Estimation in Financial Models - RiskLab

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Let P Kt (X t ) denote the orthogonal projection from L 2 (; F;P)onto the subspace K t . The following statement holds ^X t = P Kt (X t )=E(X t jG) : We will concentrate on the linear case, which allows an explicit solution in terms of a stochastic dierential equation for ^X t . This method is called the Kalman-Bucy lter. A.1.2 The linear ltering problem To focus on the main ideas consider only the one-dimensional case dX t = F (t)X t dt + C(t) dU t ; (System) (A.3) dZ t = G(t)X t dt + D(t) dV t ; (Observations) (A.4) where the real functions F (t), G(t), C(t), D(t) are bounded on bounded intervals. Furthermore assume that D(t) is bounded away from 0 on bounded intervals, Z 0 =0,X 0 is normally distributed and independent of fU t g, fV t g and E[X 0 ] = 0 (see ksendal [56], p. 48f). The ideas and techniques in the one-dimensional case can be extended in an analogous way to the multidimensional case. By means of some technical transformations of P Kt (X t ) and calculations we obtain The Kalman-Bucy Filter The solution ^X t = P Kt (X t )=E(X t jG) of the linear ltering problem (A.3), (A.4) satises the stochastic dierential equation d ^X t = " # F (t) , G2 (t)S(t) D 2 (t) ^X t dt + G(t)S(t) D 2 (t) dZ t ; ^X 0 =E[X 0 ] ; where S(t) =E[(X t , ^X t ) 2 ] satises the \Riccati-equation" (A.5) dS dt =2F (t)S(t) , G2 (t) D 2 (t) S2 (t)+C 2 (t) ; (A.6) with S(0) =E[(X 0 ,E[X 0 ]) 2 ]. 71
A.2 The discrete case For reasons of completeness we again note here the Kalman-Bucy lter for the discrete case, as already given in section 3.1.2. Consider the stochastic system X i = D i X i,1 + S i + " i ; i =1;:::;n; (System) (A.7) where fX i g n i=0 are random d 1vectors, fD i g n i=0 are non-random d d matrices, fS i g n i=1 are non-random d1 vectors, X 0 N d (x 0 ;V 0 ), " i N d (0;V i ), i = 1;:::;n and X 0 , " 1 ;:::, " n are stochastically independent. Assume the observable quantities are Y 0 ;Y 1 ;:::;Y n given by Y i = T i X i + U i + e i ; i =0; 1;:::;n; (Observations) (A.8) where fT i g n i=0 are non-random k d matrices (k d), fU i g n i=0 are nonrandom k 1 vectors, e i N k (0;W i ) and X 0 ;" 1 ;:::;" m ;e 0 ;e 1 ;:::;e n are stochastically independent, i =0; 1;:::;n. The Kalman-Bucy lter Under some assumptions (see Pedersen [59], pp. 4{5), we have for given observations y 0 ;y 1 ;:::;y n of Y 0 ;Y 1 ;:::;Y n X i jY i = y i N d i (y i ); i ; (A.9) X i jY i,1 = y i,1 N d Di i,1 (y i,1 )+S i ;R i ; (A.10) Y i jY i,1 = y i,1 N d Ti (D i i,1 (y i,1 )+S i )+U i ;T i R i T T i + W i ;(A.11) where R i = D i i,1 D T i + V i is positive denite, and where 0 (y 0 ) = x 0 + V 0 T T 0 T0 V 0 T T 0 + W 0 ,1 (y0 , T 0 x 0 , U 0 ); (A.12) 0 = V 0 , V 0 T T 0 i (y i ) = D i i,1 (y i,1 )+S i + R i T T i T0 V 0 T T ,1 0 + W 0 T0 V 0 ; (A.13) Ti R i Ti T ,1 + W i (y i , T i Di i,1 (y i,1 )+S i , Ui ) ; (A.14) i = R i , R i T T i (T i R i T T i + W i ) ,1 T i R i : (A.15) 72
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Estimation in Financial Models RISK
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5 Some Diusion Models with Explicit
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ing functions or methods based on a
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wonder why people still use the Bla
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Chapter 2 Approximations 2.1 Diusio
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1) fX kh g: the family of discrete
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By means of Theorem 1we now obtain
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2.2 Discrete time models by diusion
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Chapter 3 Parameter Estimation 3.1
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which in the case (3.6) has the for
Page 21 and 22: (see [49], p.249f). Asymptotic norm
Page 23 and 24: a convergence in probability result
Page 25 and 26: time between observations is bounde
Page 27 and 28: For xed 0 s < t, x 2 IR d , 2 an
Page 29 and 30: After these theoretical considerati
Page 31 and 32: Considering the system (3.26) we di
Page 33 and 34: An Application As an application of
Page 35 and 36: If depends on we use the same est
Page 37 and 38: The asymptotic properties of the es
Page 39 and 40: Corollary 2 For we have F (X (i,1)
Page 41 and 42: disadvantage of ~G n and G n motiv
Page 43 and 44: 3.2 Discrete models In section 1.2
Page 45 and 46: The sequence hMi n is called the sq
Page 47 and 48: of time varying variances and covar
Page 49 and 50: The Hessian is @ 2 l t @@ 0 = , 1 @
Page 51 and 52: 3.2.3 The Bayesian estimation metho
Page 53 and 54: f() with respect to (jx). If 1 ; 2
Page 55 and 56: p( 1 j 2 ) and we have the a priori
Page 57 and 58: Chapter 4 Nonparametric Estimation
Page 59 and 60: for model (4.2) and the estimator ^
Page 61 and 62: Chapter 5 Some Diusion Models with
Page 63 and 64: and thus X t = t;t0 X t0 + Z t t 0
Page 65 and 66: If we do not specialize at this poi
Page 67 and 68: where we assume the choice of b 1 b
Page 69 and 70: where h is given by (5.42). Again w
Page 71: Appendix A The Kalman-Bucy Filter A
Page 75 and 76: we can write the Euler scheme (B.1)
Page 77 and 78: Appendix C The It^o Formula C.1 The
Page 79 and 80: Appendix D The Radon-Nikodym Theore
Page 81 and 82: [10] Billingsley,P., 1968, \Converg
Page 83 and 84: [38] Ibragimov, I.A. and R.Z. Has'm
Page 85: [63] Pedersen, A.R., 1994, \Statist
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