Estimation in Financial Models - RiskLab

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Then p(s; x; t; y; ) exists, and for all 0 s
After these theoretical considerations we deal with the actual calculation of l n;N () for large values of N. In order to maximize l n;N () numerical algorithms usually require the value of l n;N () in a nite numberofpoints . For N =1 l n;N () is explicitly given, because p 1 has a closed expression, but for N 2 this is in general not the case. For calculating l n;N () for N 2, we have to know how to calculate p N (s; x; t; y; ) for all 0 s < t, x; y 2 IR d and 2 . Considering again equation (3.23) p N (s; x; t; y; ) =E P;s;x p1 ( N,1 ;Y (N) N,1 ;t;y; ) ; the idea is to calculate p N (s; x; t; y; ) by nding a good approximation to the right hand side. Denote by (Uk m ) N,1;M k=1;m=1 an i.i.d. sample from the r- dimensional standard normal distribution. Then (Y m ) M m=1 =(YN,1) m M m=1 given by the Euler approximation Y m 0 = x; m =1;:::;M Yk m = Y m k,1 + t , s N b( k,1;Yk,1; m )+ s t , s N ( k,1;Y m k,1; ) U m k for k =1;:::;N, 1 and m =1;:::;M, has the same distribution as an i.i.d. sample of Y (N) N,1 under P ;s;x .Thus we are able to approximate the right hand side of (3.23), while we calculate 1 M MX m=1 p 1 ( N,1 ;Y m ;t;y; ) (3.24) by means of the sequence (U m k ) N,1;M k=1;m=1, for M chosen suciently large, and thereby we are able to calculate p N (s; x; t; y; ) with any given accuracy. From a practical point of view it is convenient to simulate the sample (Uk m ) N,1;M k=1;m=1 once (see Kloeden and Platen [45]) and store it. Then it can be used to calculate p N (s; x; t; y; ) for all values of 0 s
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: For xed 0 s < t, x 2 IR d , 2 an
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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