Estimation in Financial Models - RiskLab

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where X 0 = 0 and t 0, the log-likelihood function l n () is unknown. The approximate log-likelihood functions (l n;N ()) 1 N=1 can be used to estimate , because (3.25) has a weak solution for all x 0 and for all >0 which is unique in law and (A4) is satised for all >0. With 0 = 5, n = 1000 and = 0:1 the Milstein-scheme (see Appendix B.2) with time-step =1000 is used to simulate (X i ) n i=0. For such a simulation the approximate log-likelihood functions l n;N () are calculated for N =1; 25; 50; 1000. The estimates that are obtained are shown in Table 3.1. N 1 25 50 1000 ^ n;N 3.98 4.76 5.20 5.04 Table 3.1: Example 1. The estimates corresponding to the functions l n;N (). Approximation of the transition densities of X by means of the Kalman lter The following approach is proposed by Pedersen [59, 63]. We consider the stochastic system X i = D i X i,1 + S i + " i ; i =1;:::;n; (3.26) where (X i ) n i=0 are random d 1 vectors, (D i ) n i=0 are non-random d d matrices, (S i ) n i=1 are non-random d 1vectors, 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. The nonrandom elements in (3.26) including x 0 and (V i ) n i=0 are given up to the parameter 2 IR p . Though this chapter is about diusion models, we nevertheless look here at stochastic dierence equations of the type (3.26), since as a particular case of (3.26) we will consider later discretely observed diusion processes given as solutions to linear stochastic dierential equations in the narrow sense. These are equations of the following kind dX t =(A t X t + a t )dt + B t dW t ; X 0 = x 0 ; t 0; (3.27) where A :[0; 1) 7! M dd , a :[0; 1) 7! IR d and B :[0; 1) 7! M dm (d m) are deterministic functions of t and W is an m-dimensional Wiener process. Why (3.27) is a particular case of (3.26), will be explained below. 29
Considering the system (3.26) we distinguish two cases depending on whether X can be observed completely or only partially, i.e. wether all or only a few coordinates of X can be observed. First, if the complete observations of X 0 ;X 1 ;:::;X n are given, we can use the log-likelihood function to estimate the parameter , see (3.15). However, the case where X 0 ;X 1 ;:::;X n can only be observed partially and possibly with measurement errors often arises in practice. As mentioned, with 'partially observed' we do not mean partially in time but in coordinates of X i . We 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; (3.28) where (T i ) n i=0 are non-random kd matrices (k d), (U i ) n i=0 are non-random 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 matrices (T i ) specify the observable parts of (X i ), the vectors (U i ) are other inputs and the vectors (e i ) are measurement errors. It is obvious that both the case of complete observations, Y i = X i , and the case of partial observations without measurement errors are contained in the general case (3.28). As an application to the case of incomplete observations think of a stochastic volatility model that can be seen as a multi-dimensional process with the volatility process as the unobservable coordinates. In the case where all non-random elements in (3.26) and (3.28) including x 0 , (V i ) n i=0 and (W i ) n i=0 are known, we want to obtain the, in some sense, best predictions of X 0 ;X 1 ;:::;X n from given observations of Y 0 ;Y 1 ;:::;Y n . The conditional expectations (E(X i jY i )) n i=0, respectively (E(X i jY i,1 )) n i=1, are the best predictors of X i given Y i (Y T 0 ;:::;Yi T ) T , respectively given Y i,1 (Y T 0 ;:::;Yi,1) T T , in the sense of minimal variance (see Appendix A.1). They can be calculated by means of an iterative procedure called the Kalman-Bucy lter. The Kalman-Bucy lter and ltering theory in general are well-studied for continuous time stochastic processes with continuous observations (see Liptser and Shiryaev [51], Kallianpur [43], ksendal [56] and Appendix A). Here we consider discrete time stochastic processes, and discretely observed continuous time stochastic processes in particular. Belowwe give the iterative procedure used for calculating (E(X i jY i )) and (E(X i jY i,1 )). The important point is that the Kalman-Bucy lter gives besides the predictors (E(X i jY i )) and (E(X i jY i,1 )) the density p iji,1 for the conditional distribution of Y i given Y i,1 . That means we are able to calculate the transition 30
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: After these theoretical considerati
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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