Dynamic Asset Allocation - Identifying Regime Shifts in Financial Time Series to Build Robust Portfolios

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1 Downloaded from orbit.dtu.dk on: Jan 22, 2019 Dynamic Asset Allocation - Identifying Regime Shifts in Financial Time Series to Build Robust Portfolios Nystrup, Peter Publication date: 2018 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Nystrup, P. (2018). Dynamic Asset Allocation - Identifying Regime Shifts in Financial Time Series to Build Robust Portfolios. DTU Compute. DTU Compute PHD-2017, Vol General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

25 (rt) (r 2 t )

31 t

32 t t

37 l 1 l 1 l 2 l 1 l 2

39 l 1

49 t

50 l 1

52 l 1

61 t

62 {S t : t N} t N (S t+1 S t,..., S 1 ) = (S t+1 S t ). (S u+t = j S u = i) = γ ij (t) u Γ (t) = {γ ij (t)} π πγ = π π = 1 π = δ δ δ i = (S 1 = i) {S t } m {X t : t N} m X (t) S (t) t ( S t S (t 1)) = (S t S t 1 ), t = 2, 3,..., ( X t X (t 1), S (t)) = (X t S t ), t N. S t X t S t X t S t {S t } {X t } X t = µ St + ε St, ε St N ( 0, σ 2 S t ),

63 µ St = { { µ 1, S t = 1, σ 2 [ µ 2, S t = 2, σ2 S t = 1, S t = 1, 1 σ2, 2 S t = 2, Γ = γ12 γ 12 γ 21 1 γ 21 k ρ Xt (k θ) = π 1 (1 π 1 ) (µ 1 µ 2 ) 2 σ 2 λ k, ρ X 2 t (k θ) = π 1 (1 π 1 ) ( µ 2 1 µ σ1 2 σ2 2 [Xt 4 θ] [Xt 2 θ] 2 λ k, θ σ 2 = [X t θ] λ = γ 11 + γ 22 1 Γ λ ) L T (θ) = (X (T ) = x (T ) θ = δ (x 1 ) Γ (x 2 ) Γ (x T ), (x) p i (x) = (X t = x S t = i) i {1, 2,..., m} X t ) 2 ].

64 ( t i ) = γ t 1 ii (1 γ ii ). i d i (u) = (S t+u+1 i, S t+u v = i, v = 0,..., u 2 S t+1 = i, S t i) γ ij = (S t+1 = j S t+1 i, S t = i) i j γ ii = 0 j γ ij = 1 t p ij ( t) = (S (t + t) = j S (t) = i) t 0 p ij (0) = 0 i j (t) =. t 0

65 p ij ( t) p p ij ( t) p ij (0) ij (0) = t 0 t = t 0 = q ij (S (t + t) = j S (t) = i) t q ii = q i = j i q ij = {q ij } q ij q i π { π = π = 1. π (t) = {p ij (t)} (t) t (0) = = (t) (t) = e t (0) = e t. i q i > 0 j i q ij /q i δ (t) = e t = (1). q i γ ii q i = ˆγ ii.

66 q i q i 1 t p ij = o ( t), i j 2, p ii ( t) = 1 q i t + o ( t), p i,i 1 ( t) = w i q i t + o ( t), p i,i+1 ( t) = (1 w i ) q i t + o ( t), i {1, 2,..., m}, t 0 o( t) t = 0 m = m (m + 1) = i i + 2 i + 1 i i + 2 q 1 (1 w 1 ) q w 1 q 1 w 2 q 2 q 2 (1 w 2 ) q =. (1 w m ) q m 0 0 w m q m q m

67 r t = (P t ) (P t 1 ) P t t

68 rt r t JB [ ; ] [0.0116; ] [ 0.75; 0.30] [8.5; 14.1] [6356; 25803] JB = T ( ) 2 + ( 3)2 T 6 24

69 ( rt θ) θ = 1 θ = 0.75 θ = 0.5 θ = 0.25 ( rt θ) θ = 1 θ = 1.25 θ = 1.5 θ = 1.75 N N t N N t N t t

70 N (2) N (2) t (2) N (3) N (3) t (3) N (3) N (4) N (4) t (4) N (4) k 0.95 (100 k) t t

71 r t [ ; ] [0.0116; ] [ 0.75; 0.30] [8.5; 14.1] N (2) N (2) t (2) N (3) N (3) t (3) N (3) N (4) N (4) t (4) N (4) N N t N N t N r t ± 4 σ

72 N (2) N (2) t (2) N (3) N (3) t (3) N (3) N (4) N (4) t (4) N (4) t = 2 L + p T T p = 2 L + 2p

73 ( rt θ) θ = 1 θ = 0.75 θ = 0.5 θ = 0.25 ( rt θ) θ = 1 θ = 1.25 θ = 1.5 θ = 1.75

74 t

76 m Γ µ 10 4 σ δ (0.002) (1.1) (0.01) (0.2) (0.004) (4.9) (0.14) (0.004) (0.001) (0.003) (0.002) (0.003) (0.010) 10.1 (1.3) 0.5 (2.5) 12.4 (12.8) 0.32 (0.01) 1.28 (0.04) 7.08 (0.50) 1.0 (0.1) 0.0 (0.1) (0.005) (0.006) (0.00) (0.005) (0.003) (0.001) (0.000) (0.006) (0.004) (0.000) (0.002) (0.026) 10.8 (1.5) 3.3 (2.7) 2.9 (5.7) 29.9 (31.2) 0.29 (0.01) 0.96 (0.05) 2.39 (0.14) (1.73) 1.0 (0.2) 0.0 (0.2) 0.0 (0.1) 0.0 m

77 m Γ 1 p r 10 µ 10 4 σ δ (0.002) (0.1) (1.3) (0.01) (0.4) (0.010) (0.2) (5.7) (0.18) (0.000) (0.013) (0.000) (0.003) (0.037) (0.052) 0.3 (0.1) 1.7 (1.7) 5.5 (33.0) 10.5 (1.3) 1.0 (3.0) 14.3 (12.7) 0.31 (0.01) 1.51 (0.07) 7.26 (0.54) 1.0 (0.3) 0.0 (0.3) (0.000) (0.000) (0.022) (0.020) (0.000) (0.161) (0.000) (0.000) (0.006) (0.024) (0.124) (0.098) 0.5 (0.5) 1.0 (0.7) 31.2 (124.8) 4.2 (68.9) 11.6 (1.4) 2.1 (2.8) 2.4 (5.2) 30.6 (31.8) 0.25 (0.01) 1.05 (0.06) 2.38 (0.17) (2.26) 1.0 (0.3) 0.0 (0.2) 0.0 (0.2) 0.0 m p r m Γ 1 p r 10 µ 10 4 σ t δ (0.002) (0.1) (1.2) (0.02) (1.2) (0.4) (0.008) (0.2) (4.8) (0.17) (1.0) (0.042) (0.144) (0.045) (0.009) (0.047) (0.055) 7.4 (7.8) 10.5 (22.2) 5.2 (28.5) 10.5 (1.1) 1.3 (2.7) 12.2 (13.1) 0.25 (0.01) 1.16 (0.07) 4.96 (0.75) 6.7 (1.4) 22.4 (467.9) 7.2 (7.4) 1.0 (0.3) 0.0 (0.3) (0.000) (0.000) (0.103) (0.106) (0.000) (0.148) (0.000) (0.000) (0.013) (0.052) (0.115) (0.110) 8.9 (10.2) 18.6 (34.2) 34.3 (115.9) 3.9 (61.9) 10.6 (1.4) 4.1 (2.2) 1.9 (5.2) 29.3 (31.6) 0.23 (0.01) 0.86 (0.05) 2.22 (0.16) 9.56 (2.44) 6.8 (1.5) 24.8 (16.4) 49.0 (98.2) 13.8 (11968) 1.0 (0.2) 0.0 (0.2) 0.0 (0.0) 0.0 m t p r t t

78 m µ 10 4 σ δ (0.003) (1.4) (0.01) (0.003) (2.5) (0.03) (0.019) (0.003) (0.003) (12.1) (0.26) (0.005) (0.001) (0.004) (0.002) (0.003) (0.002) (0.005) (0.013) 11.1 (1.6) 3.6 (2.6) 3.2 (5.2) 29.2 (25.9) 0.29 (0.01) 0.95 (0.03) 2.39 (0.10) (0.82) m N N t N N t N

79 r t [ ; ] [0.0113; ] [ 0.56; 0.24] [7.0; 10.9] N (2) N (2) t (2) N (3) N (3) t (3) N (4) N N t N N t N N (2) N (2) t (2) N (3) N (3) t (3) N (4)

83 t

84 t {S t : t N} t N (S t+1 S t,..., S 1 ) = (S t+1 S t ). (S u+t = j S u = i) = γ ij (t) u {S t } m {(S t, X t )} m S (t) X (t) t ( S t S (t 1)) = (S t S t 1 ), t = 2, 3,..., ( X t X (t 1), S (t)) = (X t S t ), t N.

85 S t X t S t {X t } {S t } {S t } {X t } X t = µ St + ε St, ε St N ( 0, σ 2 S t ), { µ 1, S t = 1, µ St = µ 2, S t = 2, { σs 2 σ 2 t = 1, S t = 1, σ2, 2 S t = 2, [ 1 γ12 γ Γ = 12 γ 21 1 γ 21 k ]. ρ Xt (k θ) = π 1 (1 π 1 ) (µ 1 µ 2 ) 2 σ 2 λ k ρ X 2 t (k θ) = π 1 (1 π 1 ) ( ) µ 2 1 µ σ1 2 σ2 2 2 [Xt 4 θ] [Xt 2 θ] 2 λ k, π 1 λ = γ 11 + γ 22 1 Γ λ Γ λ = 1

86 ( t i ) = γ t 1 ii (1 γ ii ).

87 2 f f f = 10

88 ˆθ t = θ t n=1 ( ) X w n X (n 1) n, θ = l t (θ) θ w n = 1 w n = λ t n

89 0 < λ < 1 N = 1 1 λ. l t (θ) ˆθ t 1 θ ˆθ t ˆθ t = ˆθ )] 1 ) t 1 [ θθ lt (ˆθt 1 θ lt (ˆθt 1. ) t θθ lt (ˆθt 1 = θθ λ t n X (X (n 1) n, ˆθ ) t 1 = n=1 t λ t n X θθ (X (n 1) n, ˆθ ) t 1 n=1 t ( )) λ t n I t (ˆθt 1 n=1 = 1 λt 1 λ ˆθ t ˆθ t 1 + ( I t (ˆθt 1 )), A )] 1 ) [I t (ˆθt 1 θ lt (ˆθt 1, (t, N ) A

90 1 λ 1 λ t 1 (t,n ) N t t 0 > 0 [ θθ l t ] = [ θ l t θ l t] ) I t (ˆθ = 1 t t n=1 = t 1 t ) ) θ l n (ˆθ θ l n (ˆθ 1 t 1 { t 1 n=1 ) ) θ l n (ˆθ θ l n (ˆθ } ( X + θ X (t 1) t, ˆθ ) ( X θ X (t 1) t, ˆθ ) ) = I t 1 (ˆθ + 1 [ ( X θ X (t 1) t, t ˆθ ) ( X θ X (t 1) t, ˆθ ) (ˆθ) ] It 1. r t = (P t ) (P t 1 ) P t t

91 rt r t t 2.53 t t 22.9 r t ± 4 σ r t σ

92 (r 2 t ) (r 2 t ) (r 2 t ) r t ± 4 σ

93 t

94 µ1 µ2 σ 2 1 σ 2 2 γ11 γ22

95 µ1 µ2 σ 2 1 σ 2 2 γ11 γ22 ν t

96 t t t N = 250 t 0 = 250 A = 1.25

97 -0.02 µ µ Long memory and time-varying parameters Year σ σ12 1e-04 3e-04 5e-04 Year Year Year Year 2000 γ Year γ Figure 5: Parameters of a two-state Gaussian HMM estimated adaptively using an eﬀective memory length Neff = 250. The dashed lines are the MLE for the full series and the gray areas are approximate 95% conﬁdence intervals based on the proﬁle likelihood functions.

98 (r 2 t ) (r 2 t ) r t ± 4 σ =1700 N =1700 Nt N=250 N ( ) N =250 N t ( =1700 ) Nt r t ± 4 σ

99 1: : : : =1700 =1700 N Nt N =250 N t t t t

100 N Nt N =1700 =1700 Nt N =250 N t t t t θ t X t+1 t t + 1

101 N =1700 =1700 Nt N =250 N θ t t θ t t θ t

102 t t

103 t

104

105

106

107

108

109

110

111 {X t : t N} t N (X t+1 X t,..., X 1 ) = (X t+1 X t ). (X t+1 = j X t = i) = γ ij r t = (P t) (P t 1 ) P t t

112 Y t X t N ( µ Xt, σ 2 X t ), µ Xt = { { µ 1, X t = 1, σ 2 [ µ 2, X t = 2, σ2 X t = 1, X t = 1, 1 σ2, 2 X t = 2, Γ = γ12 γ 12 γ 21 1 γ 21 ]. X t Y t X t ( t i ) = γ t 1 ii (1 γ ii ).

113 t t t + 1 (γ ii 0.5) t + 1 t t + 1 t t + 1 t+1

114

115

116

117 t

118

119

120

121

122

123

124

125

126 r t = (P t) (P t 1 ) P t t

127 p = 0.5 1/N

128 p = 0.5 p = 0.5 p = 1 p X w (p, X) = p w (X) + (1 p) w w (X) = (25, 5, 10, 5, 5, 5, 5, 0, 0, 0) /60 (X) + (0, 0, 0, 0, 0, 0, 0, 10, 10, 20) /40 (X) (X) 1 X = (X) 0 X =

129 {X t : t N} t N (X t+1 X t,..., X 1 ) = (X t+1 X t ). (X t+1 = j X t = i) = γ ij Y t X t N ( µ Xt, σ 2 X t ), µ Xt = { { µ 1, X t = 1, σ 2 µ 2, X t = 2, σ2 X t = 1, X t = 1, σ2, 2 X t = 2, Γ = [ ] 1 γ12 γ 12. γ 21 1 γ 21 X t Y t Y t X t ( t i ) = γ t 1 ii (1 γ ii ).

130 {Y t : t 1, 2,..., T } 1 1/ T T

131 t = T = 1 T ( X t = i Y 1, Y 2,..., Y T ) i t T T i ( X t = i Y 1, Y 2,..., Y T ) > X t = i ( X t = i Y 1, Y 2,..., Y T ) t = t + 1 T = (t, T ) T = T + 1 t t (γ ii 0.5) t + 1 t

132 t + 1 t t + 1 t t T t T + 1 T t t + 1 t + 1 T T + 1 t + 2 T t t + 1

133 p p p p p = 0 p p p p

134 (p = 0.5) p = 0.5 p = 0.5

135 (p = 0.5) (p = 0.8) p = 0.5 p = 0.5 p = 0.8 p = 0.5 p = 0.8 p = 0.5 t

136 (p = 0.5) p = 0.5 t X ( f t [ = Xt 20, X t 19,..., X )] t, [ ] X t = i ( X t = i Y 1, Y 2,..., Y t)

137

138 t 1/N

139

140

141

142

143

144 t r r t t = t i=t 20 (P i/p i 1 ) 2 P t t

145 t t+21 r t = (P t/p t 1 )

146 r t t r 0.24

147 τ X i { F 0 i < τ, F 1 i τ. F 0 t = τ F 1 x 1, x 2,... X 1, X 2,... X i τ t k < t k { F 0 i < k, H 0 : i X i F 0, H 1 : X i F 1 i k.

148 k D k,t D k,t t D k,t > h k,t h k,t k D k,t 1 < k < t D,t = D k,t, k t D,t > h t h t ˆτ k D k,t D k,t h t F 0 F 1 (x) = F 0 (x + δ) F 1 (x) = F 0 (δx) F 0 t F

149 n = n A + n B A B (n + 1) /2 M = (r (x i ) (n + 1) /2) 2, x i A r (x i ) x i µ M = n A ( n 2 1 ) /12, σ 2 M = n An B (n + 1) ( n 2 4 ) /180. M = (M µ M )/ σ M i t r (x i ) = t i j I (x i x j ), I

150 t r τ < t t

151 τ t t + 1 t + 1 t = λ t 1 + (1 λ) r 2 t. λ

152

153

154

155 100

156

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161

162

163

164

165

166 l 1 l 2 l 0

167 x 1,..., x T R n t = 1,..., T x t x t N (µ t, Σ t ) µ t Σ t K T b 1,..., b K T K + 1 x t K b 1,..., b K µ (1),..., µ (K+1), Σ (1),..., Σ (K+1) K + 1 b 0 b K+1 1 = b 0 < b 1 < < b K < b K+1 = T + 1, (µ t, Σ t ) = (µ (i), Σ (i) ), b i 1 t < b i, i = 1,..., K. t t (i) b i x 1,..., x T x 1,..., x T l(b, µ, Σ) = = = T t=1 K+1 ( 1 2 (x t µ t ) T Σ 1 t (x t µ t ) 1 2 Σ t n 2 (2π) ) b i 1 i=1 t=b i 1 K+1 i=1 ( 1 2 (x t µ (i) ) T (Σ (i) ) 1 (x t µ (i) ) 1 2 Σ(i) n 2 (2π) ) l (i) (b i 1, b i, µ (i), Σ (i) ),

168 l (i) (b i 1, b i, µ (i), Σ (i) ) = b i 1 ( 1 2 (x t µ (i) ) T (Σ (i) ) 1 (x t µ (i) ) t=b i Σ(i) n 2 (2π) ) b i 1 = 1 (x t µ (i) ) T (Σ (i) ) 1 (x t µ (i) ) 2 t=b i 1 b i b ( ) i 1 Σ (i) + n (2π) 2 i b = (b 1,..., b K ) µ = (µ (1),..., µ (K+1) ) Σ = (Σ (1),..., Σ (K+1) ) Σ Σ b i b i 1 i K b µ Σ K+1 ϕ(b, µ, Σ) = l(b, µ, Σ) λ (Σ (i) ) 1 = K+1 i=1 i=1 ( l (i) (b i 1, b i, µ (i), Σ (i) ) λ (Σ (i) ) 1), λ 0 K λ K ( ) T 1 K b 1,..., b K µ Σ λ = 0 λ > 0 b i

169 b i 1 µ (i) 1 = x t, b i b i 1 t=b i 1 i Σ (i) = S (i) + λ b i b i 1 I, S (i) b i 1 S (i) 1 = (x t µ (i) )(x t µ (i) ) T. b i b i 1 t=b i 1 S (i) b i b i 1 < n λ > 0 Σ (i) b b ϕ(b) = C 1 K+1 ( (b i b i 1 ) (S (i) λ + I) 2 b i=1 i b i 1 ) λ (S (i) λ + I) 1 b i b i 1 K+1 = C + ψ(b i 1, b i ), i=1 C = (T n/2)((2π) + 1) b ψ(b i 1, b i ) = 1 ( (b i b i 1 ) (S (i) λ + I) 2 b i b i 1 ) λ (S (i) λ + I) 1. b i b i 1 S (i) b i 1 b i λ = 0 ψ(b i 1, b i ) = 1 2 (b i b i 1 ) S (i).

170 K λ 1 K+1 ( (b i b i 1 ) (S (i) λ + I) 2 b i=1 i b i 1 ) λ (S (i) λ + I) 1, b i b i 1 b = (b 1,..., b K ) ( ) T 1 K bi S (i) b S (i) T n 2 K + 1 Kn 2 T n Kn T n S (i) i = 1,..., K + 1 ψ(b i 1, b i ) LL T = S (i) λ + I, b i b i 1 L n 3 n 2 n i=1 (L ii) n 3 L 1 2 F T n 2 + Kn 3 K + 1 T n K K < T T = Kn n K n T n 2 T = 1000 n = 100

171 (b i 1, b i ) b i 1 < b i T (T 1)/2 K (p, q) (q, r) n 3 KT 2 T T (b i 1, b i ) i t ψ(b i 1, t)+ψ(t, b i ) t b i 1 b i b i b i 1 > 1 i t = (b i 1, b i ) b i 1 b i ψ(b i 1, t) + ψ(t, b i ) ψ(b i 1, b i ) t t = (b i 1, b i ) b i 1 b i t

172 x bi 1,..., x bi µ Σ µ = 0 µ = µ Σ = λi Σ = Σ + λi t = b i 1 + 1,..., b i 1 µ µ Σ Σ ψ t = ψ(b i 1, t) + ψ(t, b i) t ψ t ψ t ψ(b i 1, b i) t x 1,..., x T K b 0 = 1 b 1 = T + 1 K = 0,..., K i = 1,..., K + 1 (t i, ψ ) (b i 1, b i) ψ K > 0 (b 1,..., b K) ψ () t i ψ 1 = b 0 < b 1 < < b K+1 < b K+2 = T + 1 i = 1,..., K (t i, l ) (b i 1, b i+1) t i b i b i = t i (b 1,..., b K) n 2 ψ t n 3 (b i b i 1 )n 3 K K = 1,..., K

173 K = 1,..., K (b i 1, b i+1 ) K K + 1 Kn 2 K KLn 3 T L Ln 3 T L K K = 1 K = K L (K ) 2 n 3 T K n 3 T K n 3 T 2 λ K K λ 0.9T K

174 K = (T /n)/3 3n T /n λ λ K l(x t ) = 1 2 (x t µ (i) ) T (Σ (i) ) 1 (x t µ (i) ) 1 2 Σ(i) n 2 (2π), t i 1 X X x l(x t X t) t λ K K λ K K = 0 K

175 O(K T n 3 ) O(T n 3 ) K 1 T 1 K M λ K t t T x T x 1 t = 1,..., 365

176 (Σ (i), µ (i) ) (Σ (i 1), µ (i 1) ) λ

177 n = 3 µ (i) Σ (i) n = 3 T = 4943 K = 30 λ = 10 4 K K = 8 K = 10 n

178 ϕ(b) λ = 10 4 K λ K λ λ K = K K K = 30 K λ λ = 10 4 K = 10 λ = 10 4 K λ K λ λ = 10 4 K = 10 t

179 λ = 10 6 λ = 10 5 λ = 10 4 λ = 10 3 λ K 30

180 λ = 10 4 K = 10 K = 10 λ = 10 4 K = 8 K = 11 K = 9

181 K K K = 10 K (K ) 2 LT n

182 ϕ(b) K = 10 λ = 10 4 K λ K n = K = 3 λ =

183 λ = λ = 10 2 λ = λ = 10 1 λ λ K = 10

184 K = 3 λ = K = 4 λ = 10 3 λ

185

186 λ = 10 K = 9 Σ i = A (i) A (i)t, i = 1,..., 10 A (i) R A (i) j,k K = 9 i Σ i K = 9 K + 1 = 10 λ λ = 10 λ = 10 λ λ λ λ

187 [100, 200, 300, 400, 500, 600, 700, 800, 900] [100, 200, 300, 400, 500, 600, 663, 700, 800] n K T /(K + 1) K T /K n

188 t l 1

189

190 l 1

191

192 l 1

193

194

195

196

197

198 {S t : t N} t N (S t+1 S t,..., S 1 ) = (S t+1 S t ). (S t+1 = j S t = i) = γ ij Γ = {γ ij } π π T Γ = π T T π = 1 δ = π δ δ i = (S 1 = i) Y t S t N ( µ St, σ 2 S t ),

199 µ St = { { µ 1, S t = 1, σ 2 [ ] µ 2, S t = 2, σ2 S t = 1, S t = 1, 1 σ2, 2 S t = 2, Γ = γ12 γ 12. γ 21 1 γ 21 S t Y t S t ( t i ) = γ t 1 ii (1 γ ii ).

200 ˆθ t = θ t n=1 w n (Y n Y n 1,..., Y 1, θ ) = lt (θ) θ w n = 1 w n = f t n 0 < f < 1 N = 1 1 f. l t (θ) ˆθ t 1 θ ˆθ t ˆθ t = ˆθ )] 1 ) t 1 [ θθ lt (ˆθt 1 θ lt (ˆθt 1. I t (θ) = [ θθ l t ] = [ θ l t θ lt T ]. ˆθ t ˆθ )] 1 ) t 1 + A [I t (ˆθt 1 θ lt (ˆθt 1.

201 A A 1/N t > 1 t α T T T = δt 1 (y 1 ) T t=2 Γ t t (y t ( δ T T ), 1 (y 1 ) t=2 Γ t t (y t ) i ( α T T )i = (S T = i Y T,..., Y 1 ) t (y t ) p i (y t ) = (Y t = y t S t = i, θ t ) k α T T k α T T +k T = αt T T Γk T.

202 µ = σ 2 = m µ i α i i=1 m i=1 ( µ 2 i + σi 2 ) αi µ 2 α i r t (1 + r t ) N ( µ, σ 2), µ σ 2 [r t ] = ( µ + σ 2/ 2 ) 1 [r t ] = ( ( σ 2) 1 ) ( 2µ + σ 2). λ 2 = γ 11 + γ 22 1 λ 2

203 h t R n t (h t ) i i t (h t ) i < 0 i u t R n (u t ) i > 0 i t h + t = h t + u t, t = 0,..., T 1, t + 1 V t = T h t

204 V + t = T h + t V t h t h t /V t T u t + κ T u t = 0, t = 0,..., T 1, κ T u t κ T u t T u t + κ T u t 0, t = 0,..., T 1, u 0 u 1 u 2 r 1 r 2... h t+1 = ( + r t+1 ) h + t, t = 0,..., T 1, r t+1 R n t t + 1 t h 0 h + 0 h 1 h + 1 h 2 h + 2 t = 0 t = 1 t = 2 r t [ ] [r t ] = r t, (r t r t ) (r t r t ) T = Σ t, t = 1,..., T. ϕ t : R n R n u t = ϕ t (h t ), t = 0,..., T 1. C t R n C t h t u t h + t = h t + u t C t.

205 h + t u t h t r t h t h + t h t, h t h t h = 0 H t R n y t + V t + Ht, [ ] T 1 J = V T ψ t (h t, u t ), t=0 r 1,..., r T V T = T h T ψ t : R n R n R t r t ψ t h 0 ] ( ) h + T t Σt+1 h + t ψ t (h t, u t ) = γ [ V t+1 h + t V t + = γ V + t, γ 0 V t +

206 ψ t (u t ) = ρ T u t, ρ κ l 1 V t

207 u t ˆr τ τ = t+1,..., T. V T T 1 τ=t ψ τ (h τ, u τ ) h τ+1 = ( + ˆr τ+1 ) (h τ + u τ ), τ = t,..., T 1 h t+1,..., h T u t,..., u T 1 h t u t,..., u T 1 ϕ (h t ) = u t h t+1 r t u t,..., u T 1 t,..., T K

208 K u t,..., u t+k 1 u t t Vt+K (h t+k) t+k 1 τ=t ψ τ (h τ, u τ ) h τ+1 = ( + ˆr τ+1 ) (h τ + u τ ), τ = t,..., t + K 1 h t+1,..., h t+k u t,..., u t+k 1 K Vt+K K Vt+K (h t+k) V t+k = T h t+k K K t K u t,..., ut+k 1 K = 100

209 r t = (P t) (P t 1 ) P t t

210 µ1 µ2 σ 2 1 σ 2 2 γ11 γ22 N = 260 N = 260 A = 1/N

211 γ 11 γ 22 1/ (1 0.99) = 100 K γ = 0 γ = 2 K λ 2 = 1 γ 11 γ 22

212 (γ = 0, κ = 0) (γ = 2, κ = 0) (γ = 0, κ = 0.02) (St = 1) (St = 1) (St = 1) γ κ κ = K = 100 κ = (γ = 0, ρ = 0) γ = 2

213 (γ = 0, ρ = 0) (γ = 2, ρ = 0) (γ = 0, ρ = 0.02) γ (γ = 0, ρ = 0.02) ρ = 0.02 κ = γ = 2

214 (γ = 0, κ = 0.001, ρ = 0) ut/vt ut/vt ut/vt (γ = 2, κ = 0.001, ρ = 0) (γ = 0, κ = 0.001, ρ = 0.02)

215 (γ = 0, ρ = 0) (γ = 0, ρ = 0.02) u 0 u 1 u 2 h 0 h + 0 r 1 h 1 h + 1 r 2 h 2 h + 2 t = 0 t = 1 t = 2... t t + 1

216

217 (γ = 0, κ = 0.001, ρ = 0) (κ = 0.001) (γ = 0, ρ = 0)

218

219 t

220 1/N

221

222

223

224

225

226

227

228

229 n t = 1,..., T t t t t + 1 n h t R n+1 t (h t ) i i t (h t ) i < 0 i i = 1,..., n (h t ) i 0 i = 1,..., n (h t ) n+1 (h t ) n+1 < 0 p t R n +

230 t (h t ) n+1 = 0 v t t v t = T h t v t > 0 (h t ) 1:n = ((h t ) 1,..., (h t ) n ) (h t ) 1:n 1 = (h t ) (h t ) n, (h t ) 1:n 1 /v t w t R n+1 h t w t = h t /v t v t > 0 T w t = 1 (w t ) n+1 h t = v t w t w 1:n 1 l 1 u t R n (u t ) i > 0 i (u t ) i < 0 i t i = 1,..., n (u t ) n+1 z t = u t /v t w t h + t = h t + u t, t = 1,..., T.

231 t v + t = T h + t v + t v t = T h + t T h t = T u t. (u t ) 1:n R n l 1 (u t ) 1:n 1 /2 t (u t ) 1:n 1 /(2v t ) = z 1:n 1 /2 w t = h t /v t h + t /v t = w t + z t. ϕ t (u t ) ϕ t : R n+1 R ϕ t (u t ) n+1 ϕ t ((u t ) 1:n ) ϕ t (0) = 0 ϕ t (u t ) ϕ t ϕ t (x) = n i=1 (ϕ t ) i (x i ), (ϕ t ) i R R i t (ϕ t ) i x a x + bσ x 3/2 + cx, V 1/2 a b σ V c x a

232 b V x 3/2 /V 1/2 σ b b c c = 0 x c > 0 c > a i t 3/2 3/2 v t z i i t a i z i + b i σ i z i 3/2 (V i /v) 1/2 + c iz i. V i /v V i t ϕ t (z t ) ϕ t (z t ) v t

233 z i h + t t ϕ t (h + t ) ϕ t : R n+1 R (h + t ) n+1 ϕ t (h + t ) = s T t (h + t ), (s t ) i 0 t i (z) = { z, 0} z (s t ) n+1 = 0 (s t ) n+1 > 0 ϕ t (h + t )/v t = s T t (w t + z t ). ϕ t (w t + z t )

234 ϕ t h i ϕ t (w t + z t ) = s T t (w t + z t ) + f T t (w t + z t ), f t (f t ) i i i T u t + ϕ t (u t ) + ϕ t (h + t ) = 0. T u t v t + = v t ϕ t (u t ) ϕ t (h + t ) (u t ) n+1 (u t ) 1:n (u t ) n+1 = ( T (u t ) 1:n + ϕ t ((h t + u t ) 1:n ) + ϕ t ((u t ) 1:n ) ). n + 1 n (u t ) 1:n (u t ) n+1 (u t ) 1:n ϕ t

235 ϕ t v t T z t + ϕ t (v t z t )/v t + ϕ t (v t (w t + z t ))/v t = 0, u t = v t z t h + t = v t (w t + z t ) T z t + ϕ t (z t ) + ϕ t (w t + z t ) = 0, (z t ) n+1 (z t ) 1:n (z t ) n+1 = ( T (z t ) 1:n + ϕ t ((w t + z t ) 1:n ) + ϕ t ((z t ) 1:n ) ). h t+1 = h + t + r t h + t = ( + r t ) h + t, t = 1,..., T 1, r t R n+1 t t + 1 i t (r t ) i = (p t+1) i (p t ) i (p t ) i, i = 1,..., n, + r t 0 (p t+1) i (p t ) i = (1 + (r t ) i ), i = 1,..., n.

236 (r t ) n+1 (s t ) n+1 > 0 (u t ) 1:n (r t ) 1:n (r t ) n+1 v t+1 = T h t+1 = ( + r t ) T h + t = v t + r T t h t + ( + r t ) T u t = v t + rt T h t + rt T u t ϕ t (u t ) ϕ t (h + t ). t R t = v t+1 v t v t, R t = rt T w t + rt T z t ϕ t (z t ) ϕ t (w t + z t ). t r T t w t r T t z t ϕ t (z t ) ϕ t (w t + z t ) w t+1 w t z t r t 1 w t+1 = 1 + R ( + r t ) (w t + z t ). t T w t+1 = 1 w t+1 = w t + z t r t = 0 w t+1 w t + z t

237 d T t h t d t t h t+1 = ( + r t ) (h t + u t ) h t+1 = ( + r t ) h t + (1 θ t /2)( + r t ) u t, θ t θ t > 0 u t u t

238 t = 1,..., T h 1 R n+1 (u t ) 1:n (u t ) n+1 a t R n b t R n c t R n σ t R n V t R n s t R n r t R n+1 d t R n (u t ) 1:n

239 t = 1,..., T R = 1 T R t. T t=1

240 t G t = (v t+1 /v t ) = (1 + R t ). G t t = 1,..., T G t R t P P P 250 σ = ( 1 T ) 1/2 T (R t R ) 2. t=1 1/T 1/(T 1) R t (σ ) 2 1 T T (R t ) 2. t=1 P P wt R n+1 T wt = 1 wt wt = e n+1 t Rt = rt T wt (r t ) n+1

241 R t = R t R t. R t = R t (r t ) n+1. R R t Rt = R t Rt = rt T ( wt wt ) + r T t z t ϕ t (z t ) ϕ t (w t + z t ). z t = 0 w t = wt ϕ t (wt ) = 0 Rt σ σ σ R σ = R /σ. = R /σ. t (u t ) 1:n (z t ) 1:n (z t ) n+1

242 (z t ) 1:n r t ϕ t Z t (z t ) 1:n Ẑ ˆϕ t r t ˆr t Ẑ Z t r t ˆR t = ˆr T t w t + ˆr T t z t ˆϕ t (z t ) ˆϕ t (w t + z t ), r t ˆr t ˆR t = ˆr T t (w t w t ) + ˆr T t z t ˆϕ t (z t ) ˆϕ t (w t + z t ). ˆr T t z t ˆϕ t (z t ) ˆϕ t (w t + z t ),

243 z t ˆR t γ t ψ t (w t + z t ) z t Z t, w t + z t W t T z t + (z t ) + ˆϕ t ˆϕ t (w t + z t ) = 0, z t ψ t : R n+1 R γ t > 0 Z t W t w t v t ˆR t ˆR t ˆR t z t ˆr T t z t ˆϕ t (z t ) ˆϕ t (w t + z t ). ˆϕ t ˆr t T z t (z t ) z t Z t, w t + z t W t T z t + (z t ) + ˆϕ t ˆϕ t (w t + z t ) γ t ψ t (w t + z t ) ˆϕ t (w t + z t ) = 0, z t z t w t + z t (z t ) 1:n = (zt ) 1:n zt (u t ) 1:n = v t (zt ) 1:n (z t ) n+1 (zt ) 1:n (zt ) n+1 (z t ) n+1

244 (z t ) n+1 (z t ) n+1 (zt ) n+1 w t + zt W t w t + z t w t + zt (w t + zt ) 1:n T z t + ˆϕ t (z t ) + ˆϕ t (w t + z t ) = 0 T z t = 0 T z = 0 ˆϕ t ˆϕ t (w t + z t ) γ t ψ t (w t + z t ) ˆr t T z t (z t ) T z t = 0, z t Z t, w t + z t W t. zt (z t ) n+1 (w t + z t ) n+1 T z t = 0 w t + z t T (w t + z t ) = 1 T (w t + z t ) w t+1 = w t + z t ˆϕ t ˆϕ t (w t+1 ) γ t ψ t (w t+1 ) ˆr t T w t+1 (w t+1 w t ) T w t+1 = 1, w t+1 w t Z t, w t+1 W t,

245 w t+1 ψ t r t Σ t R (n+1) (n+1) R t [R t ] = (w t + z t ) T Σ t (w t + z t ). t ψ t (x) = x T Σ t x. Σ t (r t ) n+1 Σ t r t Σ t Rt [R t] = (w t + z t w t ) T Σ t (w t + z t w t ). ψ t (x) = (x w t ) T Σ t (x w t ). x T Σ t x Σ t γ t γ t = 1/2 r t w R t = w T r t [(1 + R t )] T w = 1 w 0

246 (1 + a) a (1/2)a 2 [(1 + R t )] [ R t (1/2)(R t ) 2] = µ T w (1/2)w T (Σ + µµ T )w, µ = [r t ] Σ = [(r t µ)(r t µ) T ] r t µµ T Σ µ T w (1/2)w T Σw γ t = 1/2 γ t 1/2 n Σ t Σ t = F t Σ tft T + D t, F t R (n+1) k Σ t R k k F T r t D t R (n+1) (n+1) k n (F t ) ij i j D t O(n 3 ) O(nk 2 ) O((n/k) 2 ) n k

247 ψ t (x) = φ((x w t ) T Σ t (x w t )), φ : R R φ(x) = (x a) + a a φ(x) = (x/η) η > 0 η x ψ t (x) = (x i=1,...,m w t ) T Σ (i) t (x wt ). Σ (i) i = 1,..., M M i M Σ (i) M Σ (i)

248 ˆr ˆr + δ δ ρ ρ R n ρ ˆr i ρ i ρ i 6 ± (ˆr t + δ t ) T (w t + z t ) δ ρ δ ˆr i + ρ i ˆr i ρ i ˆR t = ˆr T t (w t + z t w t ) ρ T w t + z t w t. δ ψ t (x) = ρ T x w t. l 1

249 Σ Σ = Σ +, ij κ (Σ ii Σ jj ) 1/2, κ [0, 1) κ κ κ κ = v = x w t v T (Σ )v = v T (Σ + )v ij κ(σ iiσ jj) 1/2 ij κ(σ iiσ jj) 1/2 = v T Σv + ij κ(σ iiσ jj) 1/2 = v T Σv + κ v i v j (Σ ii Σ jj ) 1/2 ij ( 2 = v T Σv + κ Σ 1/2 ii v i ). i ij v i v j ij ψ t (x) = (x w t ) T Σ(x w t ) + κ ( σ T x w t ) 2, σ = (Σ 1/2 11,..., Σ1/2 nn ) l 1 w t +z t w t+1

250 w t+1 w t + z t w t + z t w t+1 w t+1 w t + z t w t + z t w t w t + z t w t + z t 0. (w t +z t ) 1:n 0 w t + z t ( + r t ) (h t + z t ) + r t 0 (w t + z t ) 1:n 1 L, L L C t (w t + z t ) i δ C t /v t, δ 0 / i δ i w w t + z t w, w w w = w = (0.05)

251 c (w t + z t ) n+1 c /v t. i i (w t + z t ) i = 0. β β R R Σ t [r t ] w t + z t β (w t ) T Σ t (w t + z t ) = 0. σi i ( ) σ 2 i = (wt + z t ) T (F t ) i (Σ t) ii (F t ) T i (w t + z t ). i σi (F t ) T i (w t + z t ) = 0. = 0 1,..., K i c i c i i i c T i (w t + z t ) R, i R R i

252 T h + T ˆϕ (w t + z t )/T δ T ˆϕ t ((w t + z t )/T ) δ. T ω K K (w t + z t ) [i] ω, i=1 a [i] i a K K = 20 ω = K z t (z t ) 1:n (z t ) n+1 t (z t ) 1:n 1 /2 δ (z t ) 1:n 1 /2 δ. δ V t (z t ) 1:n δ(v t /v t ),

253 i (z t ) i 0, (z t ) i 0. h(x) = 0 x γ h(x) 1 γ > 0 γ T h(x) γ h(x) h(x) 0 γ T (h(x)) + γ > 0 F T t (w t + z t ) = 0 γ F T t (w t + z t ) 1 γ > 0 γ F T t (w t + z t ) = 0 T z t + T z t + ˆϕ t (z t ) + ˆϕ t (w t + z t ) = 0, ˆϕ t (z t ) + ˆϕ t (w t + z t ) 0. T z t = 0

254 O(nk 2 ) n k k n 3/ O(nk 2 )

255 (z t ) i (z t ) i ϵ ϵ > 0 (z t ) i ϵ (z t ) i 0 z z (z t ) i = 0 ( z t ) i = 0 (z t ) i 0 ( z t ) i > 0 (z t ) i 0 ( z t ) i < 0

256 K l 1 K ϕ t γ 1/γ a t 3/2 3/2

257 γ γ κ T (w t +z t ) 2 κ 0 ( ˆr t T z t γt ˆϕ t (z t ) ) γt ˆϕ t (w t + z t ) γt ψ t (w t + z t ) T z t = 0, z t Z t, w t + z t W t. γt γt γt

258

259

260 L V t z t H t, t + 1,..., t + H 1. H = 1 t, t + 1,..., t + H 1 t Z Ẑτ t Z τ t τ t Ẑτ t = Z τ Z τ ˆr t t t t ˆr t ˆr t+2 t t t + 2 z t, z t+1,..., z t+h 1 t+h 1 τ=t ( ˆr T τ t (w τ + z τ ) γ τ ψ τ (w τ + z τ ) ) ˆϕ τ (w τ + z τ ) ˆϕ τ (z τ ).

261 w t w t+1,..., w t+h z t,..., z t+h 1 1 w t+1 = 1 + R ( + r t ) (w t + z t ), t T w t = 1 T w t+1 = 1 γ τ ψ t (w τ +z τ ) γ τ ψ τ (w τ + z τ ) w t z t w t+1 R t = 0 r t = 0 w t+1 = w t + z t T z t = 0 T w τ = 1 τ = t + 1,..., t + H T z τ = 0, τ = t + 1,..., t + H 1. w t T w t = 1 T w τ = 1 τ = t + 1,..., t + H ( t+h 1 τ=t ˆr τ t T (w τ + z τ ) γ τ ψ τ (w τ + z τ ) ˆϕ τ (w τ + z τ ) ˆϕ τ (z τ ) T z τ = 0, z τ Z τ, w τ + z τ W τ, w τ+1 = w τ + z τ, τ = t,..., t + H 1, z t, z t+1,..., z t+h 1 w t+1,..., w t+h w t H = 1 ˆr T t t w t

262 w τ+1 = w τ + z τ z τ ( t+h τ=t+1 ˆr τ t T w τ γ τ ψ τ (w τ ) ) ˆϕ τ (w τ ) ˆϕ τ (w τ w τ 1 ) T w τ = 1, w τ w τ 1 Z τ, w τ W τ, τ = t + 1,..., t + H, w t+1,..., w t+h H H z t z t+1,..., z t+h 1 z t 1 w t+1 = 1 + R ( + r t ) (w t + z t ) t w t+1 = w t + z t r t = 0

263 T z τ = 0 τ = t + 1,..., t + H 1 z τ T z τ = ϕ τ (z τ ) ϕ τ (w τ +z τ ) w t+h = w w t + H w = e n+1 H w t+h w t w t+1,..., w t+h 1 Hn H H H H = 100

264 H H H w t+h = w z τ 0 τ = t, τ = t + T, τ = t + T, τ = t + H 1, 1 < T < T < H 1. z = z t z = z t+t z = z t+t z t+h 1 T = 5 T = 21 H = 100 w = w t + z, w = w + z, w = w + z. z τ t + H 1

265 t ˆr t t t (w t + z t ) T Σ t (w t + z t ) Σ t R (n+1) (n+1) Σ t

266 w t v t z t t

267 r t V t (σ t ) i = (p t ) i (p t ) i (p t ) i ) i i t (p t a t = 0.05 s t = 0.01 b t = 1 c t = 0 d t = 0 w = ( /n, 0) w 1 = w z t = w w t t z t = 0 R σ 250 T T t=1 ϕ t (z t ), 250 T T t=1 (z t) 1:n 1 /2

268 v 1 = $100 w 1 = ( /n, 0) L = 3 t t M Σ = 1 t 1 M τ=t M r τ rτ T Σ = n i=1 λ iq i qi T λ i F = [q 1 q k ], Σ = (λ,..., λ ), = = + λ ( ) ( ), k = 15 D F Σ F T + D Σ

269 (ˆr t ) n+1 = (r t ) n+1 t t (ˆr t ) 1:n = α ((r t ) 1:n + ϵ t ), ϵ t N (0, σ 2 ϵ I) σ 2 ϵ = 0.02 α [((ˆr t ) 1:n (r t ) 1:n ) 2 ] r t σ r α = σ 2 r/(σ 2 r + σ 2 ϵ ) σ 2 r = α = ±0.3 α 0.15

270 t i ( ˆV t ) i = τ=1 (V t τ ) i γ γ γ γ = 0.1, 0.3, 1, 3, 10, 30, 100, 300, 1000, γ = 1, 2, 5, 10, 20, γ = 1, R σ γ γ γ γ γ γ γ = 4, 5, 6, 7, 8, γ γ γ = 6

271

272 γ = 0.1, 0.178, 0.316, 0.562, 1, 2, 3, 6, 10, 18, 32, 56, 100, 178, 316, 562, 1000, γ = 5.5, 6, 6.5, 7, 7.5, 8, γ = 0.1, 1, 10, 100, 1000, γ = 1

273 γ γ γ H = 2 H = 2 H = 1

274 v 1 = $100 w 1 = ( /n, 0) L = 3 ˆr t t = ˆr t, ˆr t+1 t = ˆr t+1, ˆr t ˆr t+1 ˆr t+1 = ˆr t+1 t = ˆr t+1 t+1 ˆr t+1 t ˆr t+1 t+1 γ γ γ γ = 0.1, 0.3, 1, 3, 10, 30, 100, 300, 1000, γ = 1, 2, 5, 10, 20, γ = 1, R σ γ γ = 10 γ = 1, 2, 3, 6, 10, 18, 32, 56, 100, 178, 316, 562, 1000, γ = 7, 8, 9, 10, 11, 12, γ = 0.1, 1, 10, 100, 1000,

275

276

277 3/2

278

279 1/N

280

281

282

283

284

285

286

287

288

289 w t R n+1 t (w t ) i V t i (w t ) i < 0 i (w t ) n+1 T w t = 1 T [ T 1 η t+1 ( rt+1w T t+1 γ t+1 ψ t+1 (w t+1 ) ) t=0 η t ( ϕ t (w t+1 w t ) + ϕ t (w t+1 ) )], r 1,..., r T R n+1 ψ t : R n+1 R γ t ϕ t : R n+1 R

290 ϕ t : R n+1 R η (0, 1) H ˆµ τ t τ = t+1,..., t+h ˆµ τ t t τ t+h τ=t+1 ( ˆµ T τ t w τ ˆϕ τ t (w τ w τ 1 ) ˆϕ τ t (w τ ) γ τ ˆψτ t (w τ ) T w τ = 1, τ = t + 1,..., t + H, w t+1,..., w t+h w t ˆϕ ˆϕ wt+1,..., wt+h H wt+1 w t w t+1 H T

291 H wt+1,..., wt+h w t w t+1 w t H = 15 n = 10

292 ψ t (w t ) = w T t Σ t w t. Σ t q q H = 1

293 V t t M t = V τ, τ t t D t = 1 V t M t. t D D t D (0, 1) γ 0 D t = 0 V t = M t V 0 = M 0 D t = D D γ t = γ 0 D. D t (D D t, ϵ) ϵ γ τ γ τ = γ t τ = t + 1,..., t + H

294 γ t γ 0 D γ 0 ϕ t (w t w t 1 ) = κ T 1 w t w t 1 + κ T 2 (w t w t 1 ) 2, κ 1 κ 2 l 1 l 2 l 1 l 2 l 1 l 2

295 w t w t 1 3/2 (w t w t 1 ) T Σ t (w t w t 1 ) w t t ϕ t (w t ) = s T t (w t ), (s t ) i 0 i t (w) = { w, 0} w (s t ) n+1 > 0 ϕ t (w t ) = ρ T 1 w t + ρ T 2 w 2 t, ρ 1 ρ 2 l 1 l 2 l 1 l 2

296 w w t w, w w w = 0 (w t ) 1:n 1 L, L k

297 {s t : t N} t N (s t+1 s 1,..., s t ) = (s t+1 s t ). (s t+1 = j s t = i) = γ ij Γ = {γ ij } π π T Γ = π T T π = 1 o t s t N (µ st, Σ st ). s t o t s t α t o 1,..., o t i S t (α t ) i = (s t = i, o 1,..., o t ), i S. (α 1 ) i = (δ) i (o 1 s 1 = i), i S, δ (δ) i = (s 1 = i) α j S [ ] (α t ) j = (α t 1 ) i γ ij (o t s t = j), j S. i S

298 i S t (ξ t ) i = (s t = i o 1,..., o t ) = (s t = i, o 1,..., o t ) (o 1,..., o t ) = (α t) i T α t. i j (ζ t ) ij = (s t 1 = i, s t = j o 1,..., o t ) = (s t 1 = i, o 1,..., o t 1 ) (s t = j s t 1 = i) (o t s t = j) (o 1,..., o t ) = (α t 1) i γ ij (o t s t = j) T α t. t ξ t ζ t i, j S ˆγ t ij = = ˆµ t i = t τ=2 (s τ 1 = i, s τ = j o 1,..., o τ ) t τ=2 (ξ τ ) i t 1 τ=2 (ξ τ ) i t τ=2 (ξ τ ) i ˆγ t 1 ij + (ζ t ) ij t τ=2 (ξ τ ) i t τ=1 (ξ t 1 τ ) i o τ τ=1 t τ=1 (ξ = (ξ τ ) i t τ ) i τ=1 (ξ ˆµ t 1 i + (ξ t) i o t t τ ) i τ=1 (ξ τ ) i t ˆΣ t τ=1 i = (ξ τ ) i (o τ ˆµ t i ) (o τ ˆµ t i )T t τ=1 (ξ τ ) i t 1 τ=1 = (ξ τ ) i ˆΣt 1 t τ=1 (ξ i + (ξ t) i (o t ˆµ t i ) (o t ˆµ t i )T t τ ) i τ=1 (ξ. τ ) i t = 1

299 t τ=1 ξ τ S ξ t S ξ t = λs ξ t 1 + (1 λ) ξ t, λ (0, 1) τ λ τ T = 1/ (1 λ) = (1 ν i ) ˆΣ ) i + ν i (ˆΣi n 1 I n, ˆΣ i ν i [0, 1] I n n n

300 ξ t h ˆξ t t h ˆξ t+h t T = ˆξ t t T Γh t. µ = (ξ) i µ i, i S Σ = (ξ) i Σ i + (ξ) i (µ i µ) (µ i µ) T, i S i S (ξ) i r t ( + r t ) N ( µ s t, Σ ) s t, µ s t Σ s t r t (µ s ) i = (Σ s ) ij = { { (µ ) s { (µ s i )i + ( µ s { (Σ ) s ij } ( ) Σ s 1, ii ) } 1 j }. { (Σ s )ii + ( Σ s ) jj } } i j s

301

302

303 γ = 5 γ = 5 1/n

304 T = 65, 130, 260, ν i = 0.1, 0.2,..., 0.5

305 H = 10, 15,..., 30 w = 0.2, 0.3,..., 0.5 w = 0.2 w = 0.4 (κ 1 ) 1:n = , 0.001,..., (κ 1 ) n+1 = 0 κ T 1 w t w t 1 κ T 2 (w t w t 1 ) 2 (κ 1 ) 1:n = ρ 2 = 0, ,..., ρ T 2 wt 2 ρ 2 (ρ 2 ) n+1 = 0 ρ 2 = l 1 ρ T 1 w t γ 0 = 5 γ 0 = 5 γ 0

306 Empirical results Asset weight 0.8 TBill IFL bonds CORP bonds GVT bonds EM HY bonds DM HY bonds Oil Gold Real estate EM stocks DM stocks Year 1.0 TBill IFL bonds CORP bonds GVT bonds EM HY bonds DM HY bonds Oil Gold Real estate EM stocks DM stocks Asset weight 2.0 (a) γ = 5, (κ1 )1:n = 0.004, ρ2 = , (wmax )1:n = 0.4, (wmax )n+1 = Year ) ) ( ( (b) γ = 5, (κ1 )1:n = 0.004, ρ2 = , wmin 1:n = (wmax )1:n = 0.4, wmin n+1 = (wmax )n+1 = 1, Lmax = 2. Figure 3: Asset weights over time for a long-only and a long short portfolio. mentioned in the ﬁgure captions are equal to zero. The portfolios always include multiple assets at a time due to the imposed maximum holding (wmax )1:n = 0.4. The allocations change quite a bit over the test period, especially in the LS portfolio. Leverage is primarily used between 2003 and mid-2006 and again from 2010 until mid With the exception of these two periods, the four portfolios include holdings in the risk-free asset most of the time in addition to some short positions in the LS portfolio. The impact of drawdown control on the allocation is most

307 Multi-period portfolio selection with drawdown control 1.0 TBill IFL bonds CORP bonds GVT bonds EM HY bonds DM HY bonds Oil Gold Real estate EM stocks DM stocks Asset weight Year 1.0 TBill IFL bonds CORP bonds GVT bonds EM HY bonds DM HY bonds Oil Gold Real estate EM stocks DM stocks Asset weight 2.0 ) ( (a) γ = 5, (κ1 )1:n = 0.004, ρ2 = , (wmax )1:n = 0.4, wmin n+1 = (wmax )n+1 = Year ) ( (b) γ0 = 5, (κ1 )1:n = 0.004, ρ2 = , (wmax )1:n = 0.4, wmin n+1 = (wmax )n+1 = 1, Dmax = 0.1. Figure 4: Asset weights over time for a leveraged long-only portfolio with and without drawdown control.

308 D =0.1 1/n γ 0 = 5 1/n 1/n γ 0 = 5 (1/n) 1/n 1/n L = 2 γ 0 = 5 D = 0.1

309 D =0.1 1/n γ 0 = 5 1/n

310 γ 0 D 1/n 1/n (w ) 1:n = 0.4 1/n 1/n D γ 0 = 1 γ 0 γ 0 3 D = 0.1

311 D =0.15 D =0.1 1/n D =0.15 D =0.1 1/n D γ 0 = 1, 3, 5, 10, 15, 25

312 D =0.15 D =0.1 1/n D =0.15 D =0.1 1/n D γ 0 = 1, 3, 5, 10, 15, 25

313 γ 0 D γ 0 D 1/n ( w ) 1:n = (w ) 1:n = 0.4 L = 2 γ 0

314

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316 1/N

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Single Shooting and ESDIRK Methods for adjoint-based optimization of an oil reservoir

Downloaded from orbit.dtu.dk on: Dec 2, 217 Single Shooting and ESDIRK Methods for adjoint-based optimization of an oil reservoir Capolei, Andrea; Völcker, Carsten; Frydendall, Jan; Jørgensen, John Bagterp