Estimation in Financial Models - RiskLab

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(A2) b and are local Lipschitz continuous: for all 0
For xed 0 s < t, x 2 IR d , 2 and N 2 IN, we consider for k = 0; 1;:::;N the Euler approximation (see Appendix B.1): k = s + k t , s N Y (N) s = x Y (N) k = Y (N) k,1 + t , s N b ( k,1;Y (N) k,1 ; )+a ( k,1 ;Y (N) k,1 ; ) 1=2 (W ;s k , W ;s k,1 ) : Under (A1)-(A4) we have Y (N) N in L 1 (P ;s;x )asN ,! 1(see [45], x10.2). Y (N) t ,! X t (3.21) Theorem 2 For xed 0 s < t, x 2 IR d , 2 and N 2 IN the distribution of Y (N) t under P ;s;x has a density p N (s; x; t; ; ) with respect to the d-dimensional Lebesgue measure d . For N=1 and for N 2 p 1 (s; x; t; y; ) =(2(t , s)) ,d=2 [det(a(s; x; ))] ,1=2 ( exp , 1 [y , x , (t , s)b(s; x; )]T 2(t , s) a(s; x; ) ,1 [y , z , (t , s)b(s; x; )] ; (3.22) p N (s; x; t; y; ) =E P;s;x p1 ( N,1 ;Y (N) N,1 ;t;y; ) : (3.23) For the proof we refer to Pedersen [58]. Our aim is to show that l n;N () will indeed be close to l n () for large values of N, that is we give a result on the convergence of the approximating densities p N (s; x; t; y; ) top(s; x; t; y; ) asN ,! 1. Theorem 3 In addition to (A2) and (A3) assume for all 2 that (1) b is continuous in t and in x, and (2) a(t; x; ) a() is positive denite. 26
Page 1 and 2: Estimation in Financial Models RISK
Page 3 and 4: 5 Some Diusion Models with Explicit
Page 5 and 6: ing functions or methods based on a
Page 7 and 8: wonder why people still use the Bla
Page 9 and 10: Chapter 2 Approximations 2.1 Diusio
Page 11 and 12: 1) fX kh g: the family of discrete
Page 13 and 14: By means of Theorem 1we now obtain
Page 15 and 16: 2.2 Discrete time models by diusion
Page 17 and 18: Chapter 3 Parameter Estimation 3.1
Page 19 and 20: 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: time between observations is bounde
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 and 72: Appendix A The Kalman-Bucy Filter A
Page 73 and 74: A.2 The discrete case For reasons o
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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