Multidisciplinary Faculty ( Cadi Ayyad University Multidisciplinary Faculty Safi, Morocco ITN—Roscof, Existence-uniqueness of solution for BSDE 2 / 34
2/34 1. BSDEs–Introduction (Ω , F , ( F t ) t ≤ 1 , P ) be a complete probability space F t = σ ( B s , 0 ≤ s ≤ t ) ∨ N be a filtration Consider the following terminal value problem � dY t = 0 , t ∈ [ 0 , T ] , (1) Y T = ξ ∈ L 2 (Ω , F T ; I R ) . El Hassan Essaky Multidisciplinary Faculty ( Cadi Ayyad University Multidisciplinary Faculty Safi, Morocco ITN—Roscof, Existence-uniqueness of solution for BSDE 2 / 34
2/34 1. BSDEs–Introduction (Ω , F , ( F t ) t ≤ 1 , P ) be a complete probability space F t = σ ( B s , 0 ≤ s ≤ t ) ∨ N be a filtration Consider the following terminal value problem � dY t = 0 , t ∈ [ 0 , T ] , (1) Y T = ξ ∈ L 2 (Ω , F T ; I R ) . We want to FIND F t -ADAPTED solution Y for equation (1). El Hassan Essaky Multidisciplinary Faculty ( Cadi Ayyad University Multidisciplinary Faculty Safi, Morocco ITN—Roscof, Existence-uniqueness of solution for BSDE 2 / 34
2/34 1. BSDEs–Introduction (Ω , F , ( F t ) t ≤ 1 , P ) be a complete probability space F t = σ ( B s , 0 ≤ s ≤ t ) ∨ N be a filtration Consider the following terminal value problem � dY t = 0 , t ∈ [ 0 , T ] , (1) Y T = ξ ∈ L 2 (Ω , F T ; I R ) . We want to FIND F t -ADAPTED solution Y for equation (1). This is IMPOSSIBLE, since the only solution is Y t = ξ, for all t ∈ [ 0 , T ] , (2) which is not F t − adapted. El Hassan Essaky Multidisciplinary Faculty ( Cadi Ayyad University Multidisciplinary Faculty Safi, Morocco ITN—Roscof, Existence-uniqueness of solution for BSDE 2 / 34
2/34 1. BSDEs–Introduction (Ω , F , ( F t ) t ≤ 1 , P ) be a complete probability space F t = σ ( B s , 0 ≤ s ≤ t ) ∨ N be a filtration Consider the following terminal value problem � dY t = 0 , t ∈ [ 0 , T ] , (1) Y T = ξ ∈ L 2 (Ω , F T ; I R ) . We want to FIND F t -ADAPTED solution Y for equation (1). This is IMPOSSIBLE, since the only solution is Y t = ξ, for all t ∈ [ 0 , T ] , (2) which is not F t − adapted. A natural way of making (2) F t − adapted is to redefine Y . as follows Y t = I E ( ξ |F t ) , t ∈ [ 0 , T ] . (3) Then Y . is F t − adapted and satisfies Y T = ξ , but not equation (1). El Hassan Essaky Multidisciplinary Faculty ( Cadi Ayyad University Multidisciplinary Faculty Safi, Morocco ITN—Roscof, Existence-uniqueness of solution for BSDE 2 / 34
3/34 1. BSDEs–Introduction MRT = ⇒ there exists an F t − adapted process Z square integrable s.t � t Y t = Y 0 + Z s dB s . (4) 0 It follows that � T Y T = ξ = Y 0 + Z s dB s . (5) 0 Combining (4) and (5), one has � T Y t = ξ − (6) Z s dB s , t whose differential form is � dY t = Z t dB t , t ∈ [ 0 , T ] , (7) Y T = ξ. Comparing (1) and (7), the term ” Z t dB t ” has been added. El Hassan Essaky Multidisciplinary Faculty ( Cadi Ayyad University Multidisciplinary Faculty Safi, Morocco ITN—Roscof, Existence-uniqueness of solution for BSDE 3 / 34
4/34 1. BSDEs–Introduction BSDE is an equation of the following type: � T � T Y t = ξ + f ( s , Y s , Z s ) ds − Z s dB s , 0 ≤ t ≤ T . (8) t t T : TERMINAL TIME R d : GENERATOR or COEFFICIENT R d × I R d × n → I f : Ω × [ 0 , T ] × I R d . ξ : TERMINAL CONDITION F T − adapted process with value in I R d and Z ∈ I R d × n . UNKNOWNS ARE : Y ∈ I El Hassan Essaky Multidisciplinary Faculty ( Cadi Ayyad University Multidisciplinary Faculty Safi, Morocco ITN—Roscof, Existence-uniqueness of solution for BSDE 4 / 34
5/34 1. BSDEs–Introduction R d × I R d × n –valued processes ( Y , Z ) defined on I Denote by L the set of I R + × Ω which are F t –adapted and such that: � T � � � ( Y , Z ) � 2 = I | Y t | 2 + | Z s | 2 ds < + ∞ . E sup 0 ≤ t ≤ T 0 The couple ( L , � . � ) is then a Banach space. Definition A solution of equation (8) is a pair of processes ( Y , Z ) which belongs to the space ( L , � . � ) and satisfies equation (8). El Hassan Essaky Multidisciplinary Faculty ( Cadi Ayyad University Multidisciplinary Faculty Safi, Morocco ITN—Roscof, Existence-uniqueness of solution for BSDE 5 / 34
6/34 2. BSDEs with Lipshitz coefficient Consider the following assumptions: R d × I R d × n : For all ( y , z ) ∈ I ( ω, t ) − → f ( ω, t , y , z ) is F t − progressively measurable f ( ., 0 , 0 ) ∈ L 2 ([ 0 , T ] × Ω , I R d ) f is Lipschitz : ∃ K > 0 and ∀ y , y ′ ∈ I R d , z , z ′ ∈ I R d × n and ( ω, t ) ∈ Ω × [ 0 , T ] s.t | f ( ω, t , y , z ) − f ( ω, t , y ′ , z ′ ) |≤ K � | y − y ′ | + | z − z ′ | � . ξ ∈ L 2 (Ω , F T ; I R d ) Theorem : Pardoux and Peng 1990 Suppose that the above assumptions hold true. Then, there exists a unique solution for BSDE (15). El Hassan Essaky Multidisciplinary Faculty ( Cadi Ayyad University Multidisciplinary Faculty Safi, Morocco ITN—Roscof, Existence-uniqueness of solution for BSDE 6 / 34
6/34 2. BSDEs with Lipshitz coefficient Consider the following assumptions: R d × I R d × n : For all ( y , z ) ∈ I ( ω, t ) − → f ( ω, t , y , z ) is F t − progressively measurable f ( ., 0 , 0 ) ∈ L 2 ([ 0 , T ] × Ω , I R d ) f is Lipschitz : ∃ K > 0 and ∀ y , y ′ ∈ I R d , z , z ′ ∈ I R d × n and ( ω, t ) ∈ Ω × [ 0 , T ] s.t | f ( ω, t , y , z ) − f ( ω, t , y ′ , z ′ ) |≤ K � | y − y ′ | + | z − z ′ | � . ξ ∈ L 2 (Ω , F T ; I R d ) Theorem : Pardoux and Peng 1990 Suppose that the above assumptions hold true. Then, there exists a unique solution for BSDE (15). El Hassan Essaky Multidisciplinary Faculty ( Cadi Ayyad University Multidisciplinary Faculty Safi, Morocco ITN—Roscof, Existence-uniqueness of solution for BSDE 6 / 34
7/34 3. APPLICATIONS OF BSDE : FINANCE & PDE Consider a market where only two basic assets are traded. BOND : STOCK : Consider a European call option whose payoff is ( X T − K ) + . The option pricing problem is : fair price of this option at time t = 0? Suppose that this option has a price y at time t = 0. Then the fair price for the option at time t = 0 should be such a y that the corresponding optimal investment would result in a wealth process Y t satisfying Y T = ( X T − K ) + . El Hassan Essaky Multidisciplinary Faculty ( Cadi Ayyad University Multidisciplinary Faculty Safi, Morocco ITN—Roscof, Existence-uniqueness of solution for BSDE 7 / 34
7/34 3. APPLICATIONS OF BSDE : FINANCE & PDE Consider a market where only two basic assets are traded. BOND : dX 0 t = rX 0 t dt STOCK : Consider a European call option whose payoff is ( X T − K ) + . The option pricing problem is : fair price of this option at time t = 0? Suppose that this option has a price y at time t = 0. Then the fair price for the option at time t = 0 should be such a y that the corresponding optimal investment would result in a wealth process Y t satisfying Y T = ( X T − K ) + . El Hassan Essaky Multidisciplinary Faculty ( Cadi Ayyad University Multidisciplinary Faculty Safi, Morocco ITN—Roscof, Existence-uniqueness of solution for BSDE 7 / 34
7/34 3. APPLICATIONS OF BSDE : FINANCE & PDE Consider a market where only two basic assets are traded. BOND : dX 0 t = rX 0 t dt STOCK : dX t = bX t dt + σ X t dB t Consider a European call option whose payoff is ( X T − K ) + . The option pricing problem is : fair price of this option at time t = 0? Suppose that this option has a price y at time t = 0. Then the fair price for the option at time t = 0 should be such a y that the corresponding optimal investment would result in a wealth process Y t satisfying Y T = ( X T − K ) + . El Hassan Essaky Multidisciplinary Faculty ( Cadi Ayyad University Multidisciplinary Faculty Safi, Morocco ITN—Roscof, Existence-uniqueness of solution for BSDE 7 / 34
8/34 3. APPLICATIONS OF BSDE : FINANCE & PDE Denote by R t : the amount that the writer invests in the stock Y t − R t : the remaining amount which is invested in the bond R t determines a strategy of the investment which is called a portfolio. By setting Z t = σ R t , we obtain the following BSDE dX t = bX t dt + σ X t dB t dY t = ( rY t + b − r Z t ) dt + Z t dB t , t ∈ [ 0 , T ] , σ � �� � (9) f ( t , Y t , Z t ) X 0 = x , Y T = ( X T − K ) + . � �� � ξ Pardoux & Peng result = ⇒ there exits a unique solution ( Y t , Z t ) . The option price at time t = 0 is given by Y 0 , and the portfolio is given by R t = Z t σ . El Hassan Essaky Multidisciplinary Faculty ( Cadi Ayyad University Multidisciplinary Faculty Safi, Morocco ITN—Roscof, Existence-uniqueness of solution for BSDE 8 / 34
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