Estimation in Financial Models - RiskLab

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Example 2 The solutions of the stochastic dierential equation dX t =( + X t ) dt + (X t ) dW t ; (3.54) where X 0 = x 0 and the function takes positive values in IR, are called mean-reverting processes (see also model (1.5)). The unknown parameters are and . Our aim is to be able to calculate the martingale estimating functions ~G n and G n. Lemma 1 The function f(t) E ; (X t jX 0 ) solves f 0 (t) = + f(t): (3.55) Proof: Write (3.54) in integral form X t = X 0 + Conditioning on X 0 we have and equivalently Z t 0 ( + X s )ds + Z t 0 (X s )dW s : Z t E ; (X t jX 0 ) = E ; (X 0 jX 0 )+E ; ( + X s )dsjX 0 0 (X s )dW s jX 0 ; 0 | {z } =0 +E ; Z t E ; (X t jX 0 )=X 0 + t + Z t 0 E ; (X s jX 0 )ds: We conclude dE ; (X t jX 0 ) = + E ; (X t jX 0 ); dt and the claim follows. Note that the function f r (t) =E ; (X t jX r ), 0 r t, also solves (3.55), for the proof stays the same apart from E ; Z t 0 (X s )dW s jX r = Z r 0 (X s )dW s ; which is independent of t and thus plays no role in the derivative. 2 37
Corollary 2 For we have F (X (i,1) ; ; ) =E ; Xi jX (i,1) F (x; ; ) =xe + (e , 1): (3.56) Proof: The solution of f 0 (t) = + f(t); f(t 0 )=f 0 ; t t 0 ; is f(t) =f 0 e (t,t 0) + (e(t,t 0) , 1): Hence we have for E ; (X ti jX ti,1 ) f(t i ) with constant t i , t i,1 for all i E ; (X ti jX ti,1 ) = E(X ti,1 jX ti,1 ) e + (e , 1) and the claim follows. 2 = X ti,1 e + (e , 1); With (3.56) we are now able to calculate ~G n and G n in the following way " X n 1 ~G n (; ) = 2 X i , X (i,1) e + (X (i,1) ) (1 , e ) ; nX i=1 nX i=1 G n(; ) = i=1 X (i,1) 2 (X (i,1) ) " n X i=1 X i , X (i,1) e + (1 , e )# T ; e , 1 X i , X (i,1) e + (X (i,1) ; ; ) (1 , e ) ; e (X (i,1) + )+ 2 (1 , e ) (X (i,1) ; ; ) Considering G + is not interesting since F is known. X i , X (i,1) e + (1 , e )# T : The estimation equation ~G n (; ) = 0 can be solved explicitly. Abbreviating 2 i,1 2 (X (i,1) )we obtain e ~ n = Pni=1 X (i,1) 2 i,1 Pni=1 X i 2 i,1 Pni=1 X (i,1) 2 i,1 , Pni=1 X (i,1) X i 2 i,1 2 , Pni=1 X 2 (i,1) 2 i,1 38 Pni=1 1 Pni=1 1 2 i,1 2 i,1 (3.57)
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 and 36: If depends on we use the same est
Page 37: The asymptotic properties of the es
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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