Vol. 34 ( 2014 ) No. 4. J. of Math. (PRC) : A : (2014) XJ130246).

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1 Vol. 34 ( 2014 ) No. 4 J. of Math. (PRC) (, ) :. -,,, [8].,,. : ; - ; ; MR(2010) : 91A30; 91B30 : O225 : A : (2014) ,. [1],. [2],.,,,. [3],.,,,.,,,,.., [4].,.. [5] -,. [6] Markov. [7]. [8]. HJBI,. [8]. [8] : : ( ); (XJ120106; XJ120109; XJ130246). : (1983 ),,,, :.

2 780 Vol HJBI, -. 2.,, [8] : (1) ; (2) ; (3)., (Ω, F t, P ),, F t P. [9], dc(t) = αdt βdw 1 (t),, α > 0, β > 0,, W 1 (t). dc(t) = (1 + v)αdt, v > 0. dx(t) = c(t)dt dc(t) = vαdt + βdw 1 (t)., a(t) 1. 0 a(t) 1, a(t) = 1, a(t) = 0. a(t) < 0,. η, η > v. dx(t, a(t)) = (v ηa(t))αdt + (1 a(t))βdw 1 (t).,, (), t {B(t), t 0} db(t) = r 0 B(t)dt, r 0 > 0. (), t {S(t), t 0} ds(t) = S(t)[rdt + dw 2 (t)], r r 0, > 0, W 2 (t), W 1 (t), W 2 (t).. π(t) t, X(t, u(t)) π(t), X(t, u(t)), u(t) = (a(t), π(t))., X(t, u(t)) dx(t, u(t)) = [(v ηa(t))α + π(t)(r r 0 ) + r 0 X(t, u(t))]dt +(1 a(t))βdw 1 (t) + π(t)dw 2 (t). (2.1) 1 u( ) = (a( ), π( )), u( ) {F t }, t 0 u( ) : (1) T 0 [π(t)] 2 dt < a.e. T < ;

3 No. 4 : 781 (2) a(t) 1; (3) (1) {u(t), t 0}. U. {θ(t), t 0} (Ω, F t, P ), (1) {θ(t), t 0} F t ; (2) (t, ω) [0, + ) Ω, θ(t) = θ(t, ω) < 1; (3) E T 0 θ2 (t)dt <. θ(t) Θ(t). {θ(t), t 0} Θ {Z θ (t), t 0} Z θ (t) = ex{ Ito t 0 θ(s)dw 1 (s) t 0 θ(s)dw 2 (s) t 0 θ 2 (s)ds}. dz θ (t) = Z θ (t)θ[dw 1 (t) + dw 2 (t)]. (2.2) Z θ (t) (F t, P ), {θ(t), t 0} P -, Z θ (0) = Z 0 (0 < Z 0 < 1), Z θ (t) (F t, P ), EZ θ (T ) = EZ θ (0) = 1. dp θ dp = Zθ (T ), θ Θ, P θ. 2.2 W, W > 0, W < 0, W. u( ), P θ V u,θ (t, x) = E θ [W (XT u) Xu t = x], E θ P θ.,.,, u( ). su inf V u,θ (t, x) = V u,θ (t, x) = V (t, x), (2.3) θ Θ u U u, θ., u, θ V (t, x) W (x) = 1 m e mx, m > 0., ηα β(r r0) 1 g 1 (t) g 1(t) + g 1 (t)mα(η v)e r0(t t) g 1(t) 4 (r r 0 ηα β )2 = 0, g 1 (T ) = 1, (3.1) g 1 (t) = ex{ mα(η v) r 0 [e r0(t t) 1] T t ( r r 0 ηα 4 β )2 }. (3.2)

4 782 Vol. 34 (3.1) (3.2).. 1 (2.3) π = a = 1 e r0(t t) 2m 2 e r0(t t) (r r 0 ηα ); (3.3) β (ηα β(r r 0) 2mβ 2 θ = 1 2 (r r 0 + ηα β ); (3.4) ). (3.5) V (t, x) = 1 m g 1(t) ex{ mxe r0(t t) }, (3.6) g 1 (t) (3.2). u( ), θ( ), X(t, u) (2.1), Ito, (3.1) d[ 1 m g 1(t)Z θ (t)e mx(t,u)er 0 (T t) ] 1 m zg 1(t)e mxer 0 (T t) = Z θ (t)e mx(t,u)er 0 (T t) { 1 m g r0(t t) 1(t) r 0 X(t, u)g 1 (t)e m 2 g 1(t)e 2r0(T t) [β 2 (1 a) π 2 ] +g 1 (t)e r0(t t) [(v ηa(t))α + π(t)(r r 0 ) + r 0 X(t, u(t))] g 1 (t)e r0(t t) [(1 a)β + π]θ}dt + 1 m g 1(t)θZ θ (t)e mx(t,u)er 0 (T t) [dw 1 (t) + dw 2 (t)] +g 1 (t)θz θ (t)e r0(t t) e mx(t,u)er 0 (T t) θ[(1 a)βdw 1 (t + πdw 2 (t] = Z θ (t)e mx(t,u)er 0 (T t) { m2 2 g 1(t)e 2r0(T t) [π(t) π (t)] 2 mβ2 2 g 1(t)e 2r0(T t) [a(t) a (t)] 2 + g 1(t) 2m [θ(t) θ (t)] 2 }dt + 1 m g 1(t)θZ θ (t)e mx(t,u)e r 0 (T t) [dw 1 (t) + dw 2 (t)] +g 1 (t)θz θ (t)e r0(t t) e mx(t,u)er 0 (T t) θ[(1 a)βdw 1 (t) + πdw 2 (t)], π, a, θ (3.3) (3.5), t T, Z θ (t) = z, X(t, u) = x P, Beyes, V u,θ (t, x) = 1 m g 1(t)e mxer 0 (T t) + 1 z E{ T t Z θ (s)e mx(s,u)er 0 (T s) [ m2 2 g 1(s)e 2r0(T s) [π(s) π (s)] 2 mβ2 2 g 1(s)e 2r0(T s) [a(s) a (s)] 2 + g 1(s) 2m [θ(s) θ (s)] 2 ]}ds.

5 No. 4 : 783 g 1 (t) > 0, Z θ (t) > 0,.. 1 (3.3), (3.4), a < 1,, (π < 0) ηα < β(r r0) 2 g 2 (t) g 2(t) + g 2 (t)mα(η v)e r0(t t) = 0, g 2 (T ) = 1, (3.7) mα(η v) g 2 (t) = ex{ [e r0(t t) 1]}. (3.8) r 0 (3.7) (3.8).. 2 (2.3) π = 0; (3.9) a = 1; (3.10) θ = r r 0. (3.11) V (t, x) = 1 m g 2(t) ex{ mxe r0(t t) }, (3.12) g 2 (t) (3.8). (3.4) ηα < β(r r0), a > 1, a = 1. a = 1 (2.1), 1 zg m 2(t)e mxer 0 (T t) Ito, (3.8) (3.9), (3.10), a = 1,,. 3 [8], ρ = [8] HJBI, - ( ρ = 0, ρ 0 - ). 3.2 W (x) = x (0 < < 1),, ηα β(r r0) 3 f 1 (t), h 1 (t) f 1(t) + f 1 (t)[r 0 + 4(1 ) (r r 0 ηα) 2 ] = 0, f 1 (T ) = 1, (3.13) h 1(t) r 0 h 1 (t) + (η v)α = 0, h 1 (T ) = 0, (3.14)

6 784 Vol. 34 f 1 (t) = ex{[r 0 + 4(1 ) (r r 0 ηα) 2 ](T t)}, (3.15) (η v)α h 1 (t) = [1 e r0(t t) ]. r 0 (3.16) (3.13), (3.14) (3.15), (3.16).. 3 (2.3) π = x h 1(t) 2(1 ) (r r 2 0 ηα ); (3.17) β π = 1 x h 1(t) 2(1 )β (ηα β(r r 0) ); (3.18) 2 θ = 1 2 ((r r 0) + ηα β ). (3.19) V (t, x) = f 1 (t) [x h 1(t)], (3.20) f 1 (t), h 1 (t) (3.15), (3.16). u( ), θ( ), X(t, u) (2.1), Ito, (3.13), (3.14) d[f 1 (t) [x h 1(t)] Z θ (t)] f 1 (t) [X(t, u) h 1(t)] z θ = Zθ (t) {f 1(t)[X(t, u) h 1 (t)] + f 1 (t)( [X(t, u) h 1 (t)] 1 h 1(t) +[X(t, u) h 1 (t)] 1 [(v ηa(t))α + π(t)(r r 0 ) + r 0 X(t, u(t))] ( 1)[X(t, u) h 1(t)] 2 [β 2 (1 a) π 2 ] [X(t, u) h 1 (t)] 1 [(1 a)β + π]θ)}dt f 1 (t) [x h 1(t)] Z θ (t)θ(t)[dw 1 (t) + dw 2 (t)] +f 1 (t)[x(t, u) h 1 (t)] Z θ (t)[(1 a)βdw 1 (t) + πdw 2 (t) = Zθ (t) { 1 2 ( 1)2 [X(t, u) h 1 (t)] 2 [π(t) π (t)] ( 1)β2 [X(t, u) h 1 (t)] 2 [a(t) a (t)] (1 )β [θ(t) 2 θ (t)] 2 }dt f 1 (t) [x h 1(t)] Z θ(t)θ (t)[dw 1 (t) + dw 2 (t)] +f 1 (t)[x(t, u) h 1 (t)] Z θ (t)[(1 a)βdw 1 (t) + πdw 2 (t)],

7 No. 4 : 785 π, a, θ (3.17) (3.19), t T, Z θ (t) = z, X(t, u) = x P, Beyes, V u,θ (t, x) = f 1 (t) [X(t, u) h 1(t)] + 1 z E{ T t Z θ (s) [ 1 2 ( 1)2 [X(s, u) h 1 (s)] 2 [π(s) π (s)] ( 1)β2 [X(s, u) h 1 (s)] 2 [a(s) a (s)] ( 1)β [X(s, u) h 1(s)] 2 [θ(s) θ (s)] 2 }ds, (3.17), (3.18), a < 1,, ( π < 0 ) ηα < β(r r0) 4 f 2 (t), h 2 (t) f 2(t) + r 0 f 2 (t) = 0, f 2 (T ) = 1, (3.21) h 2(t) r 0 h 2 (t) + (η v)α = 0, h 2 (T ) = 0, (3.22) f 2 (t) = e r0(t t), (3.23) (η v)α h 2 (t) = h 1 (t) = [1 e r0(t t) ]. r 0 (3.24) (3.21), (3.22) (3.23), (3.24).. 4 (2.3) π = 0; (3.25) π = 1; (3.26) θ = r r 0. (3.27) V (t, x) = f 2 (t) [x h 2(t)], (3.28) f 2 (t), h 2 (t) (3.23), (3.24). (3.4) ηα < β(r r0), a > 1, a = 1. a = 1 (2.1), f 2 (t) [X(t,u) h2(t)] z θ Ito, (3.23), (3.24) (3.25), (3.26), a = 1,,..

8 786 Vol. 34 [1] Browne S. Otimal investment olicies for a firm with a random risk rocess: exonential utility and minimizing the robability of ruin[j]. Mathematics Methods Oerator Research, 1995, 20(4): [2] Yang H, Zhang L. Otimal investment for insurer with jum diffusion risk rocess[j]. Insurance: Mathematics and Economics, 2005, 37(3): [3] Bai L, Guo J. Otimal roortional reinsurance and investment with multile risky assets and no-shorting constraint[j]. Insurance: Mathematics and Economics, 2008, 42(3): [4] Isaacs R. Differential games [M]. New York: Wiley, [5] Mataramvura S, Sendal B. Risk minimizing ortfolios and HJBI equations for stochastic differential games [J]. Stochastics An International Journal of Probability and Stochastic Processes, 2008, 4: [6] Siu T K. A game theoretic aroach to otion valuation under Markovian regime-switching models[j]. Insurance: Mathematics and Economics, 2008, 42(3): [7] Browne S. Stochastic differential ortfolio games [J]. Journal of Alied Probability, 2000, 37(1): [8],. [J]., 2101, 8: [9] Promislow D S, Young V R. Minimizing the robability of ruin when claims follow Brownian motion with drift[j]. North American Actuarial Journal, 2005, 9(3): STOCHASTIC DIFFERENTIAL GAMES WITH REINSURANCE AND INVESTMENT YANG Peng (Deartment of Basic, Xijing College, Xi an , China) Abstract: This aer investigates a stochastic differential games roblem with reinsurance and investment. Under exonential utility and ower utility, by using linear-quadratic control theory, we obtain otimal reinsurance strategies, investment strategies and otimal market strategies as well as the value function closed form exressions. We generalize the results of aer [8]. Through this research when the market is worst, we can guide insurance comany to select the aroriate reinsurance and investment strategies to maximize his wealth. Keywords: stochastic differential games; linear-quadratic control; exonential utility; ower utility 2010 MR Subject Classification: 91A30; 91B30

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