Estimation in Financial Models - RiskLab

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where (X (i,1) ; ) =E h (Xi , F (X (i,1) ; )) 2 jX (i,1) i ; i =1;:::;n: (3.51) The function G n() is within the class (3.49) in some sense "closest" to the score function based on the usually unknown exact likelihood function. It should be mentioned here that for small the martingale estimating function ~ G n (), dened in (3.48), is a rst order approximation in of G n, that means ~G n () is approximately optimal. (3) In some cases there are possibly numerical problems in calculating F _ (x; ) in (3.50). One way to solve these problems involves approximating F _ (x; ) up to the order O( 2 ). That leads to a third martingale estimating function G + n () = nX _b(X(i,1) ; ) + 1 _ 2 2 b(X(i,1) ; )b 0 (X (i,1) ; ) i=1 +b(X (i,1) ; ) _ b 0 (X (i,1) ; )+ 1 2 (_2 (X (i,1) ; )b 00 (X (i,1) ; ) + 2 (X (i,1) ; ) b _ i 00 (X i , F (X (i,1) ; )) (X (i,1) ; )) : (3.52) (X (i,1) ; ) Altogether we have now found expressions for three dierent zero-mean P - martingale estimating functions. As for the multi-dimensional case, suppose is k-dimensional, fX t g and b(X t ; ) are d-dimensional, is a dm-dimensional matrix with T positive denite and the Wiener process fW t g is m-dimensional. Then the k 1- dimensional martingale estimating functions ~G n and G n have the form and ~G n () = G n() = nX _b(X (i,1) ; ) T (X (i,1) ; )((X (i,1) ; ) T ,1 i=1 X i , F (X (i,1) ; ) nX F _ (X (i,1) ; ) T (X (i,1) ; ) ,1 X i , F (X (i,1) ; ) ; i=1 where is assumed to be positive denite and _ b and _ F denote the d k- dimensional matrices of partial derivatives with respect to the components of . 35
The asymptotic properties of the estimator ^ n we obtain from the martingale estimating functions (3.48), (3.50) and (3.52), or more generally from the class of martingale estimating functions G n of the form (3.49), are discussed by Bibby and Srensen [8]. Under natural regularity conditions (see Bibby and Srensen [8], pp. 7{9) we have Theorem 4 An estimator ^ n , which solves the equation G n (^ n )=0; exists with probability tending to one as n ,! 1 under P 0 . Moreover, as n ,! 1, ^ n ,! 0 in probability under P 0 and ^ n is asymptotically normal in distribution under P 0 . For the proof we refer to [8]. As a rst example we consider the Ornstein-Uhlenbeck process where the transition densities are well-known. Example 1 The Ornstein-Uhlenbeck process is the solution of the stochastic dierential equation dX t = X t dt + dW t ; (3.53) with X 0 = x 0 . In this case the drift coecientisb(x; ) =x and the diusion coecient (x; ) is assumed to be known. The transition probability is normal with mean F (x; ) =xe and variance () = 2 2 (e2 , 1). Hence the estimating function ~G n has the form ~G n () = 1 2 nX i=1 and we obtain ^ n as solution of ~G n () =0: X (i,1) (X i , X (i,1) e ) ; ^ n = 1 P ni=1 log X (i,1) X i Pni=1 : X 2 (i,1) The estimators ^ n we obtain from the martingale estimating functions G + and G are the same because G + and G are proportional to ~G. In the next example we consider a wider class of stochastic processes. 36
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 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: If depends on we use the same est
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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