Đang chuẩn bị nút TẢI XUỐNG, xin hãy chờ
Tải xuống
Tham khảo tài liệu 'stochastic control part 16', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 592 Stochastic Control Besides the optimal filtered wealth process XỊ n x J0 n dSu is a solution of the linear equation X x - P2u fiu 2 ẢụYụ 2 1 Pu pUYu 2 Xu dSu fiu 1 pu uYu 1 hu dS 1 P2u p2Yu 2 u. 4.7 Proof.Similarly to the case of complete information one can show that the optimal strategy exists and that VH t x is a square trinomial of the form 4.3 see e.g. Mania Tevzadze 2003 . More precisely the space of stochastic integrals fr T Ì Jt T G Jt nudSu n e n G I is closed by Proposition 2.1 since M is G-predictable. Hence there exists optimal strategy n t x e n G and UH t x E H x n t x dSu 2 Gt . Since fT n t x dSu coincides with the orthogonal projection of H x e L2 on the closed subspace of stochastic integrals then the optimal strategy is linear with respect to x i.e. n t x nu t xn1 t . This implies that the value function UH t x is a square trinomial. It follows from the equality 3.14 that VH t x is also a square trinomial and it admits the representation 4.3 . Let us show that Vt 0 Vt 1 and Vt 2 satisfy the system 4.4 - 4.6 . It is evident that Vt 0 VH t 0 ess inf E nen G WudS u Ht y n2 1 pty 2nuhu d M u Gt 4.8 and Vt 2 V0 t 1 ess inf E nen G í T 2 p T 1 It WudSu 1 n2 l 1 ptyd M u Gt 4.9 Therefore it follows from the optimality principle taking n 0 that Vt 0 and Vt 2 are RCLL G-submartingales and Vt 2 E Vt 2 Gt 1 Vt 0 E E2 H Gt Gt E H2 Gt . S nc e V . 2 Vt 0 Vt 2 V t . 4.M the process Vt 1 is also a special semimartingale and since Vt 0 2Vt 1 x Vt 2 x2 VH t x 0 for all x e R wehave V2 1 Vt 0 Vt 2 hence V2 1 E H Gt . Expressions 4.8 4.9 and 3.13 imply that Vt 0 E2 H Gt Vt 2 1 and VH T x x E H Gt 2. Therefore from 4.10 wehave Vt 1 EfH T and V 0 V 1 and V 2 satisfy the boundary conditions. Thus the coefficients Vt i i 0 1 2 are special semimartingales and they admit the decomposition Vt i Vt i At i Vs i dMis mt i i 0 1 2 4.11 Mean-variance hedging under partial information 593 where m 0 m 1 and m 2 are G-local martingales strongly orthogonal to M and A 0 A 1 and
