Backward Stochastic Differential Equations with Infinite Time Horizon

Transcription

1 Backward Stochastic Differential Equations with Infinite Time Horizon Holger Metzler PhD advisor: Prof. G. Tessitore Università di Milano-Bicocca Spring School Stochastic Control in Finance Roscoff, March 2010

2 Outline General setup and standard results 1 General setup and standard results 2

3 Outline General setup and standard results 1 General setup and standard results 2

4 General setup General setup and standard results Throughout this talk, we are given a complete probability space (Ω, F, P), carrying a standard d-dimensional Brownian motion (W t ) t 0, the filtration ( F t ) generated by W, the filtration (F t ), which is ( F t ) augmented by all P-null sets. = (F t ) satisfies the usual conditions Adapted processes are always assumed to be (F t )-adapted.

5 General setup General setup and standard results Throughout this talk, we are given a complete probability space (Ω, F, P), carrying a standard d-dimensional Brownian motion (W t ) t 0, the filtration ( F t ) generated by W, the filtration (F t ), which is ( F t ) augmented by all P-null sets. = (F t ) satisfies the usual conditions Adapted processes are always assumed to be (F t )-adapted. We denote by M 2,ϱ (E) the Hilbert space of processes X with: X is progressively measurable, with values in the Euclidean space [ E, ] E e ϱs X s 2 E ds <. 0

6 Consider the BSDE with infinite time horizon dy t = ψ(t, Y t, Z t )dt Z t dw t, t [0, T ], T 0. (1) ψ : Ω R + R n L(R d, R n ) R n is such that ψ(, y, z) is a progressively measurable process. A solution is a couple of progressively measurable processes (Y, Z) with values in R n L(R d, R n ), such that, for all t T with t, T 0, Y t = Y T + T t ψ(s, Y s, Z s ) ds T t Z s dw s.

7 Assumption (A1) General setup and standard results (A1) There exist C 0, γ 0 and µ R, such that (1) ψ is uniformly lipschitz, i.e. ψ(t, y, z) ψ(t, y, z ) C y y + γ z z ; (2) ψ is monotone in y: y y, ψ(t, y, z) ψ(t, y, z) µ y y 2 ; (3) There exists ϱ R, such that ϱ > γ 2 2µ, and E 0 e ϱs ψ(s, 0, 0) 2 ds C.

8 Set λ := γ2 2 µ. This implies ϱ > 2λ. Darling and Pardoux (1997) established the following result. Theorem If (A1) holds then BSDE (1) has a unique solution (Y, Z) in M 2,2λ (R n L(R d, R n )). The solution actually belongs to M 2,ϱ (R n L(R d, R n )). The major restriction is the structural condition in part (3) of (A1): We want to solve the equation for arbitrary bounded ψ(, 0, 0). So we need µ > 1 2 γ2. This condition is not natural in applications and, hence, is very unpleasant.

9 The one-dimensional case (n = 1) Significant improvement due to Briand and Hu (1998). Solution exists for all µ > 0, if ψ(, 0, 0) is bounded, i.e. (3 ) ψ(t, 0, 0) K. µ > 0 means, ψ is dissipative with respect to y. Theorem (n = 1) Assume parts (1) and (2) of (A1) with µ > 0, and (3 ). Then BSDE (1) has a solution (Y, Z) which belongs to M 2, 2µ (R R d ) and such that Y is a bounded process. This solution is unique in the class of processes (Y, Z), such that Y is continuous and bounded and Z belongs to M 2 loc (Rd ).

10 Idea of the proof General setup and standard results 1 Consider the equation with finite time horizon [0, m]. Call the unique solution (Y m, Z m ). 2 Establish the a priori bound Y m () K, for all. µ 3 Use this a priori bound to show that (Y m, Z m ) m N is a Cauchy sequence in M 2, 2µ (R R d ). The crucial part is to establish the a priori bound.

11 The a priori bound General setup and standard results Linearise ψ to ψ(s, Y m, Z m ) = α m (s)y m (s) + β m (s)z m (s) + ψ(s, 0, 0) with α m (s) µ and β m bounded. (Y m, Z m ) solves the equation m Y m (t) = [α m (s)y m (s) + β m (s)z m (s) + ψ(s, 0, 0)] ds t m t Z m (s) dw s.

12 Introduce Note that and ( t R m (t) := exp W m (t) := W (t) t ) α m (s) ds, 0 R m (s) e µ(s ) R m (s) ds 1 µ. β m (s) ds.

13 Apply Itô s formula to the process R m Y m : m Y m () = R m (m)y m (m) + R m (s)ψ(s, 0, 0) ds m R m (s)z m (s) dw m (s). Take into account that Y m (m) = 0: m m Y m () = R m (s)ψ(s, 0, 0) ds R m (s)z m (s) dw m (s).

14 Using Girsanov s theorem, we can consider W m as a Brownian motion with respect to an equivalent measure Q m and hence, we get, Q m -a.s., Y m () = E Qm [ Y m () F ] E Qm K µ. ψ(s, 0, 0) R m (s) ds F In the end, this estimate assures also the boundedness of the limit process Y.

15 Problem for n > 1 General setup and standard results If Y is a multi-dimensional process (n > 1), we cannot use this Girsanov trick, because each coordinate needs its own transformation, and these transformations are not consistent among each other. So we are restricted to the case µ > 1 2 γ2, whereas the case µ > 0 could have multiple interesting applications, e.g. in stochastic differential games or for homogenisation of PDEs.

16 Outline General setup and standard results 1 General setup and standard results 2

17 Let us now consider the following equation: d dy t = [AY t + Γ j Zt j + f t ]dt Z t dw t, t [0, T ], T 0. (2) j=1 A, Γ j R n n. Z j t denotes the j-th column vector of Z t R n d. f t R n is bounded by K. A is assumed to be dissipative, i.e. there exists µ > 0 such that y y, A(y y ) µ y y 2. The coefficients in equation (2) are non-stochastic and, except f t, time-independent.

18 As in the one-dimensional non-linear case, we are interested in progressively measurable solutions (Y, Z), such that Y is bounded. This can be achieved by establishing the above mentioned a priori estimate Y m () K µ. To this end, we consider the dual process to Y m, denoted by X x. This process satisfies dxt x = A Xt x dt + X x = x Rn. d j=1 (Γ j ) X x t dw j t

19 By Itô s formula and the Markov property of X x, we obtain Y m () sup E x =1 K sup E x =1 m Xt x, f t dt F X x t dt. = Question of L 1 -stability of X x with x = 1. We need E Xt x dt M. Task: Find appropriate assumptions on Γ j and µ. 0

20 Lyapunov approach Try to find Lyapunov function v C 2 (R n ) with (1) v 0, (2) v(x) c x, for some c > 0, (3) [Lv](x) δ x, for some δ > 0. Here L is the Kolmogorov operator of X x, i.e. dv(xt x ) = [Lv](Xt x )dt + martingale part. This approach was used by Ichikawa (1984) to show stability properties of strongly continuous semigroups.

21 Itô s formula and the Markov property of X x give us t E[v(Xt x ) v(x x )] = E δ E [Lv](X x s ) ds t X x s ds. By showing E[v(X x t )] 0 as t, we obtain E X x s ds 1 δ E[v(X x )] c δ x c δ =: M.

22 How to find a Lyapunov function? First idea: v(x) = x. Problem: v is not C 2, hence Itô s formula inapplicable. Second idea: Define, for ε > 0, v ε (x) = x 2 + ε. v ε (x) x.

23 How to proceed? Calculate [Lv ε ](x). Choose µ large enough, such that the coefficient in front of x 4 is negative. This choice will depend on Γ j. Find appropriate κ ε > 0, κ ε 0 and split the integral on the RHS: t Ev ε (Xt x ) Ev ε (X x ) = E = E t [Lv ɛ ](X x s ) ds [Lv ɛ ](X x s )1 { X x s κ ε} ds + E t [Lv ɛ ](X x s )1 { X x s <κ ε} ds

24 Obtain with ε 0 t E X t E X δ E Apply Gronwall s lemma to Φ(t) := E X t. = lim E X t = 0 t = E X t 1 δ =: M X s ds. So X x is L 1 -stable and equation (2) admits a bounded solution.

25 Simple example ( ) γ1 0 Assume Γ = and γ := max{ γ 0 γ 1, γ 2 }. 2 [Lv ε ](x) [ 1 8 (γ 1 γ 2 ) 2 µ] x εγ2 x 2 ( x 2 +ε) 3 2 For µ > 1 8 (γ 1 γ 2 ) 2 is X x L 1 -stable, and equation (2) has a bounded solution. The general result from the first part requires the much stronger assumption µ > 1 2 Γ 2 = 1 2 (γ2 1 + γ 2 2).

26 L 2 -stability is strictly stronger than L 1 -stability. Example We take n = d = 1 and consider the following equation: { dxt = µx t dt + γx t dw t X 0 = 1. The solution is a geometric Brownian motion X t = e µt e γwt 1 2 γ2 t and E X t = e µt, E X t 2 = e 2µt e γ2t. So X is L 1 -stable for each µ > 0, but L 2 -stable only for µ > 1 2 γ2.

27 References General setup and standard results P. Briand and Y. Hu, Stability of BSDEs with Random Terminal Time and Homogenization of Semilinear Elliptic PDEs, Journal of Functional Analysis, 155 (1998), R. W. R. Darling and R. Pardoux, Backwards SDE with random terminal time and applications to semilinear elliptlic PDE, Annals of Probability, 25 (1997), M. Fuhrman, G. Tessitore, Infinite Horizon Backward Stochastic Differential Equations and Elliptic Equations in Hilbert Spaces, Annals of Probability, Vol. 32 No. 1B (2004), A. Ichikawa, Equivalence of L p Stability and Exponential Stability for a Class of Nonlinear Semigroups, Nonlinear Analysis, Theory, Methods & Applications, Vol. 8 No. 7 (1984), A. Richou, Ergodic BSDEs and related PDEs with Neumann boundary conditions, Stochastic Processes and their Applications, 119 (2009),