Estimation in Financial Models - RiskLab

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Inserting U in (5.30) gives h(x) =a 1 h(x)+a 2 and hence a 1 = and a 2 =0. We have dY t = Y t dt + dW t with its solution Z t Y t = e t Y 0 + e t e ,s dW s : With C =0wehave Y t = h(Y t ), especially Y 0 = h(X 0 ), and we obtain (5.52). The applications of example 1 we gave can all be modied to represent applications to example 3. For instance, consider dX t = q 2a X t = e at X 0 + X t X t +1 Z t 0 0 q dt +2 X t dW t ; e ,as dW s : (5.53) We remark that some examples of nonlinear reducible stochastic dierential equations that are not included in the preceding three cases are listed in [45], p.124-126. Finally, we again put stress on the fact that the two models (1.4) and (1.5) considered in chapter 1 are not reducible and cannot be solved explicitly. 69
Appendix A The Kalman-Bucy Filter A.1 The continuous case For the Kalman-Bucy lter in the continuous case we refer to e.g. Kallianpur [43], x10. In our notation we follow ksendal [56], x4, pp. 46{68. A.1.1 The general ltering problem Consider dX t = b(t; X t ) dt + (t; X t ) dU t ; (System) (A.1) dZ t = c(t; X t ) dt + (t; X t ) dV t ; (Observations) (A.2) where U and V are independent Brownian Motions, U t p-dimensional, V t r-dimensional. The ltering problem is the following. Given the observations Z s satisfying (A.2) for 0 s t, what is the best estimate ^X t of the state X t of (A.1) based on these observations? To solve this problem we formulate it mathematically. Let G t be the -algebra generated by fZ s (); s tg, (; F;P) the probability space corresponding to the (p + r)-dimensional Brownian motion (U t ;V t ) and K = fS : 7! IR n ; S 2 L 2 (; F;P); SG,measurableg : \Based on the observations Z s " means that the estimate ^X t is G t -measurable. ^X t shall be \the best estimate based on the observations Z s " which means that E[jX t , ^X t j 2 ]= inf S2K fE[jX t , Sj 2 ]g: 70
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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
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 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: where h is given by (5.42). Again w
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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