Leader-Follower Stochastic Differential Game with Asymmetric Information and Applications

Transcription

1 Leader-Follower Stochastic Differential Game with Asymmetric Information and Applications Jie Xiong Department of Mathematics University of Macau Macau Based on joint works with Shi and Wang 2015 Barrett Lecture, Knoxville, 14/05/2015]

2 Outline 1 Motivation 2 Problem formulation 3 LQ Leader-Follower Stochastic Differential Game with Asymmetric Information 4 Follower s optimization problem 5 Complete Information LQ Stochastic Optimal Control Problem of the Leader

3 1. A motivation Example: Continuous Time Newsvendor Problems D( ) is the demand rate { dd(t) = a(µ kr(t) D(t))dt + σdw (t) + σd W (t), D(0) = d 0 R, retail price R(t).

4 The retailer will obtain an expected profit J 1 (q( ), R( ), w( )) T [ = E (R(t) S) min[d(t), q(t)] 0 ] (w(t) S)q(t) dt. order rate q(t); wholesale price w(t); salvaged at unit price S 0.

5 the manufacturer has a fixed production cost per unit M 0, an expected profit J 2 (q( ), R( ), w( )) = E T 0 (w(t) M)q(t)dt.

6 The information G 1 t, G 2 t available to the retailer and the manufacturer at time t, respectively. G 1 t G 2 t. For any w( ), retailer selects a G 1 t -adapted processes pair (q ( ), R ( )) for the retailer such that J 1 (q ( ), R ( ), w( )) J 1 ( q ( ; w( )), R ( ; w( )), w( ) ) = max q( ),R( ) J 1(q( ), R( ), w( )),

7 Manufacturer selects a G 2 t -adapted process w ( ) for the manufacturer such that J 2 (q ( ), R ( ), w ( )) ( J 2 q ( ; w ( )), R ( ; w ( )), w ( ) ) = max J ( 2 q ( ; w( )), R ( ; w( )), w( ) ), w( )

8 Example: Cooperative Advertising and Pricing Problems dx(t) = [ ρu(t) 1 x(t) δx(t) ] dt + σ(x(t))dw (t) + σ(x(t))d W (t), x(0) = x 0 [0, 1], u(t) advertisement effort rate, x(t) awareness share, ρ response constant, and δ the rate at which potential consumers are lost. Manufacturer decide: wholesale price w(t), cooperative participation rate θ(t). Retailer decides: effort u(t), retail price p(t).

9 Given w(t) and θ(t), retailer solves an optimization problem to maximize his expected profit = E J 1 (w( ), θ( ), u( ), p( )) T D is demand function. 0 e rt[ (p(t) w(t))d(p(t))x(t) ] (1 θ(t))u 2 (t) dt,

10 The manufacturer s optimization problem is to maximize his expected profit J 2 (w( ), θ( ), u( ), p( )) T = E e rt[ ] (w(t) c)d(p(t))x(t) θ(t)u 2 (t) dt, 0 where c 0 is the constant unit production cost.

11 The information G 1 t, G 2 t available to the retailer and the manufacturer at time t, respectively. G 1 t G 2 t. Given w and θ, retailer chooses G 1,t -adapted pair (u ( ), p ( )), J 1 (w( ), θ( ), u ( ), p ( )) J 1 ( w( ), θ( ), u ( ; (w( ), θ( )), p ( ; (w( ), θ( )) ) = max u( ),p( ) 0 J 1(w( ), θ( ), u( ), p( )).

12 Select a G 2,t -adapted pair (w ( ), θ ( )) for the manufacturer J 2 (w ( ), θ ( ), u ( ), p ( )) ( J 2 w ( ), θ ( ), u ( ; (w ( ), θ ( )), p ( ; (w ( ), θ ( )) ) = max J ( 2 w( ), θ( ), u ( ; (w( ), θ( )), w( ),0 θ( ) 1 p ( ; (w( ), θ( )) ),

13 2. Problem formulation State equation: dx u 1,u 2 (t) = b ( t, x u 1,u 2 (t), u 1 (t), u 2 (t) ) dt + σ ( t, x u 1,u 2 (t), u 1 (t), u 2 (t) ) dw (t) + σ ( t, x u 1,u 2 (t), u 1 (t), u 2 (t) ) d W (t), x u 1,u 2 (0) = x 0, (2.1)

14 Gt 1 Gt 2 F t. The admissible control { U 1 := u u1 1 : Ω [0, T ] U 1 is Gt 1 -adapted and sup E u 1 (t) i <, i = 1, 2, 0 t T { U 2 := u 2 u 2 : Ω [0, T ] U 2 is Gt 2 -adapted and sup E u 2 (t) i <, i = 1, 2, 0 t T }, }. (2.2) (2.3)

15 Problem of the follower. For any chosen u 2 ( ) U 2 by the leader, choose a G 1 t -adapted control u 1 ( ) = u 1 ( ; u 2( )) U 1, such that J 1 (u 1( ), u 2 ( )) J 1 ( u 1 ( ; u 2 ( )), u 2 ( ) ) = inf u 1 U 1 J 1 (u 1 ( ), u 2 ( )), u 1 ( ) = u 1 ( ; u 2( )), (partial information) optimal control. (2.4)

16 The leader would like to choose a Gt 2 -adapted control u 2 ( ) to minimize J 2 (u 1( ), u 2 ( )) [ T ( = E g 2 t, x u 1,u 2 (t), u 1(t; u 2 (t)), u 2 (t) ) dt 0 ] + G 2 (x u 1,u 2 (T )). (2.5)

17 Problem of the leader. Find a G 2 t -adapted control u 2 ( ) U 2, such that J 2 (u 1( ), u 2( )) = J 2 ( u 1 ( ; u 2( )), u 2( ) ) = inf u 2 U 2 J 2 ( u 1 ( ; u 2 ( )), u 2 ( ) ), (2.6)

18 Issacs (1954-5) differential games Basar and Olsder (1982) Monograph to summarize Stackelberg (1934) Leader-follower game introduced Yong (2002) LQ, complete info. Bensoussan et al (2012) Stoch. Max. Principle

19 3. LQ Leader-Follower Stochastic Differential Game with Asymmetric Information; Stochastic maximum principle under partial information. The state equation dx u 1,u 2 (t) = [ A(t)x u 1,u 2 (t) + B 1 (t)u 1 (t) + B 2 (t)u 2 (t) ] dt + [ C(t)x u 1,u 2 (t) + D 1 (t)u 1 (t) + D 2 (t)u 2 (t) ] dw (t) + [ u C(t)x 1,u 2 (t) + D 1 (t)u 1 (t) + D 2 (t)u 2 (t) ] d W (t), x u 1,u 2 (0) = x 0, (3.7)

20 Given u 2 U 2, follower minimizes his cost functional J 1 (u 1 ( ), u 2 ( )) = 1 [ T ( Q1 2 E (t)x u 1,u 2 (t), x u 1,u 2 (t) 0 + N 1 (t)u 1 (t), u 1 (t) ) dt (3.8) + G 1 x u 1,u 2 (T ), x u 1,u 2 (T ) ], Info: G 1 t = σ{ W (s); 0 s t}

21 Stochastic maximum principle under partial information. Probability space (Ω, F, ˆP). State equation: FBSDE dx = b(t, x, v)dt + σ(t, x, v)dw + σ(t, x, v)d W, dy = g(t, x, y, z, z, v)dt zdw zdy, x(0) = x 0, y(1) = f(x(1)). (3.9) Observation { dy = h(t, x)dt + d W, Y (0) = 0. (3.10)

22 Admissible controls v A v is G t -adapted, where G t = F Y t. Control problem: Find u A s.t. where ( 1 J(v) = Ê 0 J(u) = min v A J(v), ) l(t, x v, y v, z v, z v, v)dt + φ(x v (1)) + γ(y v (0)).

23 Hypothesis (H1): Coeff. conti. diff. w/ bounded 1st order partial derivatives. σ, h bounded. Hypothesis (H2): l, φ, γ conti. diff. ( 1 ) Ê l(t, x v, y v, z v, z v, v) dt + φ(x v (1)) + γ(y v (0)) <. 0

24 when b = σ = σ = 0, studied by Huang-Wang-Xiong (SICON 2009) General form by Wang-Wu-Xiong (SICON 2013). Detail on LQ case by Wang-Wu-Xiong (IEEE TAC 2015).

25 Replace b, W in state equation by b, Y, resp, where b = b σh. State equation: dx = b(t, x, v)dt + σ(t, x, v)dw + σ(t, x, v)dy, dy = g(t, x, y, z, z, v)dt zdw zdy, x(0) = x 0, y(1) = f(x(1)). (3.11)

26 Let So { t Z v (t) = exp h(s, x v )dy t 0 } h 2 (s, x v )ds. dz v = Z v h(t, x v )dy. (3.12) Let P ˆP be s.t. dˆp = Z v (1)dP. Then, under P, (W, Y ) is a B.M. ( 1 ) J(v) = E Z v l(t, x v, y v, z v, z v, v)dt + Z v (1)φ(x v (1)) + γ(y v (0)). 0

27 Hypothesis (H3): For ξ bdd, G t -measurable, then v A. v(s) ξ1 [t,t+τ) (s), For any u A and v being G s -adapted and bdd, δ > 0, if ɛ < δ, then u + ɛv A.

28 Adjoint equations: dp = (g y (t, x, y, z, z, u)p l y (t, x, y, z, z, u)) dt + (g z (t, x, y, z, z, u)p l z (t, x, y, z, z, u)) dw { + (g z (t, x, y, z, z, u)p h(t, x)) d W, dq = (b x (t, x, u) σ(t, x, u)h x (t, x))q + σ x (t, x, u)k + σ x (t, x, u) k g x } (t, x, y, z, z, u)p l x (t, x, y, z, z, u) dt kdw kd W, p(0) = γ y (y(0)) q(1) = f x (x(1))p(1) + φ x (x(1)), (3.13)

29 and { dp = l(t, x, y, z, z, u)dt QdW Qd W P (1) = φ(x(1)). (3.14) Hamitonian H(t, x, y, z, z, v; p, q, k, k, Q) = bq + σk + σ k + h Q + l (g h z)p.

30 Theorem If u is a local minimum for J( ), then ( ) E H u (t, x, y, z, z, v; p, q, k, k, Q) G t = 0. Idea of proof Set d dɛ J(u + ɛv) ɛ=0 = 0.

31 Remark (3.14) is for Z v (t). If h = 0, this eq. is not needed. Suppose h = b = σ = σ, it is proved in Huang-Wang-Xiong (2008, SICON)

32 4. Follower s optimization problem Step 1. (Optimal control) Hamiltonian function: H 1 ( t, x, u1, u 2 ; q, k, k ) = q, A(t)x + B 1 (t)u 1 + B 2 (t)u 2 + k, C(t)x + D 1 (t)u 1 + D 2 (t)u 2 + k, C(t)x + D1 (t)u 1 + D 2 (t)u 2 (4.15) 1 Q1 (t)x, x 1 N1 (t)u 1, u

33 Then, 0 = N 1 (t)u 1(t) B1 (t)e [ q(t) ] G 1 t D1 (t)e [ k(t) G t 1 ] D 1 (t)e [ k(t) (4.16) Gt 1 ], a.e.t [0, T ], where F t -adapted (q( ), k( ), k( )) satisfies the BSDE [ dq(t) = A (t)q(t) + C (t)k(t) + C (t) k(t) ] Q 1 (t)x u 1,u 2 (t) dt k(t)dw (t) k(t)d W (t), q(t ) = G 1 x u 1,u 2 (T ). (4.17)

34 Step 2. (Optimal filtering) Let ˆf(t) := E [ f(t) G t 1 ], f = q, k, k. How to derive filtering equation? Direct from (4.17) is impossible. Solve (4.17) first! Guess: q(t) = P 1 (t)x u 1,u 2 (t) ϕ(t), t [0, T ], (4.18) Set { dϕ(t) = α(t)dt + β(t)d W (t), ϕ(t ) = 0. (4.19)

35 Applying dq(t) and comparing, we get [ ] k(t) = P 1 (t) C(t)x u 1,u 2 (t) + D 1 (t)u 1(t) + D 2 (t)u 2 (t), (4.20) k(t) = P1 (t)[ C(t)x u 1,u 2 (t) + D 1 (t)u 1(t) + D ] 2 (t)u 2 (t) β(t), [ α(t) = P 1 (t) P 1 (t)a(t) A (t)p 1 (t) ] Q 1 (t) x u 1,u 2 (t) P 1 (t)b(t)u 1(t) P 1 (t)b 2 (t)u 2 (t) A (t)ϕ(t) + C (t)k(t) + C (t) k(t). (4.21) (4.22)

36 Hence, ˆq(t) = P 1 (t)ˆx u 1,û 2 (t) ϕ(t), (4.23) [ ] ˆk(t) = P 1 (t) C(t)ˆx u 1,û 2 (t) + D 1 (t)u 1(t) + D 2 (t)û 2 (t), (4.24) [ ˆ k(t) = P 1 (t) C(t)ˆx u 1,û 2 (t) + D 1 (t)u 1(t) + D ] 2 (t)û 2 (t) β(t) (4.25) with û 2 (t) := E [ u 2 (t) G 1,t ].

37 Using filtering technique (Lemma 5.4 in Xiong (2008)), we get dˆx u 1,û 2 (t) = [ A(t)ˆx u 1,û 2 (t) + B 1 (t)u 1(t) + B 2 (t)û 2 (t) ] dt + [ u C(t)ˆx 1,û 2 (t) + D 1 (t)u 1(t) + D 2 (t)û 2 (t) ] (4.26) d W (t), ˆx u 1,û 2 (0) = x 0, and { dˆq(t) = A (t)ˆq(t) + C (t)ˆk(t) + C (t)ˆ k(t) } Q 1 (t)ˆx u 1,û 2 (t) dt ˆ k(t)d W (t), ˆq(T ) = G 1ˆx u 1,û 2 (T ), (4.27)

38 Step 3. (Optimal state feedback control) By (4.16) we arrive at UNIVERSIDADE DE MACAU [ ( u 1(t) = Ñ1(t) 1 B1 (t)p 1 (t) + D1 (t)p 1 (t)c(t) + D 1 (t)p 1 (t) C(t) ) ˆx u 1,û 2 (t) ( + D1 (t)p 1 (t)d 2 (t) + D 1 (t)p 1 (t) D ) 2 (t) û 2 (t) ] + B1 (t)ϕ(t) + D 1 (t)β(t), (4.28) where Ñ 1 (t) := N 1 (t) + D 1 (t)p 1 (t)d 1 (t) + D 1 (t)p 1 (t) D 1 (t) (4.29)

39 Substituting (4.28) into (4.22), we obtain Riccati equation P 1 (t) + P 1 (t)a(t) + A (t)p 1 (t) + C (t)p 1 (t)c(t) + C (t)p 1 (t) C(t) + Q 1 (t) ( P 1 (t)b 1 (t) + C (t)p 1 (t)d 1 (t) )Ñ 1 + C (t)p 1 (t) D 1 (t) 1 (B (t) 1 (t)p 1 (t) + D1 (t)p 1 (t)c(t) + D 1 (t)p 1 (t) C(t) ) = 0, P 1 (T ) = G 1. Suppose it admits a unique differentiable solution P 1 ( ) (4.30)

40 [ α(t) = A (t) S ] 1 (t)ñ1(t) 1 B1 (t) ϕ(t) [ + C (t) S ] 1 (t)ñ1(t) 1 D 1 (t) β(t) [ + S ] 1 (t)ñ1(t) 1 S(t) + S2 (t) û 2 (t), (4.31) which is Gt 1 -adapted, where S(t) := D1 (t)p 1 (t)d 2 (t) + D 1 (t)p 1 (t) D 2 (t), S 1 (t) := P 1 (t)b 1 (t) + C (t)p 1 (t)d 1 (t) + C (t)p 1 (t) D 1 (t), S 2 (t) := P 1 (t)b 2 (t) + C (t)p 1 (t)d 2 (t) + C (t)p 1 (t) D 2 (t), t [0, T ].

41 Then, { [ dϕ(t) = A (t) S ] 1 (t)ñ1(t) 1 B1 (t) ϕ(t) [ + C (t) S ] 1 (t)ñ1(t) 1 D 1 (t) β(t) [ + S ] } 1 (t)ñ1(t) 1 S(t) + S2 (t) û 2 (t) dt (4.32) ϕ(t ) = 0, β(t)d W (t), which admits a unique G 1 t -adapted solution (ϕ( ), β( )).

42 Finally, dˆx u 1,û 2 u (t) = [Ã(t)ˆx 1,û 2 (t) + F 1 (t)ϕ(t) + B 1 (t)β(t) + B ] 2 (t)û 2 (t) dt [ u + C(t)ˆx 1,û 2 (t) + B 1 (t)ϕ(t) + F ] 3 (t)β(t) + D2 (t)û 2 (t) d W (t), (4.33) ˆx u 1,û 2 (0) = x 0, solves ˆx u 1,û 2, where

43 Ã(t) := A(t) B 1 (t)ñ1(t) 1 S 1 (t), B 2 (t) := B 2 (t) B 1 (t)ñ1(t) 1 S(t), F 1 (t) := B 1 (t)ñ1(t) 1 B 1 (t), B 1 (t) := B 1 (t)ñ1(t) 1 D 1 (t), C(t) := C(t) D 1 (t)ñ1(t) 1 S 1 (t), F 3 (t) := D 1 (t)ñ1(t) 1 D 1 (t), D 2 (t) := D 2 (t) D 1 (t)ñ1(t) 1 S(t) F 4 (t) := S 1 (t)ñ1(t) 1 S(t) + S2 (t).

44 Theorem 3.1 Let Riccati equation admit a differentiable solution P 1 ( ) and let matrix be convertible. For chosen u 2 ( ) of the leader, Problem of the follower admits an optimal control u 1 ( ) of the state feedback, where processes triple (ˆx u 1,û 2 ( ), ϕ( ), β( )) is the unique Gt 1 -adapted solution to the FBSDE (4.32)-(4.33).

45 5. Complete Information LQ Stochastic Optimal Control Problem of the Leader The leader knows the complete information F t at any time t. The state equations faced by the leader consist of BSDE (4.32) and the SDE (3.7). By (4.28), it can be written as

46 [ dx u 2 (t) = A(t)x u 2 (t) + ( Ã(t) A(t) )ˆxû2 (t) + F 1 (t)ϕ(t) + B 1 (t)β(t) + B 2 (t)u 2 (t) + ( B2 (t) B 2 (t) ) ] û 2 (t) dt [ + C(t)x u 2 (t) + F 5 (t)ˆxû2 (t) + B1 (t) ϕ(t) + D1 (t)β(t) + D 2 (t)u 2 (t) + F ] 2 (t)û 2 (t) dw (t) [ + C(t)x u 2 (t) + ( C(t) û C(t) )ˆx 2 (t) + B 1 (t) ϕ(t) + F 3 (t)β(t) + D 2 (t)u 2 (t) + ( ] D(t) D(t) )û2 (t) d W (t), dϕ(t) = [ Ã (t)ϕ(t) + C (t)β(t) + F 4 (t)û 2 (t) ] dt β(t)d W (t), x u 2 (0) = x 0, ϕ(t ) = 0. (5.34)

47 The leader s cost functional J 2 (u 2 ( )) := J 2 (u 1( ), u 2 ( )) = 1 [ T ( Q2 2 E (t)x u 1,u 2 (t), x u 1,u 2 (t) 0 + N 2 (t)u 2 (t), u 2 (t) ) dt + G 2 x u 1,u 2 (T ), x u 1,u 2 (T ) ] (5.35) This complete info conditional mean-field LQ problem with state (5.34) and cost (5.34) can be solved explicitly using a general result.

48 General conditional mean-field SMP. State dx u 2 (t) = b(t)dt + σ(t)dw (t) + σ(t)d W (t), { d 1 dq(t) = b x (t)q(t) + σx(t)k j j (t) d 2 + j=1 j=1 } σ x(t) k j j (t) g 1x (t) dt k(t)dw (t) k(t)d W (t), x u 2 (0) = x 0, q(t ) = G 1x (x u 2 (T )), (5.36) where b is a function of (t, x, ˆx, u 2, û 2, q, k, k, ˆk, ˆ k).

49 Cost: [ T ] J 2 (u 2 ( )) = E g 2 (t)dt + G 2 (x u 2 (T )) 0 Problem of the leader. Find a Gt 2 -adapted control u 2 ( ) U 2, such that J 2 (u 2( )) = inf J 2 (u 2 ( )), (5.37) u 2 U 2 subject to (5.36).

50 Define Hamiltonian H 2 ( t, x, u2, ˆx, û 2, q, k, k; y, z, z, p ) = y, b() + tr { z σ(t) } + tr { z σ(t) } + g 2 (t) p, b x (t)q + d 1 j=1 d 2 σx(t)k j j + j=1 σ x(t) k j j g 1x (t). (5.38)

51 Let (y( ), z( ), z( ), p( )) be the unique F t -adapted solution to the adjoint conditional mean-field FBSDE of the leader { dp(t) = b x(t)p(t) + E [ b ˆx (t)p(t) ] } G 1 t dt d 1 { + σx j (t)p(t) + E [ σ j ˆx (t)p(t) G t 1 ] } dw j (t) j=1 d 2 { + σ x j j (t)p(t) + E [ σ ˆx (t)p(t) G t 1 ] } d W j (t), j=1 (5.39)

52 dy = j=1 UNIVERSIDADE DE MACAU { b x y + E [ bˆx y G t 1 ] d 1 [ + σxz j j + E [ σ jˆx zj G t 1 ] ] d 2 + [ σ x(t)z j j (t) + E [ σ jˆx (t)zj (t) ] ] G 1 t j=1 n i=1 { bx [ bx (t)q(t)p i (t) + E (t)q(t)p i (t) x i ˆx i n { i=1 x i n { i=1 x i ( d1 j=1 ( d1 j=1 + g 1xx p + E [ g 1xˆx p G 1 t [ ( d1 σxk )p j j i + E ˆx i j=1 [ ( d1 σ x k )p j j i + E ˆx i j=1 G 1 t ]} σ j xk j ) p i G 1 t ]} σ j x k j ) p i G 1 t ]} ] + g2x + E [ g 2ˆx Gt 1 ] } dt zdw zd W,

53 with boundary p(0) = 0, y(t ) = G 1xx (x (T ))p(t ) + G 2x (x (T )), (5.41) where we have used φ φ ( t, x (t), ˆx (t), u 2 (t), û 2 (t)) for φ = b, σ, σ, g 1, g 2 and all their derivatives. Related work: Anderson and Djehiche (2011), Li (2012), Yong (2.13).

54 Proposition (Maximum principle of conditional mean-field FBSDE with partial information) Let u 2 ( ) U 2 be the optimal control for Problem of the leader and (x ( ), q ( ), k ( ), k ( )) be the corresponding optimal state which is the unique F t -adapted solution to (5.36). Let (y( ), z( ), z( ), p( )) be the unique F t -adapted solution to the adjoint equation. Then { [ ] } H 2 H2 E + E u 2 û 2 G1 t G2 t = 0, a.e.t [0, T ], (5.42) u 2 =u 2 (t)

55 Now, we come back to LQ problem. Define Hamiltonian H 2 ( t, x u 2, u 2, ϕ, β; y, z, z, p ) = y, A(t)x u 2 + ( Ã(t) A(t) )ˆxû2 + F 1 (t)ϕ + B 1 (t)β + B 2 (t)u 2 + ( B2 (t) B 2 (t) ) û 2 + z, C(t)x u 2 + F 5 (t)ˆxû2 + B 1 (t)ϕ + D1 (t)β + D 2 (t)u 2 + F 2 (t)u 2 + z, C(t)x u 2 + ( C(t) û C(t) )ˆx 2 + B 1 (t)ϕ + F 3 (t)β + D 2 (t)u 2 + ( D2 (t) D 2 (t) ) û 2 + 1[ p, Ã ϕ + C β + F4 û 2 + Q2 x u 2, x u 2 + ] N 2 u 2, u 2. 2 (5.43)

56 Applying the general thm, if there exists an F t -adapted optimal control u 2 ( ) U 2 for the leader, then 0 = N 2 (t)u 2(t) + F 4 (t)ˆp(t) + B 2 (t)y(t) + ( B2 (t) B 2 (t) ) ŷ(t) + D 2 (t)z(t) + F 2 (t)ẑ(t) + D 2 (t) z(t) (5.44) + ( D2 (t) D 2 (t) ) ˆ z(t), a.e. t [0, T ], where F t -adapted processes quadruple (p( ), y( ), z( ), z( )) satisfies the conditional mean-field FBSDE

57 dp(t) = [ Ã(t)p(t) + F 1 (t)y(t) + B1 (t)z(t) + B 1 (t) z(t) ] dt + [ C(t)p(t) + B 1 (t)y(t) + D 1 (t)z(t) + F 3 (t) z(t) ] d W (t), dy(t) = [ A (t)y(t) + ( Ã(t) A(t) ) ŷ(t) + C (t)z(t) + F 5 (t)ẑ(t) + C (t) z(t) + ( C(t) C(t) ) ˆ z(t) + Q2 (t)x (t) ] dt z(t)dw (t) z(t)d W (t), p(0) = 0, y(t ) = G 2 x (T ). (5.45)

58 Thanks!