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

                

Y(t) = ξ +

T

t

g(s, Y(s), Z(s))ds −

T

t

Z(s)dBs + K(T) − K(t); Y(t) ≥ L(t), 0 ≤ t ≤ T,

T (Y(t) − L(t))′dK(t) = 0.

(1)

Multi-Dimensional Reflective BSDE

Applications

Lipschitz Growth

entry by entry,

                                                

Y1(t) = ξ1 +

T

t

g1(s, Y(s), Z(s))ds −

T

t

Z1(s)dBs + K1(T) − K1(t); Y1(t) ≥ L1(t), 0 ≤ t ≤ T,

T (Y1(t) − L1(t))dK1(t) = 0, · · ·

Ym(t) = ξm +

T

t

gm(s, Y(s), Z(s))ds −

T

t

Zm(s)dBs + Km(T) − Km(t Ym(t) ≥ Lm(t), 0 ≤ t ≤ T,

T (Ym(t) − Lm(t))dKm(t) = 0.

(2)

Multi-Dimensional Reflective BSDE

Applications

Lipschitz Growth

Seek solution (Y, Z, K) in the spaces Y = (Y1, · · · , Ym)′ ∈ M2(m; 0, T)

:={m-dimensional predictable process φ s.t. E[sup

[0,T]

φ2

t ] ≤ ∞};

Z = (Z1, · · · , Zm)′ ∈ L2(m × d; 0, T)

:={m × d-dimensional predictable process φ s.t. E[ T φ2

t dt] ≤ ∞};

K = (K1, · · · , Km)′ = continuous, increasing process in M2(m; 0, T). (3)

Multi-Dimensional Reflective BSDE

Applications

Lipschitz Growth

Assumption A 3.1 (1) The random field g = (g1, · · · , gm)′ : [0, T] × Rm × Rm×d → Rm (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,

E[ T

g(t, 0, 0)2dt] < ∞. (6) (2) The random variable ξ is FT-measurable and square-integrable. The lower reflective boundary L is progressively measurable, and satisfy E[sup

[0,T]

L+(t)2] < ∞. L ≤ ξ, P-a.s.

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

      

Xt,x(s) = x, 0 ≤ s ≤ t; dXt,x(s) = f(s, Xt,x(s))ds + σ(s, Xt,x(s))dBs, t < s ≤ T. (7) m-dim backward equation

                            

Yt,x(s) =ξ(Xt,x(T)) +

T

s

g(r, Xt,x(r), Yt,x(r), Zt,x(r))dr

− T

s

Zt,x(r)dBr + K t,x(T) − K t,x(s); Yt,x(s) ≥L(s, Xt,x(s)), t ≤ s ≤ T,

T

t

(Yt,x(s) − L(s, Xt,x(s)))′dK t,x(s)

(8)

Multi-Dimensional Reflective BSDE

Applications

Linear Growth, Markovian System

Assumption A 4.1 (1) Drift f : [0, T] × Rl → Rl, and volatility σ : [0, T] × Rl → Rl×d, are deterministic, measurable mappings, locally Lipschitz in x uniformly for all t ∈ [0, T]. And for all (t, x) ∈ [0, T] × Rl,

|f(t, x)|2 + |σ(t, x)|2 ≤ C(1 + |x|2), for some constant C.

(2) g : [0, T] × Rl × Rm × Rm×d → Rm is deterministic, measurable, and for all (t, x, y, z) ∈ [0, T] × Rl × Rm × Rm×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 : Rl → Rm deterministic, measurable. L : [0, T] × Rl → Rm deterministic, measurable, continuous. E[ξ(X(T))2] < ∞;

E[sup

[0,T]

L+(t, X(t))2] < ∞. L ≤ ξ, P-a.s.

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

Applications

THAT’S ALL THANK YOU

Multi-Dimensional Reflective BSDE