212 1 Chinese Journal of Applied Probability and Statistics Vol.28 No.5 Oct. 212 (,, 3387;,, 372) (,, 372)., HJB,. HJB, Legendre.,. :,,, Legendre,,,. : F83.48, O211.6. 1.,.,,. 199, Sharpe Tint (199),., [2 6]. Leippold, Trojani Vanini (24), ; Chiu Li (26) ; Xie Li (28),, ; (25), ; (11YJC796) (21821) (9JCYBJC18). 211 2 18, 211 5 5.
458 (25),. [1 4], [5, 6].,, ( ).,,, ;,,,. Xie Li (28),,. [9] HJB,,,, Legendre [7, 8],.,. 2. ( ), x = x x = x 2 1 + x2 2 + + x2 n x = (x 1, x 2,..., x n ), E( ), [, T ]. (W 1 t, W 2 t,..., W n t ) (Ω, F, P, {F t } t T ) n, {F t } t T (W 1 t, W 2 t,..., W n t ). L 2 (Ω, F, P) Hilbert. [, T ] n + 1,,, t P t, P t : dp t = P t r t dt, P = p >, (2.1) r t > t. n,, t i Pt i, i = 1, 2,..., n, Pt i : dpt i = Pt [b i i tdt + n ] σ ij t dw j t, P i = p i >, (2.2) j=1 b t = (b 1 t, b 2 t,..., b n t ) σ t = (σ ij ) n n., σ t = (σ ij ) n n : σ t σ t >, t [, T ], r t, b t, σ t [, T ] Borel-.
: 459 t = w > l (l R), t = x = w l >. L t t, L t dl t = L t [u t dt + v t dw t ], W t (Ω, F, P, {F t } t T ), u t, v t t. W t W j t, ρj t, j = 1, 2,..., n, ρ t = (ρ 1 t, ρ 2 t,..., ρ n t ), W t = (Wt 1, Wt 2,..., Wt n ) ρ tρ t = 1, L t : dl t = L t [u t dt + v t ρ tdw t ], L = l >. (2.3) t i π i t, i = 1, 2,..., n, t [, T ], π t = (πt 1, πt 2,..., πt n ), X t t, t πt = X t n πt. i,, π t X t : dx t = (X t n πt i ) dp t P t + n πt i dpt i Pt i dl t L t. dx t = [r t X t + B tπ t u t ]dt + (π tσ t v t ρ t)dw t, X = x >, (2.4) B t = (b 1 t r t, b 2 t r t,..., b n t r t ). θ t θ t = σ 1 t B t. π t Γ = {π t : t T }., : Max E(U(X T )), (2.5) π t Γ U( ) (, + ), π t = (π 1 t, π 2 t,..., π n t ).,. :,, U(x) = x β /β, β < 1, β ;,, U(x) = e βx, β > ;, U(x) = ln x.
46 1 : (i),, Xie Li (28) ; (ii) HJB Legendre, Xie Li (28). 3. [9], H(t, x) H(t, x) = H(t, x) HJB : sup E(U(X T ) X t = x), {π t Γ} { H t + sup (r t x + B tπ t u t )H x + 1 } π t 2 (π tσ t v t ρ t) 2 H xx =. H(T, x) = U(x), H t, H x, H xx H(t, x) t x x. HJB HJB 3.1 π t = (σ t σ t) 1( H x H xx B t + σ t ρ t v t ). (3.1) H t + (r t x u t + θ tρ t v t )H x 1 2 θ t 2 H2 x =, (3.2) H xx ( dxt = r t X t u t θ t 2 H ) x + θ H tρ t v t dt θ H x t dw t. (3.3) xx H xx, HJB (3.2) : H(t, x) = (x g(t))β f(t), (3.4) β g(t ) =, f(t ) = 1. : H t = (x g(t))β β H x = (x g(t)) β 1 f(t), f(t) (x g(t)) β 1 ġ(t)f(t), H xx = (β 1)(x g(t)) β 2 f(t).
: 461 (3.2) (x g(t)) β β [ f(t) β ] + βr t f(t) 2(β 1) θ t 2 f(t) (x g(t)) β 1 [ġ(t)f(t) (r t g(t) u t + θ tρ t v t )f(t)] =. f(t) β + βr t f(t) 2(β 1) θ t 2 f(t) =, f(t ) = 1, ġ(t)f(t) (r t g(t) u t + θ tρ t v t )f(t) =, g(t ) =. { ( β f(t) = exp βr s t 2(β 1) θ s 2) } ds, (3.5) { t } g(t) = exp r z dz (u s θ sρ s v s )ds. (3.6) t,, (2.5) : 3.1 g(t) (3.6). s U(x) = x β /β, β < 1 β, (2.5) π t = (σ t σ t) 1( X t g(t) 1 β B t + σ t ρ t v t ), (3.7) (3.7) (3.3), EXt t = X e [rz [1/(β 1)] θz 2 ]dz + t, [ t e s [rz [1/(β 1)] θz 2 ]dz (θ sρ s v s u s ) + g(s) β 1 θ s 2] ds. EXT = X e [rz [1/(β 1)] θz 2 ]dz + [ T e s [rz [1/(β 1)] θz 2 ]dz (θ sρ s v s u s ) + g(s) β 1 θ s 2] ds. (3.8) 3.1 U(x) = x β /β, β < 1 β,, u t = v t = ρ t =, (2.5) π t = X t 1 β (σ tσ t) 1 B t, EXT = X e [rt [1/(β 1)] θt 2]dt. (3.9)
462 2 (3.7), (2.5), [X t /(1 β)](σ t σ t) 1 B t,, (σ t σ t) 1 {[ g(t)/(1 β)]b t + σ t ρ t v t },,., (3.7) X t, t. 3.2, (3.2) : H(t, x) = exp { βxe t r zdz + h(t) }, h(t ) =. H t = H ( βxr t e t r zdz + ḣ(t)), H x = H ( βe t r ) zdz, H xx = H ( βe t r ) zdz 2. (3.2) 1 H(t, x) [ḣ(t) 2 θ t 2 + (u t θ tρ ] T t v t )βe t r zdz =. ḣ(t) 1 2 θ t 2 + (u t θ tρ t v t )βe t r zdz =, h(t ) =. h(t) =, : 3.2 t (u s θ sρ T T s v s )βe s rzdz ds t 1 2 θ s 2 ds. U(x) = e βx, β >, (2.5) πt = (σ t σ t) 1( 1 β B te t r zdz + σ t ρ t v t ). (3.1) (3.3) dxt = (r t Xt u t + 1 β θ t 2 e ) T t r zdz + θ tρ t v t dt + 1 β θ te t r zdz dw t. X t = X e t rzdz + t t 1 + β θ se t r zdz dw s., EX T = X e rzdz + (θ sρ s v s u s )e s t rzdz ds + 1 β θ s 2 ds + t 1 β θ s 2 e t r zdz ds (θ sρ s v s u s )e s T rzdz ds. (3.11)
: 463 3.2 U(x) = e βx, β >, u t = v t = ρ t =, πt = 1 β (σ tσ t) 1 B t e T t r zdz, EXT T T = X e rzdz + 1 β θ s 2 ds. (3.12) 3, (2.5) (3.1),,., (3.1) t X t, u t. 3.3 (3.2), Legendre [7, 8],, (3.2),,., Legendre Ĥ(t, z) = sup{h(t, x) zx}, x> g(t, z) = inf {x H(t, x) zx + Ĥ(t, z)}, x> z > x. g(t, z) Ĥ(t, z) g(t, z) = Ĥz(t, z), g(t, z) H(t, x). H x (t, x) = z : g(t, z) = x, Ĥ(t, z) = H(t, g) zg. H(T, x) = U(x), T Legendre : Ĥ(T, z) = sup{u(x) zx}, x> : g(t, z) = inf {x U(x) zx + Ĥ(T, x)}. x> g(t, z) = (U ) 1 (z). H x = z, H t = Ĥt, H xx = 1. (3.13) Ĥ zz (3.13) (3.2), z, g t (r t g u t + θ tρ t v t ) r t zg z + 1 2 θ t 2 z 2 g zz + θ t 2 zg z =, (3.14) g t, g z, g zz g(t, z) t z z.
464 g(t, z) = 1 z., (3.14) : g(t, z) = 1 a(t) + b(t), a(t ) = 1, b(t ) =. z g(t, z) : g t (t, z) = 1 z ȧ(t) + ḃ(t), g z(t, z) = 1 z 2 a(t), g zz(t, z) = 2 z 3 a(t). (3.14) z 1 z ȧ(t) + ḃ(t) r tb(t) + u t θ tρ t v t =. ȧ(t) =, a(t ) = 1, ḃ(t) r t b(t) + u t θ tρ t v t =, b(t ) =. a(t) = 1, b(t) = t { t exp s } r z dz (u s θ sρ s v s )ds. π t = (σ t σ t) 1[ H x H xx B t + σ t ρ t v t ] = (σ t σ t) 1 [zĥzzb t + σ t ρ t v t ] = (σ t σ t) 1 [ zg z B t + σ t ρ t v t ] = (σ t σ t) 1[ 1 z a(t)b t + σ t ρ t v t ] = (σ t σ t) 1 [(g b(t))b t + σ t ρ t v t ] = (σ t σ t) 1 [(x b(t))b t + σ t ρ t v t ]. : 3.3 π t = (σ t σ t) 1[( X t U(x) = ln x, (2.5) t { t exp, (3.15), X t s } ] r z dz (u s θ sρ s v s )ds )B t + σ t ρ t v t. (3.15) dxt = [r t X t u t + θ t 2 (X t b(t)) + θ tρ t v t ]dt θ t dw t. H xx { EXt t } = X exp (r s + θ s 2 )ds + t H x { t } exp (r z + θ z 2 )dz (θ sρ s v s u s θ s 2 b(s))ds. s
: 465, { EXT T } = X exp (r s + θ s 2 )ds 3.3 + exp { s } (r z + θ z 2 )dz (θ tρ s v s u s θ s 2 b(s))ds. (3.16), u t = v t = ρ t =, (2.5) { πt = X t (σ t σ t) 1 B t, EXT T } = X exp (r s + θ s 2 )ds. (3.17) 4 (3.15),, (2.5) β =. 4., (2.5),. 1 x = 1,, T = 1., u t =.25, v t =.5, ρ 1 t =.3, ρ 2 t =.5, ρ 3 t =.81. r t =.2, b t = {.12,.22,.18},.23.34.46 σ t =.34.58.61..46.61.25, β =.2. β, u t, v t. 1, β β -.5 -.2.2.3.35.4 π 1 t 18.5 225.67 338.59 386.99 416.77 451.52 π 2 t 119.2 148.64 222.69 254.43 273.96 296.75 πt 3 14.62 13.78 196.17 224.2 241.45 261.57 πt i 44.15 55.1 757.47 865.62 932.18 19.83 π t 595.85 494.9 242.53 134.37 67.82-9.83 EX T 181.5 197.44 1138.31 1156.29 1167.5 118.71
466 2, u t u t -.5 -.2.25.5 3. 5. πt 1 338.85 338.75 338.59 338.51 337.67 337. πt 2 222.86 222.79 222.69 222.64 222.9 221.65 πt 3 196.32 196.27 196.18 196.13 195.64 195.25 πt i 758.3 757.8 757.47 757.28 755.4 753.9 πt 241.97 242.2 242.53 242.72 244.6 246.1 EX T 1139.2 1138.85 1138.31 1138.2 1135.6 1132.69 3, v t v t -2.5 -.2.25 2.5 5. 8. πt 1 339.43 338.79 338.66 338.3 337.33 336.49 πt 2 219.32 221.91 222.41 224.94 227.75 231.12 πt 3 196.13 196.17 196.17 196.2 196.24 196.28 πt i 754.9 756.87 757.25 759.18 761.32 763.89 πt 245.11 243.13 242.75 24.82 238.68 236.11 EX T 1137.68 1138.16 1138.26 1138.74 1139.27 1139.91 : (a1) 1, 3 πt β i ;, π t β ; EX T β ; 1, β =.4, {451.52, 296.75, 261.57}, πt = 9.83 <, 9.83. (a2) 2, 3 πt u i t ; πt u t ; EX T u t. (a3) 3, πt v i t ; πt v t ; EX T v t,. 2 x, T, r t, b t, σ t, u t, v t, ρ 1 t, ρ 2 t, ρ 3 t 1,, β =.1, β, u t, v t. : (b1) 4, 3 πt β i ;, π t β ; EX T
: 467 4, β β 1/2 1/15 1/12.1.3.1 πt 1 531.26 398.42 318.71 265.57 88.45 26.45 πt 2 349.3 261.92 29.65 174.8 58.65 17.99 πt 3 37.66 23.73 184.57 153.8 51.22 15.32 πt i 1187.96 891.6 712.93 594.17 198.32 59.77 πt -187.96 18.94 287.7 45.83 81.69 94.23 EX T 1195.58 1151.7 1125.37 117.81 149.3 128.82 5, u t u t -2. -.25.25 1. 5. 1. πt 1 265.57 265.57 265.57 265.57 265.57 265.57 πt 2 174.8 174.8 174.8 174.8 174.8 174.8 πt 3 153.8 153.8 153.8 153.8 153.8 153.8 πt i 594.17 594.17 594.17 594.17 594.17 594.17 πt 45.83 45.83 45.83 45.83 45.83 45.83 EX T 111.9 118.32 117.81 117.6 113.2 197.96 6, v t v t -1-2.5 3. 5. 1. πt 1 267.98 266.15 265.57 264.1 264.54 263.39 πt 2 162.8 171.94 174.8 177.66 179.95 185.66 πt 3 155.26 154.15 153.8 153.45 153.17 152.47 πt i 586.4 592.24 594.17 596.11 597.66 61.53 πt 413.96 47.76 45.83 43.89 42.34 398.47 EX T 115.87 117.35 117.81 118.28 118.65 119.57 β ; β = 1/2, πt = 187.96 <, 187.96. (b2) 5,, πt π i t u t ; EX T u t,. (b3) 6,, πt v i t ; πt v t ; EX T v t.
468 3 x, T, r t, b t, σ t, u t, v t, ρ 1 t, ρ 2 t, ρ 3 t 1,, u t, v t, T. 7, u t u t -1. -5..25 3. 5. 1. πt 1 273.64 272.31 27.9 27.16 269.62 268.28 πt 2 18.1 179.22 178.29 177.81 177.46 176.58 πt 3 158.47 157.7 156.88 156.45 156.14 155.37 πt i 612.22 69.22 66.7 64.42 63.22 6.22 πt 387.78 39.78 393.93 395.58 396.78 399.77 EX T 1125.37 1119.64 1113.61 111.46 118.16 112.42 8, v t v t -1-5.5 2 5 1 πt 1 272.79 271.9 27.9 27.63 27.9 269.18 πt 2 165.95 171.83 178.29 18.6 183.58 189.46 πt 3 158.4 157.49 156.88 156.72 156.38 155.83 πt i 596.79 61.21 66.7 67.4 61.5 614.47 πt 43.21 398.79 393.93 392.6 389.95 385.53 EX T 1111.4 1112.45 1113.61 1113.93 1114.56 1115.61 9, T T.25.5 1 2 5 1 πt 1 27.93 27.92 27.9 27.86 27.73 27.55 πt 2 178.32 178.31 178.29 178.27 178.19 178.7 πt 3 156.9 156.89 156.88 156.86 156.79 156.68 πt i 66.14 66.12 66.7 65.98 65.71 65.3 πt 393.86 393.88 393.93 394.2 394.29 394.7 EX T 127.27 155.28 1113.61 124.12 1712.56 2932.85 : (c1) 7, 3 πt u i t ;, πt u t ; EX T u t.
: 469 1, r t r t.1.1.2.5.1.14 πt 1 321.82 297.67 27.9 19.35 56.19-51.13 πt 2 191.82 185.4 178.29 156.85 121.17 92.62 πt 3 18.5 169.11 156.88 12.44 59.62 1.96 πt i 693.7 652.18 66.7 467.64 236.98 52.45 πt 36.3 347.82 393.93 532.36 763.2 947.55 EX T 1119.96 1116.48 1113.61 1111.16 1127.72 116.25 (c2) 8, 3 πt v i t ; πt v t ; EX T v t. (c3) 9,. πt π i t T, EX T T (c4) 1, r t. r t ;, π t r t ; EX T r t. 5. πt i,,, HJB Lengdre, Xie Li (28).,.,.! [1] Sharpe, W.F. and Tint, L.G., Liabilities - A new approach, The Journal of Portfolio Management, 16(2)(199), 5 1. [2] Leippold, M., Trojani, F. and Vanini, P., A geometric approach to multiperiod mean variance optimization of assets and liabilities, Journal of Economic Dynamics and Control, 28(6)(24), 179 1113.
47 [3] Chiu, M.C. and Li, D., Asset and liability management under a continuous-time mean-variance optimization framework, Insurance: Mathematics and Economics, 39(3)(26), 33 355. [4] Xie, S., Li, Z. and Wang, S., Continuous-time portfolio selection with liability: Mean-variance model and stochastic LQ approach, Insurance: Mathematics and Economics, 42(3)(28), 943 953. [5],,,, 25(9)(25), 42 48. [6],,,, 25(8)(25), 5 54. [7] Xiao, J., Zhai, H. and Qin, C., The constant elasticity of variance (CEV) model and the Legendre transform-dual solution for annuity contracts, Insurance: Mathematics and Economics, 4(2)(27), 32 31. [8] Gao, J.W., Stochastic optimal control of DC pension funds, Insurance: Mathematics and Economics, 42(3)(28), 1159 1164. [9] Yong, J. and Zhou, X.Y., Stochastic Controls: Hamiltonian Systems and HJB Equations, New York: Springer-Verlag, 1999. Optimal Control for Utility Portfolio Selection with Liability Chang Hao (Department of Mathematics, Tianjin Polytechnic University, Tianjin, 3387 ) (School of Management, Tianjin University, Tianjin, 372 ) Rong Ximin (School of Science, Tianjin University, Tianjin, 372 ) In this paper we use stochastic optimal control theory to investigate a dynamic portfolio selection problem with liability process, in which the liability process is assumed to be a geometric Brownian motion and completely correlated with stock prices. We apply dynamic programming principle to obtain Hamilton-Jacobi-Bellman (HJB) equations for the value function and systematically study the optimal investment strategies for power utility, exponential utility and logarithm utility. Firstly, the explicit expressions of the optimal portfolios for power utility and exponential utility are obtained by applying variable change technique to solve corresponding HJB equations. Secondly, we apply Legendre transform and dual approach to derive the optimal portfolio for logarithm utility. Finally, numerical examples are given to illustrate the results obtained and analyze the effects of the market parameters on the optimal portfolios. Keywords: Liability process, dynamic portfolio selection, dynamic programming principle, Legendre transform, power utility, exponential utility, logarithm utility. AMS Subject Classification: 93E2, 91B28, 6H3.