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

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.. 465 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.

(rt) (r 2 t )

t t

l 1 l 1 l 2 l 1 l 2

l 1

{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 ),

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 2 2 + 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 ].

( 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

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.

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 1 0 0 w 1 q 1 w 2 q 2 q 2 (1 w 2 ) q 2 0 0 =. (1 w m ) q m 0 0 w m q m q m

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

rt r t JB [0.00001; 0.00068] [0.0116; 0.0127] [ 0.75; 0.30] [8.5; 14.1] [6356; 25803] JB = T ( ) 2 + ( 3)2 T 6 24

( 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

10 3 10 3 10 3 10 3 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

r t [0.00001; 0.00068] [0.0116; 0.0127] [ 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

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

( 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

m Γ µ 10 4 σ 2 10 4 δ 0.990 0.010 8.3 0.52 1.0 (0.002) (1.1) (0.01) (0.2) 0.021 0.979 6.9 3.47 0.0 (0.004) (4.9) (0.14) 0.982 0.018 0.000 (0.004) (0.001) 0.015 0.978 0.006 (0.003) (0.002) 0.000 0.030 0.970 (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) 0.0 0.979 0.021 0.000 0.000 (0.005) (0.006) (0.00) 0.020 0.970 0.009 0.001 (0.005) (0.003) (0.001) 0.000 0.017 0.976 0.007 (0.000) (0.006) (0.004) 0.000 0.000 0.049 0.951 (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) 12.16 (1.73) 1.0 (0.2) 0.0 (0.2) 0.0 (0.1) 0.0 m

m Γ 1 p r 10 µ 10 4 σ 2 10 4 δ 0 1 0.994 0.4 9.0 0.48 1.0 (0.002) (0.1) (1.3) (0.01) (0.4) 1 0 0.975 0.5 10.8 4.00 0.0 (0.010) (0.2) (5.7) (0.18) 0 1.000 0.000 (0.000) 0.972 0 0.028 (0.013) 0.000 1.000 0 (0.000) 0.993 (0.003) 0.945 (0.037) 0.983 (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) 0.0 0 1.000 0.000 0.000 (0.000) (0.000) 0.970 0 0.022 0.008 (0.022) (0.020) 0.000 0.675 0 0.325 (0.000) (0.161) 0.000 0.000 1.000 0 (0.000) (0.000) 0.985 (0.006) 0.965 (0.024) 0.937 (0.124) 0.977 (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) 11.81 (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 2 10 4 t δ 0 1 0.997 0.2 9.5 0.34 7.2 1.0 (0.002) (0.1) (1.2) (0.02) (1.2) (0.4) 1 0 0.984 0.5 6.0 2.10 5.6 0.0 (0.008) (0.2) (4.8) (0.17) (1.0) 0 1.000 0.000 (0.042) 0.630 0 0.370 (0.144) 0.000 1.000 0 (0.045) 0.990 (0.009) 0.979 (0.047) 0.983 (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) 0.0 0 1.000 0.000 0.000 (0.000) (0.000) 0.610 0 0.296 0.094 (0.103) (0.106) 0.000 0.724 0 0.276 (0.000) (0.148) 0.000 0.000 1.000 0 (0.000) (0.000) 0.987 (0.013) 0.957 (0.052) 0.931 (0.115) 0.981 (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

m µ 10 4 σ 2 10 4 δ 0.014 0.014 0 0 10.6 0.32 1.0 (0.003) (1.4) (0.01) 0 0.020 0.020 0 0.8 1.29 0.0 (0.003) (2.5) (0.03) 0.048 0 0.068 0.020 0.8 1.29 0.0 (0.019) 0.005 0.019 0 0.024 14.6 7.12 0.0 (0.003) (0.003) (12.1) (0.26) 0.018 0.017 0 0.001 (0.005) (0.001) 0.015 0.020 0.005 0 (0.004) (0.002) 0 0.010 0.015 0.005 (0.003) (0.002) 0.005 0 0.029 0.034 (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) 12.29 (0.82) 1.0 0.0 0.0 0.0 m N N t N N t N

r t [ 0.00002; 0.00063] [0.0113; 0.0123] [ 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)

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.

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 2 2 + 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

( t i ) = γ t 1 ii (1 γ ii ).

2 f f f = 10

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

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

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

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

(r 2 t ) (r 2 t ) (r 2 t ) r t ± 4 σ

µ1 µ2 σ 2 1 σ 2 2 γ11 γ22

µ1 µ2 σ 2 1 σ 2 2 γ11 γ22 ν t

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

-0.02 µ2-0.004 0.000 µ1 0.00 Long memory and time-varying parameters 0.004 76 1940 1960 1980 2000 1940 1980 2000 Year 0.001 σ22 0.003 σ12 1e-04 3e-04 5e-04 Year 1960 2000 1940 1960 1980 Year 2000 1940 1960 1980 Year 2000 1940 1960 1980 Year 2000 γ22 0.2 0.4 0.6 0.8 1.0 1960 1980 Year γ11 0.2 0.4 0.6 0.8 1.0 1940 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.

(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

1:250 10 3 251:500 10 3 1:250 10 3 251:500 10 3 =1700 =1700 N Nt N =250 N t t t t

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

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

t t

{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

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 ).

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

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

p = 0.5 1/N

25.0 33.3 12.5 5.0 6.7 2.5 10.0 13.3 5.0 5.0 60 6.7 80 2.5 5.0 6.7 2.5 5.0 6.7 2.5 5.0 6.7 2.5 10.0 10.0 20.0 40 5.0 5.0 10.0 20 17.5 17.5 35.0 30 70 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 =

{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 ).

{Y t : t 1, 2,..., T } 1 1/ T 0.9998 T

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

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

p p p p p = 0 p p p p

(p = 0.5) p = 0.5 p = 0.5

(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

(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)

t 1/N

t r r t t = 100 12 t i=t 20 (P i/p i 1 ) 2 P t t

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

r t t r 0.24

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.

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

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

t r τ < t t

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

100

l 1 l 2 l 0

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) ),

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 1 1 2 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

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).

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

(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

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

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

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

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

(Σ (i), µ (i) ) (Σ (i 1), µ (i 1) ) λ

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

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

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

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

K K 309 4782 K = 10 K (K ) 2 LT n

ϕ(b) K = 10 λ = 10 4 K λ K n = 309 309 309 3 3 K = 3 λ = 5 10 2 8

λ = 5 10 3 λ = 10 2 λ = 5 10 2 λ = 10 1 λ λ K = 10

K = 3 λ = 5 10 2 300 1282 K = 4 λ = 10 3 λ 10 6 10 3

λ = 10 K = 9 Σ i = A (i) A (i)t, i = 1,..., 10 A (i) R 25 25 A (i) j,k K = 9 i Σ i 25 1000 K = 9 K + 1 = 10 λ λ = 10 λ = 10 λ λ 10 3 10 3 λ λ 10 3 10 3

[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

t l 1

l 1

{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 ),

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 ).

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.

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.

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

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

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.

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 +

ψ t (u t ) = ρ T u t, ρ κ l 1 V t

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

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

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

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

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

(γ = 0, κ = 0) (γ = 2, κ = 0) (γ = 0, κ = 0.02) (St = 1) (St = 1) (St = 1) γ κ κ = 0.02 0.75 1 0.17 K = 100 κ = 0.001 (γ = 0, ρ = 0) γ = 2

(γ = 0, ρ = 0) (γ = 2, ρ = 0) (γ = 0, ρ = 0.02) γ (γ = 0, ρ = 0.02) ρ = 0.02 κ = 0.001 γ = 2

(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)

(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

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

1/N

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 +

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 ) 1 + + (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.

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

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

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 )

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

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.

(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

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

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

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

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

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

(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 ),

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

(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,

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

(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

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)

ˆr ˆr + δ δ ρ ρ R n ρ ˆr i ρ i ρ i 6 ± 2 4 8 (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

Σ Σ = Σ +, ij κ (Σ ii Σ jj ) 1/2, κ [0, 1) κ κ κ κ = 0.02 0.05 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

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)

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

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 = 0.4 40 20 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 ),

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

O(nk 2 ) n k k n 3/2 12 100 O(nk 2 )

(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

K l 1 K ϕ t γ 1/γ a t 3/2 3/2

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

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 ).

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

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

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

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

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

w t v t z t t

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

0.07 0.07 0.12 0.11 0.10 0.36 0.25 0.19 0.20 0.17 0.13 0.36 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

(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 = 0.0005 α = 0.024 ±0.3 α 0.15

t i ( ˆV t ) i = 1 10 10 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

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

γ γ γ H = 2 H = 2 H = 1

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,

3/2

1/N

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

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

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

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

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

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

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

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

{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

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

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

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 + 1 2 )i + ( µ s { (Σ ) s ij } ( ) Σ s 1, ii ) } 1 j + 1 2 }. { (Σ s )ii + ( Σ s ) jj } } i j s

γ = 5 γ = 5 1/n

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

H = 10, 15,..., 30 w = 0.2, 0.3,..., 0.5 w = 0.2 w = 0.4 (κ 1 ) 1:n = 0.0005, 0.001,..., 0.0055 (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 = 0.004 ρ 2 = 0, 0.0005,..., 0.002 ρ T 2 wt 2 ρ 2 (ρ 2 ) n+1 = 0 ρ 2 = 0.0005 l 1 ρ T 1 w t γ 0 = 5 γ 0 = 5 γ 0

285 1.0 4 Empirical results 0.6 0.4 0.0 0.2 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 2000 2002 2004 2006 2008 2010 2012 2014 2016 Year 1.0 TBill IFL bonds CORP bonds GVT bonds EM HY bonds DM HY bonds Oil Gold Real estate EM stocks DM stocks 0.0-1.0 Asset weight 2.0 (a) γ = 5, (κ1 )1:n = 0.004, ρ2 = 0.0005, (wmax )1:n = 0.4, (wmax )n+1 = 1. 2000 2002 2004 2006 2008 2010 2012 2014 2016 Year ) ) ( ( (b) γ = 5, (κ1 )1:n = 0.004, ρ2 = 0.0005, 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-2013. 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

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 0.0-1.0 Asset weight 2.0 286 2000 2002 2004 2006 2008 2010 2012 2014 2016 Year 1.0 TBill IFL bonds CORP bonds GVT bonds EM HY bonds DM HY bonds Oil Gold Real estate EM stocks DM stocks 0.0-1.0 Asset weight 2.0 ) ( (a) γ = 5, (κ1 )1:n = 0.004, ρ2 = 0.0005, (wmax )1:n = 0.4, wmin n+1 = (wmax )n+1 = 1. 2000 2002 2004 2006 2008 2010 2012 2014 2016 Year ) ( (b) γ0 = 5, (κ1 )1:n = 0.004, ρ2 = 0.0005, (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.

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

D =0.1 1/n γ 0 = 5 1/n

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

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

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

1/N