Multi-Dimensional Reflective BSDE July 29 2010, Cornell University - PowerPoint PPT Presentation

Applications Multi-Dimensional Reflective BSDE July 29 2010, Cornell University By Qinghua Li, Columbia University Multi-Dimensional Reflective BSDE

Applications Multi-Dim Reflective BSDE Multi-Dimensional Reflective BSDE

Applications Lipschitz Growth m -dim reflective BSDE � T � T   Y ( t ) = ξ + g ( s , Y ( s ) , Z ( s )) ds − Z ( s ) dB s + K ( T ) − K ( t );      t t   � T    ( Y ( t ) − L ( t )) ′ dK ( t ) = 0 .  Y ( t ) ≥ L ( t ) , 0 ≤ t ≤ T ,     0 (1) Multi-Dimensional Reflective BSDE

Applications Lipschitz Growth entry by entry, � T � T   Y 1 ( t ) = ξ 1 + g 1 ( s , Y ( s ) , Z ( s )) ds − Z 1 ( s ) dB s + K 1 ( T ) − K 1 ( t );       t t   � T      Y 1 ( t ) ≥ L 1 ( t ) , 0 ≤ t ≤ T , ( Y 1 ( t ) − L 1 ( t )) dK 1 ( t ) = 0 ,      0     · · ·     � T � T     Y m ( t ) = ξ m + g m ( s , Y ( s ) , Z ( s )) ds − Z m ( s ) dB s + K m ( T ) − K m ( t      t t    � T      Y m ( t ) ≥ L m ( t ) , 0 ≤ t ≤ T , ( Y m ( t ) − L m ( t )) dK m ( t ) = 0 .     0 (2) Multi-Dimensional Reflective BSDE

Applications Lipschitz Growth Seek solution ( Y , Z , K ) in the spaces Y = ( Y 1 , · · · , Y m ) ′ ∈ M 2 ( m ; 0 , T ) φ 2 := { m -dimensional predictable process φ s.t. E [ sup t ] ≤ ∞} ; [ 0 , T ] Z = ( Z 1 , · · · , Z m ) ′ ∈ L 2 ( m × d ; 0 , T ) � T φ 2 := { m × d -dimensional predictable process φ s.t. E [ t dt ] ≤ ∞} ; 0 K = ( K 1 , · · · , K m ) ′ = continuous, increasing process in M 2 ( m ; 0 , T ) . (3) Multi-Dimensional Reflective BSDE

Applications Lipschitz Growth Assumption A 3.1 (1) The random field g = ( g 1 , · · · , g m ) ′ : [ 0 , T ] × R m × R m × d → R m (4) is predictable in t , and is uniformly Lipschitz in y and z , i.e. there exists a constant b > 0, such that | g ( t , y , z ) − g ( t , ¯ y , ¯ z ) | ≤ b ( || y − ¯ y || + || z − ¯ z || ) , ∀ t ∈ [ 0 , T ] . (5) Further more, � T g ( t , 0 , 0 ) 2 dt ] < ∞ . E [ (6) 0 (2) The random variable ξ is F T -measurable and square-integrable. The lower reflective boundary L is progressively L + ( t ) 2 ] < ∞ . L ≤ ξ , P -a.s. measurable, and satisfy E [ sup [ 0 , T ] Multi-Dimensional Reflective BSDE

Applications Lipschitz Growth Results: ◮ existence and uniqueness of solution, via Picard iteration ◮ 1-dim Comparison Theorem (El Karoui et al, 1997) ◮ continuous dependency property Multi-Dimensional Reflective BSDE

Applications Linear Growth, Markovian System l (elle)-dim forward equation X t , x ( s ) = x , 0 ≤ s ≤ t ;     (7) dX t , x ( s ) = f ( s , X t , x ( s )) ds + σ ( s , X t , x ( s )) dB s , t < s ≤ T .    m -dim backward equation � T  Y t , x ( s ) = ξ ( X t , x ( T )) + g ( r , X t , x ( r ) , Y t , x ( r ) , Z t , x ( r )) dr       s    � T     Z t , x ( r ) dB r + K t , x ( T ) − K t , x ( s );  −   s    � T     Y t , x ( s ) ≥ L ( s , X t , x ( s )) , t ≤ s ≤ T , ( Y t , x ( s ) − L ( s , X t , x ( s ))) ′ dK t , x ( s )      t (8) Multi-Dimensional Reflective BSDE

Applications Linear Growth, Markovian System Assumption A 4.1 (1) Drift f : [ 0 , T ] × R l → R l , and volatility σ : [ 0 , T ] × R l → R l × d , are deterministic, measurable mappings, locally Lipschitz in x uniformly for all t ∈ [ 0 , T ] . And for all ( t , x ) ∈ [ 0 , T ] × R l , | f ( t , x ) | 2 + | σ ( t , x ) | 2 ≤ C ( 1 + | x | 2 ) , for some constant C . (2) g : [ 0 , T ] × R l × R m × R m × d → R m is deterministic, measurable, and for all ( t , x , y , z ) ∈ [ 0 , T ] × R l × R m × R m × d , | g ( t , x , y , z ) | ≤ b ( 1 + | x | p + | y | + | z | ) , for some positive constant b ; (3) for every fixed ( t , x ) ∈ [ 0 , T ] × R , g ( t , x , · , · ) is continuous. (4) ξ : R l → R m deterministic, measurable. L : [ 0 , T ] × R l → R m deterministic, measurable, continuous. E [ ξ ( X ( T )) 2 ] < ∞ ; L + ( t , X ( t )) 2 ] < ∞ . L ≤ ξ , P -a.s. E [ sup [ 0 , T ] Multi-Dimensional Reflective BSDE

Applications Linear Growth, Markovian System Results ◮ existence of solution, via Lipschitz approximation ◮ 1-dim Comparison Theorem ◮ continuous dependency property Multi-Dimensional Reflective BSDE

Applications What for? Multi-Dimensional Reflective BSDE

Applications Connections with ◮ Multi-dim variational inequalities (Feynman-Kac formula) ◮ Non-zero-sum stoch. differential games (esp. Dynkin games) ◮ Financial market sensitive to large traders’ transactions Multi-Dimensional Reflective BSDE

Applications References ◮ Pardoux, E. and S. Peng (1990). Adapted Solution of a Backward Stochastic Differential Equation. Systems & Control Letters 14 55-61. ◮ Hamad` ene, S., J-P . Lepeltier, and S. Peng (1997). BSDEs with Continuous Coefficients and Stochastic Differential Games. Pitman Research Notes in Mathematics Series 364. ◮ Karatzas, I. and Q. Li (2009). A BSDE Approach to Non-Zero-Sum Stochastic Differential Games of Control and Stopping. Submitted . http://stat.columbia.edu/ ∼ qinghua Multi-Dimensional Reflective BSDE

Applications References ◮ Cvitani´ c, J., and I. Karatzas (1996). Backward Stochastic Differential Equations with Reflection and Dynkin Games. The Annals of Probability . Vol. 24, No. 4, 2024-2056. ◮ El Karoui, N., C. Kapoudjian, E. Pardoux, S. Peng, M. C. Quenez (1997). Reflected Solutions of Backward SDE’S, and Related Obstacle Problems for PDE’s. The Annals of Probability . Vol. 25, No. 2, 702-737. ◮ Hu, Ying, and Shige Peng. On the Comparison Theorem for Multidimensional BSDEs. C. R. Acad. Sci. Paris, Ser. I 343 (2006) 135-140. Multi-Dimensional Reflective BSDE

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