Optimal financing and dividend control of the insurance company with proportional reinsurance policy

Transcription

1 Inurance: Mathematic and Economic Optimal financing and dividend control of the inurance company with proportional reinurance policy Lin He a, Zongxia Liang b, a Zhou Pei-Yuan Center for Applied Mathematic, Tinghua Univerity, Beijing 84, China b Department of Mathematical Science, Tinghua Univerity, Beijing 84, China Received July 27; received in revied form November 27; accepted 25 November 27 Abtract We conider the optimal control problem of the inurance company with proportional reinurance policy. The management of the company control the reinurance rate, dividend payout a well a the equity iuance procee to imize the expected preent value of the dividend minu the equity iuance until the time of bankruptcy. Thi i the firt time that the financing proce in an inurance model ha been conidered, which i more realitic. To find the olution of the mixed ingular-regular control problem, we firtly contruct two categorie of uboptimal model, one i the claical model without equity iuance, the other never goe bankrupt by equity iuance. Then we identify the value function and the optimal trategie correponding to the uboptimal model depending on the relationhip between the coefficient. c 27 Elevier B.V. All right reerved. MSC: primary 9B28; 9B6; econdary 6H5; 6H Keyword: Optimal dividend control; Optimal financing control; Proportional reinurance; HJB equation; Tranaction cot; Regular-ingular control. Introduction In thi paper, we conider an inurance company in which the dividend payout, equity iuance and the rik expoure are controlled by the management. We aume that the company can only reduce it rik expoure by proportional reinurance policy for implicity. Moreover, there exit a minimal reerve requirement. We aociate the value of the company with the expected preent value of the dividend payout minu the equity iuance. The tranaction cot i alo taken into conideration in the model. Thi i a mixed ingular-regular control problem on diffuion model. Diffuion model for companie that control their rik expoure by mean of dividend payout have attracted a lot of interet recently. We refer the reader to Radner and Sheep 996, Paulen and Gjeing 997, Højgaard and Takar 999; 2 and Amuen et al. 2. Optimizing dividend Correponding author. addree: hel4@mail.tinghua.edu.cn L. He, zliang@math.tinghua.edu.cn Z. Liang. payout i a claical problem tarting from the early work of Borch, 969; 967 and Gerber 972. For ome application of control theory in inurance mathematic, ee Harrion and Takar 983, Højgaard and Takar 998a; 998b, Martin- Löf 983, Amuen and Takar 997 and Cadenilla et al. 26. A urvey can be found in Takar 2. Guo et al. 23 i a theoretical work on contrained nonlinear ingularregular tochatic control problem. Unfortunately, there are very few reult concerned with the equity iuance of the inurance company. In the real financial market, equity iuance i an important approach for the inurance company to earn profit and reduce rik. Harrion and Takar 983 conider the optimal control problem with a lower and an upper reflecting barrier. Their work provide a good idea to olve thi kind of problem. Sethi and Takar 22 recently conidered the model for the company that can control it rik expoure by iuing new equity a well a by paying dividend. Løkka and Zervo 26 ue the above approache to tudy the problem with the poibility of bankruptcy. By thee innovative idea, we olve the optimal control problem of the inurance company /$ - ee front matter c 27 Elevier B.V. All right reerved. doi:.6/j.inmatheco.27..3

2 L. He, Z. Liang / Inurance: Mathematic and Economic effectively. The equity iuance could be conidered a the aborbing or reflecting boundary of the reerve proce. It turn out that the optimal control problem i aociated with different optimal trategie depending on the relationhip between the coefficient. Firtly, we tudy the olution of two model: one i the claical diffuion control model a in Højgaard and Takar 999, the other tand for the model with equity iuance to meet the minimal reerve requirement, o it never goe bankrupt. Then we prove that the value function and the optimal trategie are the olution of the two control problem. The paper i organized a follow: In Section 2, we etablih the control model of the inurance company. In Section 3, we give ome preliminary mathematical reult related to the problem. In Section 4, we contruct olution of two categorie of uboptimal model. One i the claical model without equity iuance, the other never goe bankrupt by equity iuance. In Section 5, we prove that the value function and the optimal trategie are the olution of the two control problem, repectively. We give the concluion of thi paper in the lat ection. 2. Control model of the inurance company with proportional reinurance policy We conider an inurance company with proportional reinurance policy. In thi cae, the company management can accommodate the profit and the rik by chooing the amount of equity iuance, dividend payout and the reinurance rate. In thi paper we will conider the linear Brownian motion model. In thi model, if there are no equity iuance and dividend payout to control the rik, then the liquid reerve of the company evolve according to the following tochatic differential equation, dr t = µatdt σ atdw t, 2. where W t i a tandard Brownian motion, at [, ] i the proportional reinurance rate. To give a mathematical foundation of the optimization problem, we fixed a filtered probability pace Ω, F, {F t t, P and {W t, t i a tandard Brownian motion on thi probability pace, where F t repreent the information available at time t and any deciion i made baed on thi information. In our model, we denote L t a the cumulative amount of dividend paid from time to time t, G t denote the total amount raied by iuing equity from time to time t. We aume that the procee {L t, t and {G t, t are {F t, t -adapted, increaing and right-continuou with left limit. A control policy π i decribed by the three tochatic procee {a π, L π, G π. Given a control policy π, we aume that the liquid reerve of the inurance company are modeled by the following equation, dr π t = µa π tdt σ a π tdw t dl π t dg π t, Rπ = x. 2.2 On the one hand, we uppoe the liquid reerve of the inurance company mut atify ome minimal reerve requirement. In thi cae, we aume that the company need to keep it reerve above m, m > i the minimal reerve requirement. The company i conidered bankrupt a oon a the reerve fall below m. We define the time of bankruptcy by τ = inf{t : Rt π < m. τ i an F t -topping time. If the company iue ome equity, then the time of bankruptcy could be infinite. On the other hand, we alo conider the tranaction cot in the model. To implify the problem, we conider the proportional tranaction cot. If the company pay l a dividend, then the hareholder can get β l, β <. In the meanwhile, the hareholder mut pay out β 2 g, β 2 > to meet the amount of g a new equity of the company. So the management of the inurance company hould imize the expected preent value of the dividend payout minu the equity iuance by control policy π. Our main objective i to imize the expected preent value of the dividend payout minu the equity iuance before bankruptcy, [ τ τ ] Jx, π = E e c β dl π e c β 2 dg π, 2.3 V x = up π Π Jx, π, 2.4 where Π = {π denote the et of all admiible policie, c denote the dicount rate. In addition, the minimal reerve requirement ak for V x =, for x < m. We aim at finding ome condition on coefficient µ, σ, c, β, β 2 of Eq. 2.2 and 2.3 uch that V x atifie a type of HJB equation. Then we get V x and the optimal trategy π aociated with V x. 3. Preliminarie of the problem In thi ection we will give two lemma before proving the main reult in Section 4 and 5. γ σ 2 Lemma 3.. Let x = µ and m < x, then there exit a unique x = x c, µ, σ, β > x atifying the following equation in x, γ x γ x β d e d x x d γ d d γ x γ x d γ β d d d e d x x =, 3. where d = µ µ 2 2cσ 2 σ 2, d = µ µ 2 2cσ 2 σ 2, γ = c c µ2 2σ 2. Proof. Clearly, d <, d >. Denote the left-hand ide of 3. by kx. Differentiating kx with repect to x, we have [ k x = β d d d d γ x γ x e d x x d γ ] γ x γ x e d x x. d γ

3 978 L. He, Z. Liang / Inurance: Mathematic and Economic Since x x, e d x x and e d x x. Uing m < x, γ x γ γ x γ x γ <, d x γ <. We claim that k x <. The claim i trivial if prove the claim for γ x γ d x γ <. γ x γ Becaue of γ x γ x d γ > γ x γ d x γ x d γ, >. It uffice to k x i negative. Therefore k x i alway negative, kx reache it imum at x on [x,. We deduce from kx, a x and kx = = = γ x γ γ x γ γ x γ x γ β x γ µ c µ 2c d d β d d β β > that Eq. 3. ha a unique olution x and x > x. x γ γ σ 2 Lemma 3.2. Let x = µ and m < x, there exit a unique x = x c, µ, σ, β, β 2 > x atifying the following equation in x, β d e d x x β d e d x x = β 2, d d d d where d, d and γ are the ame a in Lemma Proof. Denote the left-hand ide of Eq. 3.2 by hx. Differentiating hx with repect to x, we get h x = β d d e d x x e d x x. d d x γ Since x x, h x >, hx i a trictly increaing function of x. It reache it minimum at x and hx = mγ x γ β < β 2 becaue of m < x. Thu, uing hx a x, we know that Eq. 3.2 ha a unique olution x > x. 4. Two categorie of uboptimal olution In thi ection, we conider two categorie of uboptimal control problem. π p = {a p, L p, Π tand for the control proce for the company in which equity iuance i not permitted. The optimal return function aociated with π p i defined by V p x = up Jx, π p, for x m. π p Π π q = {a q, L q, G q Π i the control proce for the company with iuance procedure, i.e., it will never go bankrupt. The optimal return function aociated with π q i V q x = up Jx, π q, for x m. π q Π Since π p, π q Π, V x {V p x, V q x via 2.4. The two uboptimal olution will play an important role in contructing the overall optimal control olution. In the next two ubection, we will etablih olution for each category of the control model. 4.. The olution to the problem without equity iuance In thi ubection, we conider the company without equity iuance. Our objective i to imize the expected dicounted dividend payout. Theorem 4.. Aume that the coefficient are uch that x x, where x and x are defined in Lemma 3. and 3.2. Then the function f defined by f x = C x x γ C 2 x, m x < x, f = f 2 x = C 3 x e dx C 4 x e d x, x x < x, f 3 x = β x x f 2 x, x x 4. atifie the following HJB equation and the boundary condition: for x m, a [,] 2 σ 2 a 2 f x µa f x cf x, β f x =, 4.2 f m =. 4.3 Moreover, for x m, f x β 2, 4.4 where d, d, γ, x are the ame a in Lemma 3., C x, C 2 x, C 3 x and C 4 x are defined by β d C x = γ x γ e d x x d d β d e d x x, d d

4 L. He, Z. Liang / Inurance: Mathematic and Economic C 2 x = x β d e d x x d γ d d x β d e d x x, d γ d d β d C 3 x = e d x d d d, β d C 4 x = e d x d d d. Proof. By the tandard optimal control theory and the ame method a in Højgaard and Takar 999, Wendell and Fleming 993, we get that f atifie the following HJB equation, a [,] [ 2 σ 2 a 2 f x µa f x cf x ] =, for m x < x, 4.5 f x = β, for x x, 4.6 f x =, for x x, 4.7 where x will be pecified later. The imum in the left-hand ide of 4.5 i attained in the interior of the control region. Then, by differentiating w.r.t a, we can find the imizing function ax = µf x σ 2 f x. 4.8 If ax belong to the interval [, ], putting the expreion 4.8 into 4.5, we get the olution f x = C x x γ C 2 x 4.9 and ax = µx σ 2 γ. 4. The validity of the olution require ax [, ], which mean γ σ 2 µ. that x < x = On the other hand, if x x < x, we have ax = and 4.5 become 2 σ 2 f x µf x cf x =, 4. the olution of Eq. 4. i f 2 x = C 3 x e d x C 4 x e d x for x x < x. 4.2 For x x, the olution ha the following form, f 3 x = β x x f 2 x. 4.3 Continuity of the function f and f at point x implie that C 3 x d e d x C 4 x d e d x = β, C 3 x d 2 ed x C 4 x d 2 ed x =, i.e, β d C 3 x = e d x d d d <, β d C 4 x = e d x d d d >. Alo, ince the olution f and f are continuou at x, f x = f 2 x, f x = f 2 x, which imply that β d C x = γ x γ e d x x d d β d e d x x >, 4.4 d d C 2 x = x β d e d x x d γ d d x β d e d x x. 4.5 d γ d d Putting 4.9, 4.4 and 4.5 together, we have x = x and f m = by Lemma 3.. The problem remaining i to prove that the olution f atifie Noticing that β <, β 2 >, it uffice to prove the following condition: f β ; f β 2, for m x < x, 2 σ 2 a 2 f µa f cf, a [,] f β, f β 2, for x x < x, 2 σ 2 a 2 f µa f cf, for x x. a [,] The proof i a follow. For x x, f 3 = β x x f 2 x, we have a [,] 2 σ 2 a 2 f 3 µa f 3 cf 3 = µaβ cβ x x f 2 x µβ cf 2 x cβ x x becaue of x x and µβ cf 2 x =. Uing the ame way a in Højgaard and Takar 999, it i eay to prove that a [,] [ 2 σ 2 a 2 f 2 µa f 2 cf 2 ] hold for x x x ee Højgaard and Takar 999 for detail. Since f x = C x γ γ x γ 2, f 2 x = β d d d d e d xx e d xx, f x i a decreaing function on [m, x ], it imum and minimum are reached at m and x repectively. Moreover, f x = β and f x β are obviou. So it uffice to

5 98 L. He, Z. Liang / Inurance: Mathematic and Economic prove that f m β 2. Since f m = C x γ = mγ x γ β d d d e d x x β d e d x x d d = hx, x x and hx i a trictly increaing function, we have f m = hx hx = β 2 by Lemma 3.2. Thu the proof ha been done The olution to the problem that never goe bankrupt In thi ubection we conider the problem that aim at imizing the expected dicounted dividend payout minu the expected dicounted equity iuance over all reinurance, dividend payout and equity iuance trategie. Thi kind of inurance companie will never go bankrupt. Theorem 4.2. Aume that the coefficient are uch that x x, x, x are defined in Lemma 3. and 3.2. Then the function g defined by g x = C x x γ C 2 x, m x < x, g g = 2 x = C 3 x e dx C 4 x e d x, 4.6 x x < x, g 3 x = β x x g 2 x, x x atifie the following HJB equation and the boundary condition: a [,] 2 σ 2 a 2 g x µag x cgx, β g x, g x β 2 = 4.7 gm, 4.8 where γ, x, d and d are the ame a in Theorem 4., C x, C 2 x, C 3 x and C 4 x are defined a in Theorem 4. and by replacing x with x. Proof. Conidering the time value of money lead u to the concluion that it i optimal to potpone the new equity iuance a long a poible. We conjecture that it i optimal to iue equity only when the reerve become m. One point i that: if we iue equity at the reerve u prior to m, g u = β 2 and g x i a decreaing function, we conjecture that g u mut be to meet the requirement g β 2. Unfortunately, it i not compatible with a [, ]. Uing the ame way a in Section 4., thi trategy i aociated with a olution to the HJB equation 4.7. It hould be characterized by g m = β 2, 4.9 a [,] 2 σ 2 a 2 g x µag x cgx =, for m x < x, 4.2 g x = β, for x x, 4.2 g x =, for x x 4.22 where x i an unknown variable and x will be pecified later. Doing the ame procedure a in Section 4., we can prove that the olution gx of Eq ha the ame form a f x in , and x i the olution of the following equation γ x γ β d d d e d x x β d d d e d x x By Lemma 3.2, we have x = x and x x. γ = β 2. The problem remaining i to prove that the olution g atifie the condition mentioned in Theorem 4.2. For thi, it uffice to prove the following g β, g β 2, for m x < x, a [,] 2 σ 2 a 2 g µag cg, g β, g β 2, for x x < x, 2 σ 2 a 2 g µag cg, for x x. a [,] Uing the imilar approach a in Section 4., we can prove the above claim. Now, we will check the boundary condition. Baed on our aumption, reinvetment i not compulory and bankruptcy at m i an option, o we need only to prove gm, i.e., gm = γ x γ x d γ x d γ β d e d x x d d β d e d x x d d γ x γ By the proof of Lemma 3., kx = mγ γ x γ d x γ β d d d e d x x mγ γ x γ a decreaing function of x and kx =. Since x x, d x γ β d d d e d x x i gm = kx kx =, the inequality 4.23 hold. Thu we complete the proof. 5. The olution of the optimal control problem Theorem 5.. Let Wx atify the following HJB equation and boundary condition, { a [,] 2 σ 2 a 2 W µaw cw,

6 L. He, Z. Liang / Inurance: Mathematic and Economic which, together with 5.4, implie that β W, W β 2 =, for x m, 5. { E {e ct τ W Rt τ π E e c β dl π {W m, W m β 2 =. 5.2 { E e c β 2 dg π W x. 5.5 Then W x Jx, π for any admiible policy π. Proof. Fix a policy π, let Λ = { : L π Lπ, Λ = { : G π Gπ, ˆL = Λ, t Lπ Lπ be the dicontinuou part of L π and L π t = L π t ˆL π t be the continuou part of L π. Similarly, Ĝ and G tand for dicontinuou and continuou part of G π t. Let τ be the firt time that the correponding reerve R t defined by Eq. 2.2 hit, m. Then, by generalized Itô formula, e ct τ W Rt τ π = W x e c LW R π d a π σ e c W R π dw e c W R π dlπ e c W R π dgπ e c [W R π W Rπ Λ Λ, t τ W R π Rπ Rπ ] = W x where e c LW R π d a π σ e c W R π dw e c W R π d L π e c W R π d G π Λ Λ, t τ L = 2 a2 σ 2 d2 dx 2 aµ d dx c. e c [W R π W Rπ ], 5.3 In view of 5., the econd term on the right-hand ide i nonpoitive. Since β W R π β 2, the third term i a quare integrable martingale. Taking expectation on both ide of Eq. 5.3, {e ct τ W Rt τ π E { W x E e c W R π d L π { E e c W R π d G π E { Λ Λ, t τ Since β W R π β 2, e c [W R π W Rπ ] W R π W Rπ β 2G π Gπ β L π Lπ,. 5.4 By the definition of τ and β W x β 2 for x m, it i eay to prove that lim inf t ect τ W R π t τ = ecτ W mi {τ< lim inf t ect W R t I {τ= e cτ W mi {τ<. 5.6 So, we deduce from 5.5 and 5.6 that [{ τ τ ] Jx, π = E e c β dl π e c β 2 dg π W x. 5.7 Thu the proof ha been done. Let ax = { µx σ 2 γ, x < x,, x x where γ = c and x = γ σ 2 c µ2 µ. The main reult of thi 2σ paper are the following. 2 Theorem 5.2. If x x, then V x = f x = V p x. The optimal policy π = a π, L π, G π atifie the following R π t = x R π t x, t µar π d I {t:r π t <x tdlπ t =, G π t =, t σ ar π dw L π t, 5.8 where a π t = ar π t, x i given in Lemma 3., V x and f x are defined by 2.4 and 4., repectively. V p x i defined in Section 4. If x x, then V x = gx = V q x. The optimal policy π = a π, L π, G π atifie the following R π t = x t L π t µar π d t σ ar π dw G π t, m R π t x, I {t:r π t <x tdlπ t =, I {t:r π t >m tdgπt =, 5.9 where a π t = ar π t, x i given in Lemma 3.2, V x and gx are defined by 2.4 and 4.6, repectively. V q x i defined in Section 4.

7 982 L. He, Z. Liang / Inurance: Mathematic and Economic Remark that uing Theorem 3. in Lion and Sznitman 984 the procee R π, L π, G π and R π, L π, G π are uniquely determined by Eq. 5.8 and 5.9. Proof. If x x, then the function f x atifie the HJB equation and boundary condition It i not hard to ee that f x alo atifie condition 5. and 5.2 in Theorem 5.. So f x V x V p x by Theorem 5.. Next, we will prove V x = f x correponding to π. Applying generalized Itô formula, we deduce from 4.5 and 4.8 that L f R π t τ = and e ct τ f R π t τ = f x ar π t e c f R π σ e c f R π dw dl π e c [ f R π Λ, t τ f R π Rπ = f x R π ] e c β dl π e c L f R π d f R π ar π t σ e c f R π dw, 5. where τ = inf{t : R π t < m. Since lim inf t e ct τ f R π t τ = ecτ f m =, by taking expectation at both ide of 5., we get [ ] t τ f x = E lim up t e c β dl π = Jx, π. So f x i the return function correponding to π, and f x V p x. Uing the reult f x V x V p x, we have f x = V x = V p x under the circumtance x x. If x x, then gx defined in 4.6 atifie the HJB equation and boundary condition 4.7 and 4.8. Thu gx atifie condition 5. and 5.2 in Theorem 5.. So gx V x V q x by Theorem 5.. Next, we will prove V x = gx correponding to π. By generalized Itô formula, we deduce from 4. and 4.2 that LgR π t τ = and e ct τ gr π t τ = gx ar π t σ e c g R π dw e c g R π dl π e c g R π dg π e c [gr π Λ Λ, t τ e c LgR π d gr π g R π Rπ = gx R π ]. e c β dl π e c β 2 dg π ar π t σ e c g R π dw, 5. where τ = inf{t : R π t < m. Since lim inf t e ct τ gr π t τ = lim t e ct gr π t =, ee Højgaard and Takar 2 for detail, by taking mathematical expectation at both ide of 5., we get [ t τ gx = E e c β dl π = Jx, π. e c β 2 dg π ] So gx i the return function correponding to π, gx V p x. Uing the reult gx V x V q x, we have gx = V x = V q x under the circumtance x x. The proof ha been done. 6. Concluion In thi paper, we conider the optimal control problem of the inurance company with proportional reinurance policy. The management of the company control the reinurance rate, dividend payout and the equity iuance to imize the expected preent value of the dividend payout minu the equity iuance before bankruptcy. To be more realitic, we require the minimal reerve retriction and alo take tranaction cot into conideration. Thi i the firt time that the financing proce in an inurance model ha been conidered, and we finally find that it act a aborbing or reflecting boundary of the reerve proce. To find the olution of the mixed ingular-regular control problem, we contruct two categorie of uboptimal model, one i the claical model without equity iuance, the other never goe bankrupt by equity iuance. At lat, we identify the value function and the optimal trategy with the correponding olution in either category of uboptimal model, depending on the relationhip between the coefficient. Acknowledgement We are very grateful to the referee for bringing the paper Harrion and Takar 983 to our attention. We alo expre our deep thank to the referee and the editor for their careful reading of the manucript, correction of error, improvement of the written language and valuable uggetion which made the main reult of thi paper much better. Thi work i upported by Project 774 of NSFC, Project 263 of SRFDP, and SRF for ROCS, SEM. We would like to thank the intitution for their generou financial upport.

8 L. He, Z. Liang / Inurance: Mathematic and Economic Reference Amuen, S., Højgaard, B., Takar, M., 2. Optimal rik control and dividend ditribution policie: example of exce-of-lo reinurance for an inurance corporation. Finance and Stochatic 4, Amuen, S., Takar, M., 997. Controlled diffuion model for optimal dividend pay-out. Inurance: Mathematic and Economic 2, 5. Borch, K., 969. The capital tructure of a firm. Swedih Journal of Economic 7, 3. Borch, K., 967. The theory of rik. Journal of the Royal Statitical Society. Serie B 29, Cadenilla, A., Choulli, T., Takar, M., Zhang, L., 26. Claical and impule tochatic control for the optimization of the dividend and rik policie of an inurance firm. Mathematical Finance 6, Gerber, H.U., 972. Game of economic urvival with dicrete and continuou income procee. Operation Reearch 2, Guo, X., Liu, J., Zhou, X., 23. A contrained nonlinear regular-ingular tochatic control problem. Stochatic Procee and their Application 9, Harrion, J.M., Takar, M.J., 983. Intant control of Brownian motion. Mathematic of Operation Reearch 8, Højgaard, B., Takar, M., 998a. Optimal proportional reinurance policie for diffuion model. Scandinavian Actuarial Journal 2, Højgaard, B., Takar, M., 998b. Optimal proportional reinurance policie with tranaction cot. Inurance: Mathematic and Economic 22, 4 5. Højgaard, B., Takar, M., 999. Controlling rik expoure and dividend payout cheme: Inurance company example. Mathematical Finance 9 2, Højgaard, B., Takar, M., 2. Optimal rik control for a large corporation in the preence of return on invetment. Finance and Stochatic 5, Lion, P.-L., Sznitman, A.S., 984. Stochatic differential equation with reflecting boundary condition. Communication on Pure and Applied Mathematic 37, Løkka, A., Zervo, M., 26. Optimal Dividend and Iuance of Equity Policie in the Preence of Proportional Cot Preprint. Martin-Löf, A., 983. Premium control in an inurance ytem, an approach uing linear control theory. Scandinavian Actuarial Journal, 27. Paulen, J., Gjeing, H.K., 997. Optimal choice of dividend barrier for a rik proce with tochatic return of invetment. Inurance: Mathematic and Economic 2, Radner, R., Sheep, L., 996. Rik v. profit potential: a model for corporate trategy. Journal of Economic Dynamic & Control 2, Sethi, S.P., Takar, M., 22. Optimal financing of a corporation ubject to random return. Mathematical Finance 2 2, Takar, M., 2. Optimal rik/dividend ditribution control model: application to inurance. Mathematical Method of Operation Reearch, 42. Wendell, H., Fleming, H., 993. Mete Soner: Controlled Markov procee and vicoity olution. Springer-Verlag, New York, ISBN