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Kory McCoy
5 years ago
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5 l t

33 1

35 2

36 2 2

37 2 2

39 B B

40 m t t m 0 ṁ t = e ref a t m t B e ref a t t t x t a t δ a t0 = 0 ȧ t = x t δa t c(x t ) c x t, c (x t ) 0 c (x t ) 0 c(x t ) 0 c (x t ) c

41 c (x t ) x t t a t c (x t ) x t c(x t ) e ref min x t 0 e rt c(x t ) dt m t B ϕ t ṁ t = e ref a t ȧ t = x t δa t a t e ref µ t ν t λ t a t x t a t ν t µ t λ t e ref

42 B T T e ref t T, m t = B = a t = e ref = x t = δe ref T µ t r t < T, µ t = µe rt µ a t < e ref t < T, (r + δ) c (x t ) dc (x t ) dt = µe rt c (x t ) t c i (x i,t ) dt δ

43 (+) >0 <0 + (r + δ)c (x t ) x t t + dt c i (x i,t) + d dt c i (x i,t)dt r

44 t < T, dc (x t ) dt = (r + δ) c (x t ) µe rt µe rt (r + δ) c t (r + δ) c > µe rt dc (x t )/dt > 0 (r + δ) c < µe rt dc (x t )/dt < 0 µ (r + δ) c (δe ref ) µ (r + δ) c (δe ref ) µ = (r + δ)c (x 0 ) x 0

45 a t e ref c (x t ) t < T, c (x t ) = T t µe rθ e (δ+r)(θ t) dθ + e (δ+r)(t t) c (δe ref ) e (δ+r)(θ t) µe rθ T c (δe ref ) t T x, c (x) = C t µe rt = (r + δ)c

46 T T c (x t ) 2 / c (x t ) 2 / c (x t ) x t dt 2

47 l t l t = (r + δ) c (x t ) t $ 2 / 1/δ = 10 5%/ r + δ = 15%/ 2 l t = (r + δ) c (x t ) = µe rt + dc (x t ) dt $/( / ) = $ 2 / $ 2 /= x t l t x t = c 1 ( l t r+δ ).

48 l t µe rt dc (x t) ( dt ) dc (x t) > 0 dt i ā i i āi = e ref δ i c i min e rt (c i (x i,t )) dt x i,t 0 i ȧ i,t = x i,t δ i a i,t ν i,t a i,t ā i ṁ t = i λ i,t (ā i a i,t ) µ t m t B ϕ t ν i,t λ i,t i µ t

49 i, t < T i, (r + δ i ) c i (x i,t ) dc i (x i,t ) dt = µe rt i T i i, t < T i, c i (x i,t ) = Ti t µe rθ e (δ i+r)(θ t) dθ + +e (δ+r)(t i t) c i (δ i ā i ) ( δi = δ j, ā i > ā j y, c i(y) = c j(y) ) = t, c i (x i,t ) > c j (x j,t ) ( δi = δ j, ā i = ā j y, c i(y) > c j(y) ) = t, c i (x i,t ) > c j (x j,t )

50 ā i c i

51 γ i i, a i,t [0, ā i ] γ i (a i,t ) = 1 2 γm i a 2 i,t γ i (a i,t ) = γ m i a γi m 2 ā i 2

52 [ ] [ ] ā i 2 γi m 2 2 δ / i [%/ ] c m 2 i 2 3 i, x i,t 0, c i (x i,t ) = 1 2 cm i x 2 i,t c i (x i,t ) = c m i x i,t c m i (i, j), c m i c m j = γm i γ m j c m i T = 23 r = 4%

53 T 0 (ā i a i,t ) B i B 2 i āi B = δ i 2 2

54 δ i x i,t 2 x i,t x i,t a i,t

56 2 2

57 2 2

58 H(x t, a t, m t ) = e (c(x rt t ) + λ t (a t e ref ) + ν t (δa t x t ) ) + µ t (e ref a t ) + ϕ t (m t B) H = 0 x t c (x t ) = ν t H d (e rt ν t ) = 0 a t dt ν t (δ + r)ν t = λ t µ t H m + d (e rt µ t ) = 0 dt µ t rµ t = ϕ t ν t µ λ t (r + δ) c (x t ) dc (x t ) dt = (µ t λ t ) e ref e ref

59 t, (m t B)ϕ t = 0 m t B = µ t = µe rt t, (a t e ref )λ t = 0 T T t T, ṁ t = 0 = a t = e ref = x t = δe ref T a t < e ref x t a t e ref T

60 a t < e ref t a t < e ref, dc (x t ) dt = (r + δ) c (x t ) µe rt θ t > θ a t < e ref (r + δ) c (x θ ) < µe rθ (r + δ) c (x t ) e ref θ (r + δ) c (x θ ) = µe rθ dc (x θ ) = 0 c θ dθ θ c a t t < T a t a t t ȧ t = 0 ä t 0 ȧ t = x t δa t ä t = ẋ t δȧ t = ẋ t ẋ t 0 x t a t τ > t ẋ τ 0 m t B a t t < T, a t < e ref T < + c (x t ) = c (x 0 )e (r+δ)t µert ( e δt 1 ) δ + e ref t T t T, a t = e ref a t T e ref,t T

61 T c t < T, a t < e ref λ t = 0 t T, a t = e ref λ t 0 T t < T, (r + δ) c (x t ) dc (x t ) dt = µe rt t < T, T c (x t ) = e (r+δ)t e (r+δ)θ µe rθ dθ + e (r+δ)t C t C C T T a t x t c (x T ) = c (δe ref ) t < T, T c (x t ) = µe rt e δ(θ t) dθ + e (δ+r)(t t) c (δe ref ) t

62 c (x t ) = T t µe rθ e (δ+r)(θ t) dθ + T (r + δ) c (δe ref ) e (δ+r)(θ t) dθ T c (δe ref ) T c (x t ) E O K t < T, c (x t ) = µe rt e δ(θ t) dθ t }{{} E µe rt e δ(θ t) dθ T }{{} O + e (r+δ)(t t) c (δe ref ) } {{ } K O e ref T O T K c (δe ref ) T (r+δ)c (δe ref ) dc (x t )/dt = 0 T

63 min x t 0 ) e (c(x rt t ) + d(m t ) dt ṁ t = e ref a t µ t ȧ t = x t δa t a t e ref ν t λ t d(m t ) H(x t, a t, m t ) = e (c(x rt t ) + d(m t ) + λ t (a t e ref ) + ν t (δa t x t ) ) + µ t (e ref a t ) H = 0 x t c (x t ) = ν t H d (e rt ν t ) = 0 a t dt ν t (δ + r)ν t = λ t µ t H m + d (e rt µ t ) = 0 dt µ t = rµ t d (m t )

64 B λ t = 0 µ t t T, c (x t ) = T t µ θ e (δ+r)(θ t) dθ + T (r + δ) c (δe ref ) e (δ+r)(θ t) dθ µ t min x t 0 ) e (c(x rt t ) + µ t (e ref a t ) dt ȧ t = x t δa t ν t a t e ref λ t l t h t i 2 / h t δ i A 2 A = θ=t e r(θ t) h e δ i(θ t) dθ = h r + δ

65 h x t (x t + h) C $ C = c(x t + h) c(x t ) = h 0 h c (x t ) l t l t = C A l t = (r + δ) c (x t ) (x t ) (a t ) θ 1 δ dθ θ + dθ ( x t ) (ã t ) θ θ + dθ (x t ) ( x t ) P = 1 dθ [ ] c (1 δ dθ) (x θ ) (1 + r dθ) c (x θ+ dθ ) P dθ 0 (r + δ) c (x θ ) dc (x θ ) dθ P θ

66 (x t ) (a t ) θ e ref T T θ (ã t ) θ (1 δ dθ) θ + dθ ( x t ) H(x i,t, a i,t, m t ) = e rt ( i c i (x i,t ) + λ i,t (a i,t ā i ) + ν i,t (δ i a i,t x t ) i i ) +µ t (ā i a i,t ) + ϕ t (m t B) i

67 (i, t) c i (x i,t ) = ν i,t ( x i,t ) ν i,t (δ i + r)ν i,t = λ i,t µ t ( a i,t ) µ t rµ t = ϕ t ( m t ) ν i,t µ λ i,t T m ṁ t = 0 = i, a i,t = ā i = x i,t = δ i ā i T i i i T i, a i,t = ā i T m = max i (T i ) t < T m, m t < B ϕ t = 0 = µ t = µe rt T i t < T i, a i,t < ā i λ i,t = 0 (i, t), (r + δ i ) c i (x i,t ) dc i (x i,t ) dt = µe rt λ i,t

68 Ti c i (x i,t ) = µe rt e δi(θ t) dθ + e (δ+r)(ti t) c i (δ i ā i ) = Ti t t µe rθ e (δ i+r)(θ t) dθ + T i (r + δ i ) c i (δ i ā i ) e (δ i+r)(θ t) dθ T i i, t < T i, dc i (x i,t ) dt = (r + δ i )c i (x i,t ) µe rt δ i (i, j), δ i = δ j t min(t i, T j ) c i(x i,t ) = c j(x j,t ) = t min(t i, T j ), c i(x i,t ) = c j(x j,t ) {1, 2}

69 (+) (+) 1 ( ) ( ) x > 0, c 1(x) = c 2(x), δ 1 = δ 2 = δ, ā 1 > ā 2 t max(t 1, T 2 ), (r + δ)c 1(δā 1 ) > (r + δ)c 2(δā 2 ) t [T 1, T 2 ] { t < t, c 1(x 1,t ) c 2(x 2,t ) t > t, c 1(x 1,t ) > c 2(x 2,t ) = t < T 1, x 1,t x 2,t = T1 0 x 1,t e δ(t 2 t) dt = ā 1 < T1 0 x 2,t e δ(t 2 t) dt = a 2,T1 a 2,T2 ā 2 ā 1 < ā 2 c 1(x 1,t ) c 2(x 2,t ) t, c 1(x 1,t ) > c 2(x 2,t ) dci (x i,t) dt < 0 T i T 1 > T 2

70 x > 0, c 1(x) < c 2(x) ( ā 1 = ā 2 δ 1 = δ 2 ) ā 2 = a 2,T2 > ā 1 = ā 2 c 1 > c 2 ā 1 < ā 2 δ 1 δ 2 i µe rt i T i τ τ τ c i(x i,τ ) t τ δ i c i(x i,t ) V i (τ) V i (τ) = µτ + e rτ c i(x i,τ ) + τ e rt δ i c i(x i,t )dt

71 V τ ( V i (τ) = µ + e rτ c i(x i,τ ) r d ) dτ c i(x i,τ ) e rτ δ i c i(x i,τ ) = (r + δ i ) c i(x i,τ ) d dτ c i(x i,τ ) µe rτ V i (τ) = λ i,τ V i (τ) τ T i µe rt t a t γ γ a t, γ (a t ) > 0 γ (a t ) > 0 γ(a t ) > 0 γ (a t ) a t γ(a t )

72 t 0 r min a t 0 e rt γ (a t ) dt a t e ref λ t ṁ t = e ref a t m t B µ t ϕ t µe rt e ref T 0 t t 0 γ (a t ) = µe rt t 0 < t < T γ (e ref ) t T t 0

73 γ(a t ) a t µe rt γ ) H(a t, m t ) = e (γ(a rt t ) + λ t (a t e ref ) + µ t (e ref a t ) + ϕ t (m t B) ( a t ) γ (a t ) = (µ t λ t ) ( m t ) µ t rµ t = ϕ t

74 T ṁ t = 0 e ref t < T, a t < e ref m t < B t T, a t = e ref m t = B ϕ t t, ϕ t (m t B) = 0 = t < T, ϕ t = 0 µ t t < T, µ t = µe rt λ t t, λ t (a t e ref ) = 0 = t < T, λ t = 0

75 2

79 h l z t x i,t i k t δ eq

80 k i,t = x i,t δk t, i x i,t 0 k l,t0 = k z,t0 = 0 c i c i (x) > 0, c i(x) > 0, c i (x) > 0, x

81 c l(x) < c z(x), x c i(0) = 0 c i(0) i c i (x i,t ) i k t 0 c i (x i,t ), i c i (x i,t ) k t F i i c (x) c (0) c (x) c (0)

82 F h > F l > F z = 0 B m t B, t m t F i q j,t ṁ t = i F i q j,t S i,t S i,t0 Ṡ i,t = c i (x i,t ) S i,t 0 S z,t0 = i S h,t0 = u ( i c i (x i,t )) u x i,t c i (x i,t )

83 B ( ) max e [u rt c i (x i,t ) ] c i (x i,t ) dt x i,t,c i (x i,t ) 0 i i ki,t = x i,t δk t (ν t ) q j,t k j,t (γ i,t ) q j,t 0 (λ i,t ) x i,t 0 (ξ i,t ) ṁ t = F i q j,t i (µ t ) m t B (η t ) Ṡ i,t = c i (x i,t ) (α i,t ) S i,t 0 (β i,t ) r ν t γ i,t µ t α i,t i α h,t α l,t µ t

84 α h,t = α h e rt, α l,t = α l e rt, µ t = µe rt α h,t α l,t µ t α h = 0 α l,t = 0 α h,t α l,t α h,t α l,t α h,t α l,t γ i,t ν t c i (x i,t ) c i (x i,t ) ν t ξ i,t c i (x i,t ) = ν t + ξ i,t ν t c i(0) ξ i,t = 0 ν t > c i(0) c i (x i,t ) = ν t x i,t > 0 ν t c i(0) ν t γ i,t γ i,t = (δ + r) ν t ν t, t

85 γ i,t ν t γ i,t t ν t dt γ i,t δ t + dt ν t + ν t dt r ν t γ i,t ν t x i,t > 0 γ i,t = (δ + r) c i (x i,t ) d dt c i (x i,t ) r c i (x i,t ) = e (r+δ)t γ i,t dt γ i,t c i (x i,t ) > 0 u t = γ i,t + α i,t + µ t F i u t u t u ( i c i (x i,t )) γ i,t α i,t F i i µ t t

86 γ i,t i c i (x i,t ) < k t = γ i,t = 0 i j t 0 < c i (x i,t ) < k t & 0 < q j,t < k j,t = u t = α i,t + F i µ t = α j,t + F j µ t i α i,t + F i µ t t t t u t 0 < q h,t < k h,t = u t = α h,t + F h µ t

87 0 < q l,t < k l,t = u t = α l,t + F l µ t c i (x i,t ) = k t = γ i,t > 0 u t = γ i,t + α i,t + µ t F i t γ i

88 $/ $/ + = + l + l 0 l + l + l + / l + l 0 l / = + l + l + $/ $/ + l + l = l + l 0 + l / 0 = + ( ) = / z l h α i µf i p q u γ i = p α i µf i

90 γ i,t ν t ν t > c i(0) i

91 γ z,t = u t ν z,t = t e (r+δ)(θ t) u θ dθ, t (u θ ) (e (r+δ)(t θ) ) τ z + c z(x z,t ) = t e (r+δ)(θ t) u θ dθ, t τ + z T γ t > T γ = γ l,t = 0 ν l,t = Tγ t e (r+δ)(θ t) (u θ µ θ F l α l,θ )dθ ν l,t > c l (0) τl Tγ τ l e (r+δ)(θ τ l ) (u θ µ θ F i α i,θ )dθ = c l(0)

92 Tγ ν z,t ν l,t = e (r+δ)(θ t) (µ θ F l + α l,θ ) dθ + c z(x z,tγ )e (r+δ)(t Tγ) 0 t } {{ } } {{ } γ ν γ t τl ν T γ T γ ν z,t = c z(x z,tγ ) τ z + τ + l Th Th < τ z + τ + l Th < τ + l τ z + Th c l (0) = 0 τ l = T γ

93 τ + l Th < τ z + τ + l τ z + Th τ z + τ + l Th τ z + Th τ + l ( c i (x i,t ) = C i (1 A) x i,t + 1 ) 2 Ax2 i,t = c i (x i,t ) = C i ((1 A) + Ax i,t ) A (0, 1) C i C z > C l A = 0 A = 1 lim xi,t 0;A=0 c i (x i,t ) = 0 A (0, 1)

95 T hl = Th = τ + l T lz = T γ = τ z + τ z + = τ + l = t 0 Th

96 α i,t α h,t ( ) max e [u rt c i (x i,t ) ] (c i (x i,t ) + c i (x i,t )α i,t ) dt x i,t,c i (x i,t ) 0 i i ki,t = x i,t δk t (ν t ) q j,t k j,t (γ i,t ) q j,t 0 (λ i,t ) x i,t 0 (ξ i,t ) ṁ t = F i q j,t i (µ t ) m t B (η t ) T hl T lz T lz T hl

97 Ci m X i H i F i 2

98 r B D 0 G P e 2 δ A 1

99 D t P P = 90 e =.1 D t q(p) = D t e (p P ) p u t(q) ( u(q) = P + D ) t q q2 e 2e D 0 D 0 = G = D t = D 0 + t G k t c i (x i,t ) H i

100 D 0 k z,0 = k l,0 = 0; k h,0 = D 0 H h (X i ) Ci m c i (x i,t ) = C m i X i ( (1 A) x i,t X i + A 2 ( xi,t X i ) 2 ) = c i (X i ) = C m i e ref,t = F h = B = 22 2 B F l = r = 5 % δ δ = 1/

101

102 2 A B 2 A > 0.5 $/ A =.125 4$/

103

104

105 ( ) H = e [u rt c i (x i,t ) c i (x i,t ) ν t (δ k t x i,t ) i i i µ t F i q j,t η t (m t B) (α i,t c i (x i,t ) β i,t S i,t ) i i γ i,t (q j,t k j,t ) + λ i,t q j,t + ] ξ i,t x i,t i i i H = 0 x i c i (x i,t ) = ν t + ξ i,t H = 0 q i λ i,t µ t R i α i,t + u t = γ i,t H = d (e rt ν t ) k i dt (δ + r) ν t ν t = γ i,t H = d (e rt µ t ) m t dt µ t r µ t = η t H = d (e rt α i,t ) S i dt α i r α i,t = β i,t i, t, ξ i,t 0, x i,t 0 ξ i,t x i,t = 0 i, t, λ i,t 0, c i (x i,t ) 0 λ i,t c i (x i,t ) = 0 i, t, η t 0, B m t 0 η t (B m t ) = 0 i, t, β i,t 0, S i,t 0 β i,t S i,t = 0 i, t, γ i,t 0, k t c i (x i,t ) 0 γ i,t (k t c i (x i,t )) = 0

106 η t β i,t α h,t = α h e rt α l,t = α l e rt µ t = µe rt

107 i h l z h l z k t i t c i (x i,t ) i t x i,t i t c i (x i,t ) i t ν t k t µ t 2 α i,t i γ i,t i u t m t 2 δ 1 r 1 F i i 2 D t t B 2 u ( i c i (x i,t ))

108

109 3 12 2

110

111

112 2

113 2 2 2

114 2 / 2 / 2 2 2

115

116 2 2

117 2

118 2 2

119

120

121

122

123

124

125 4

126 Marginal cost $/tco 2 Y N-1 N c i A i 4 X 5 Abatement potential MtCO 2 /yr N i A i c i 1 4

127

128

129 2 2 2

130 N i t t E ref t a i,t i t e t = E ref t N i=1 a i,t M t M t = e t t + M t t i A i 2

131 a i,t A i i c i a i,t i t I i,t I i,t = a i,t c i v i 2 t t t a i,t a i,t t + v i t 2 a i,0 = 0) A i A i = A i a i,0

132 T Ãi A i v i Ã i = min (v i T, A i ) A i /v i C r C = T t=t 0 N i=1 I i,t (1 + r) t t M t M obj M t M obj

133 c 2 A 2 v 2 2 m e tm E obj m N = 2 v r = 5% = 5 2 E ref t

134 2 M obj 2

135

136 2 2 m {1}, t 1 = 2050 E obj 1

137

138

139 Marginal abatement cost $/tco Marginal abatement cost $/tco GtCO 2 /yr GtCO2 /yr t 1 Ã 2 t 2 Ã 2

140 E obj 1 = E(2020) =

141 E obj 1 = E(2020) = E obj 2 = E(2050) =

142

143

144

145 5

146 Marginal cost $/tco 2 Y 5 c A X Abatement potential at T MtCO 2 /yr

147

148 Emissions GtCO 2 /yr Marginal cost $/tco 2 T Wedge curve time Potential at T MtCO 2 /yr Flipped MAC curve

149

150

151

152

153

154 2 2

155

156 +

157

158 2

159

160 MtCO Emission reductions in Metro Rail and Waterways In the 2030 strategy Bullet train Refineries Heat Integration In the 2020 strategy New processes Marginal cost $/tco Marginal cost $/tco Potential MtCO 2 /yr in Potential MtCO 2 /yr in 2030 D 2020 D 2030 D 2020 D 2030

161

162

163 i A i,t c i v i a T T r a i,t i t min e rt c i a i,t a i,t i,t (i, t), a i,t A i,t v i a i,t+1 a i,t + v i a i,t a T i

164 2 2

165

166

167 6

168

169

170

171

172

173 r t c(x t ) ȧ t = r t a t c(x t ) t c(x t ) u (c(x t )) u > 0 u < 0 W W = ρ 0 e ρt u(c(x t )) dt y t k p k c c(x t ) i p,t i c,t y t = c(x t ) + i p,t + i c,t i p,t i c,t δ k p,t = i p,t δ k p,t k c,t = i c,t δ k c,t

174 i p,t 0 i c,t 0 q t k t y t y t = F (A t, q p,t, q c,t ) q p,t k p,t q c,t k c,t A t q t k t t G q p,t m t ε ṁ t = G q p,t εm t t, q c,t = k c,t

175 ċ c = u (c) c u (c) (r t ρ) u (c) cu (c) > 0 r t ρ R p,t R c,t Π t = F (A t, q p,t, q c,t ) R c,t k c,t R p,t k p,t R c,t R p,t t qb F (q p,t, q c,t ) = R p,t qg F (q p,t, q c,t ) = R c,t R p,t = R c,t = r t + δ

176 m t m m t m max c,i,k 0 e ρt u(c(x t )) dt F (q p, k c ) c(x t ) i p,t i c,t = 0 k p,t = i p,t δk p,t k c,t = i c,t δk c,t ṁ t = G q p,t εm t m t m i p,t 0 q p,t k p,t λ t ν t χ t µ t d(m t ) ψ t β t λ t ν t χ t µ t t

177 u (c t ) = λ t = ν t + ψ t = χ t kc F = 1 λ ((δ + ρ)χ t χ t ) β t = 1 λ ((δ + ρ)ν t ν t ) qp F = β t + τ t G τ τ t = µ t λ t r t t, m t < m = τ t = τ t (r t + ε) m t = m k p,t = m ε/g ṁ t = 0 A t R c,t R p,t R c,t := 1 λ [(δ + ρ)χ t χ t ] R p,t := 1 λ [(δ + ρ)ν t ν t ] χ t ν t (R c,t, R p,t ) (χ t, ν t ) δ ρ χ t ν t

178 kc F = R c,t qp F = R p,t + τ t G β t R p,t r t r t = R c,t δ R p,t = R c,t l t l t l t = 1 λ t ((ρ + δ)ψ t ψ t ) [0, R c,t ] ψ t ν t χ t ψ t R p,t = β t 0 l t = R c,t R p,t R c,t

179 l t τ t τ t }{{} = q p F kc F G } {{ } l t [0, kc F ] + l t G }{{} 0 < l t R c,t R p,t < R c,t i p,t = 0

180 t 0 t 0 t i t (t 0, t i ), i b = 0 q p,t < k p,t k p,t q p,t t ss l t = 0 R p,t = R c,t i p,t > 0 l t kc F (= R c,t ) t 0

181 l t = R c,t R τ t G > kp F (k p,t, k c,t ) = p,t = 0 q p,t < k p,t qp F (q p, k c ) = τ t G R p,t R p,t β t m k b,t0 F u δ ρ τ t G

182 k b,0 m

183 θ c,t > 0 θ p,t > 0 π t t π t = F (q p,t, q c,t ) (λ t θ c,t ) i c,t (λ t + θ p,t ) i p,t λ t θ c,t θ p,t t ss,2 t ss,1 q p,1,t0 < k p,t0 q p,2,t0 = k p,t0 t 0 F (q p,2,t0, k c,t0 ) F (q p,1,t0, k c,t0 )

184 qp F = kc F 1 ((ρ + δ)ψ t λ ) ψ t + 1 ((δ + ρ)(θ c,t + θ p,t ) ( θ } t λ c,t + ) θ p,t ) {{}} t {{} l t τ t,2 G l t τ t,2 τ t,2 }{{} = q p F kc F G } {{ } + l t G }{{} l t τ t,2 t

185 y c y t ss t ss,2 < t ss,1

186 t > t 0, ν t = χ t + θ c,t + θ p,t ψ t ν t χ t (θ c,t + θ p,t ψ t 0)

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189 k b,t0 m m t0 ρ δ

190 δ ε m 0 k 0 m < m 0 + G k 0 δ

191 H h (c t, a t ) = e ρt {u(c t ) + λ t [r t a t + y t c t ]} λ t t W t, c H h = 0 λ t = u (c t ) t, a H h + d(e ρt λ t ) dt = 0 λ t = (ρ r t )λ t λ

192 c t = u (c t ) c t c t u (c t ) (r t ρ) H t = e ρt { u(c(x t )) + λ t [F (q p, k c ) c(x t ) i p,t i c,t ] + ν t [i p,t δk p,t ] +χ t [i c,t δk c,t ] µ t [G q p,t εm t ] + d(m t ) [ m m t ] } + ψ t i p,t + β t [k p,t q p,t ] t ψ t 0 ψ t i p,t = 0 t β t 0 β t (k p,t q p,t ) = 0 t d(m t ) 0 d(m t ) ( m m t ) = 0 H t = 0 c t u (c t ) = λ t H t = 0 i p,t λ t = ν t + ψ t H t = 0 i c,t λ t = χ t H t = d(e ρt ν t ) k p,t dt ν t δ + β t = ν t + ρν t H t = d(e ρt χ t ) k c,t dt λ t kc F (k p,t, k c,t ) χ t δ = χ t + ρχ t

193 H t = 0 q p,t λ t qp F (q p,t, k c,t ) µ t G = β t H t = d(e ρt µ t ) m t dt d(m t ) + εµ t = µ t ρµ t λ t λ t c t u (c t ) u (c t ) c t = (ρ + δ R c,t ) c t r t r t := R c,t δ µ t µ t = ( λ t τ t + λ t τ t ) τ t = τ t [ε + r t ] d(m t) λ t t ss t t ss, m t = m ṁ t = 0 = G q p,t = ε m k p,t = m ε/g t ss d(m t ) = 0 τ t = τ t [ε + r t ]

194 r t + ε t 0 lim t t 0 lim t t 0 q p,t = k p,t qp F (q p,t, q c,t ) = kc F (q p,t, q c,t ) t + 0 ψ t = 0 lim qp F (q p,t, q c,t ) = kc F (q p,t, q c,t ) + τ t0 G t t + 0 lim q p,t = k p,t t t + 0 τ t0 > 0 lim qp F (q p,t, q c,t ) lim qp F (q p,t, q c,t ) t t + 0 t t + 0 qp F q p,t lim t t + q p,t 0 lim t t + q p,t 0

195 t Π t = F (q p,t, k c,t ) R c,t k c,t R p,t k p,t τ t G q p,t R p,t R c,t τ t ȧ t = r t a t + y t c(x t ) + τ t G q p,t L(t) = Π t + β t (k p,t q p,t ) + γ t (k c,t q c,t ) qg L = 0 qc F (q p,t, q c,t ) = γ t qb L = 0 qp F (q p,t, q c,t ) = β t + τ t G kg L = 0 γ t = R c,t kb L = 0 β t = R p,t t γ t 0 γ t (k c,t q c,t ) = 0 β t 0 β t (k p,t q p,t ) = 0 γ t = qc F (q p,t, q c,t ) > 0 q c,t = k c,t t kc F (q p,t, k c,t ) = R c,t qp F (q p,t, k c,t ) = R p,t + τ t G

196 R c,t = r t + δ R p,t = R c,t = r t + δ t 0 H t = e ρt {F (q p,t, q c,t ) (λ t θ c,t ) i c,t (λ t + θ p,t ) i p,t +ν t [i p,t δk p,t ] + χ t [i c,t δk c,t ] + ψ t i p,t + β t [k p,t q p,t ]} H t = 0 i p,t λ t + θ p,t = ν t + ψ t H t = 0 i c,t λ t θ c,t = χ t H t = d(e ρt ν t ) k p,t dt ν t δ + β t = ν t + ρν t H t = d(e ρt χ t ) k c,t dt ρχ t χ t = λ t kc F (k p,t, k c,t ) χ t δ H t = 0 q p,t λ t qp F (q p,t, k c,t ) = β t t, β t [k p,t q p,t ] = 0

197 F t, k p,t = q p,t ν t + ψ t = χ t + θ c,t + θ p,t kc F = 1 λ ((δ + ρ)χ t χ t ) qp F = 1 λ ((δ + ρ)ν t ν t ) qp F = kc F + 1 ((δ + ρ)(θ c,t + θ p,t ) ( θ λ c,t + ) θ p,t ) } t {{} θ t 1 ((ρ + δ)ψ t λ ) ψ t } t {{} l t l t θ t (θ c,t + θ p,t ) θ t = τ t,2 G τ t,2 θ t G θ c,t + θ p,t ψ t kc F qp F l t θ t (= τ t,2 ) ψ t θ c,t + θ p,t

198 l t = 0 θ t θ t t t ss, θ t = τ t G t ss τ t,2 τ t,2 }{{} = q p F kc F G } {{ } l t [0, τ t,2 ] + l t G }{{} l t τ t,2 i p,t σ t

199 t, i p,t = sd t σ t sd t sd t = 0 H t = e ρt {u(c(x t )) + λ t [F (q p, k c ) c(x t ) i p,t i c,t ] + ν t [i p,t δk p,t ] +χ t [i c,t δk c,t ] + σ t (sd t i p,t ) + β t [k p,t q p,t ]} λ t ν t χ t u (c t ) = λ t = ν t σ t = χ t λ t kc F = (δ + ρ)χ t χ t λ t qp F = β t β t = (δ + ρ)ν t ν t σ t (θ c,t + θ p,t ψ t ) R p,t = R c,t + n t n t = 1 ((ρ + δ)σ t σ t ) λ t n t = θ t l t n t

200 max c,i,k 0 e ρt u(c(x t )) dt F (q p, k c ) c(x t ) i p,t i c,t = 0 k p,t = i p,t δk p,t k c,t = i c,t δk c,t ṁ t = G q p,t εm t m t m i p,t 0 q p,t k p,t q p,t = k p,t λ t ν t χ t µ t d(m t ) ψ t β t α t H t = e ρt {u(c(x t )) + λ t [F (q p, k c ) c(x t ) i p,t i c,t ] + ν t [i p,t δk p,t ] +χ t [i c,t δk c,t ] µ t [G q p,t εm t ] + d(m t ) [ m m t ] + ψ t i p,t + β t [k p,t q p,t ] + α t [q p,t k p,t ]}

201 β t α t = 1 λ ((δ + ρ)ν t ν t ) qp F = β t α t + τ t G β t α t qp F = β t α t + τ t G β t > 0 α t = 0 α t > 0 β t = 0 β t = 0 qp F = α t + τ t G α t qp F = kc F l t + τ t G 0 < l t < τ t G l t > 0 l t = 0 i b > 0 i b = 0 ψ t > 0 R p,t < R c,t R p,t

202 k p,t = k 0 e δt ṁ = k 0 e δt ε m m t = G k 0 δ ε e δt + ( m 0 + G k 0 δ ε ) e εt m max = G k 0 δ t max = 1 δ ln(m max ε G k 0 ) e δt m m max = m m = G k ( m ε 0 δ ε eln( G k ) 0 + m 0 + G k ) 0 e ε m ε ln( ) δ G k 0 δ ε m = [ ( m 0 + G k ) ( ) ε 0 ε δ ε G k 0 δ ( δ ε δ ) ] δ δ ϵ

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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