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1、精品论文退休金计划定价中含非局部项的变分不等式及自由边界研究 边保军 ，袁泉 同济大学理学院，上海 200092摘要：本文研究了一个含非局部项的抛物型变分不等式，它来自于给允许提前退休的指定收益型退休金计划的定价模型。由于退休金给付额依赖员工退休时的工资，首先假定工资服从跳扩 散过程，可以用一个最优停时问题来刻画该退休金计划的价值，从而得到该值函数满足的含抛 物型积分 -微分算子的变分不等式和一个等价的自由边界问题，随后讨论了该变分不等式解的 存在唯一性，和自由边界的相关性质。 关键词：含非局部项的变分不等式，退休金计划，跳扩散过程，自由边界，最优提前退休策略 中图分类号： O29A Nonl
2、ocal Variational Inequality and Related Free Boundary arising from Pricing Retirement BenetsBIAN Bao-Jun , YUAN QuanDepartment of Mathematics, Tongji University, Shanghai 200092Abstract: This paper is concerned with a nonlocal parabolic variational inequality which arises from the nancial valuation
3、of dened benets retirement pension plan that allows early retirement. The paid benets on retirement depend on the salary at that time. Theunderlying salary is assumed to follow a jump-diusion process. We characterize the nancial value of the retirement benets as the solution of an optimal stopping t
4、ime problem, which corresponds to a variational inequality or a free boundary problem of an integro-dierential operator of the parabolic type. The existence and uniqueness of the solution to thevariational inequality are proved and the properties for related free boundary are discussed.Key words: No
5、nlocal variational inequality, retirement benets, jump-diusion process, free boundary, optimal retirement strategy.基金项目： This work was supported by National Science Foundation(No.11071189, No.71090404)作者简介： Bian Baojun （1962-），male，professor，major research direction：partial dierential equation, nanc
6、ialmathematics. Correspondence author：Yuan Quan （1985-），female，phD，major research direction：nancial mathematics.- 23 -0 IntroductionConsider the following Cauchy problem of nonlocal parabolic variational inequalityminLu f, u = 0, a.e. (x, t) QTu(x, 0) = (x, 0), x R(1)for a second linear parabolic in
7、tegro-dierential operator L, where QT = R (0, T . In thispaper, we will prove the existence and uniqueness of the solution to the above problem and discuss the properties for the related free boundary.The variational inequality (1) is from the nancial valuation for a dened benets retire- ment pensio
8、n plan that allows early retirement. The related free boundary corresponds to the optimal retirement strategy for the member of the pension plan. A dened benets pension plan (DB in short) is a type of pension plan in which the benets on retirement are prede- termined by a formula based on the employ
9、ees salary history, age, duration of employment and some other personalized factors, rather than investment returns. Its dierent from any pension plan where payout is somewhat dependent on the performance of the portfolio in which the contributions are invested. In a dened benets plan the company sp
10、onsoring the plan is in charge of the portfolio management and bears the investment risk. The benets are usually distributed through life annuities. In a life annuity, employees receive equal periodic benet payments (monthly, quarterly, etc.) for the rest of their lives.More recently option pricing
11、techniques have been adopted for the nancial valuation of retirement benets. A number of researchers have made important contributions in this research eld. In 1, the method of ”pricing contingent claims” is rstly applied to price dened benets plans with the type of ”Greater of Benets”. In 2, author
12、s give a more in-depth and more general study to this type of pension plan. They establish existence and uniqueness of the solution to a variational inequality which is satised by the value of the retirement benets and then study the continuity and dierentiability of the free boundary which correspo
13、nds to the optimal time to retire. Other related works can be found in 3, 4 and references in 2. All the above works are based on the assumption that the underlying salary follows a continuous Markov process. But indeed the salary is not always continuous. Instead, it may jump up or down due to some
14、 unexpected reasons, such as a bonus for an excellent employee from the company or a penalty for bad behaviors.Our paper is based on the research in 2, yet using the jump-diusion model as in 5 and 6 to describe the dynamics of the underlying salary St . We generalize the results in 2 to a jump-diusi
15、on model and obtain some interesting results of the free boundary. We overcome and obtain explicity cha.Assume that the dynamics of the salary process (St )t0 followstdSt = dt + dB StNt+ d( Uj ) (2)j=1where , are positive constants called accrual rate and volatility. The process (Bt )t0 is a risk-ad
16、justed Brownian motion. The process (Nt )t0 is a Poisson process with intensity , and (Uj )j1 is a sequence of square integrable, independent, identically distributed random variables, with values in (1, +). The parameter of the Poisson process represents the frequency of jumps, and the random varia
17、ble Uj represents the relative amplitude of jumps. Furthermore, we assume stochastic processes (Bt )t0 , (Nt )t0 , (Uj )j1 are independent. Let P (z) be the distribution function of U1 and k = E(U1 ), then +1dP (z) = 1, +1zdP (z) = k. (3)To derive the mathematical model for the value of the pension
18、plan, we rst adjust the salary process (St )t0 to risk-neutral state. If jump occurs in the interval t, t + dt, set dqt = 1 with probability dt. Otherwise dqt = 0 with probability (1 dt). Let the random variable U represents the relative jump amplitude having the same distribution function as U1 . W
19、e can rewrite (2):tdSt = (k)dt + dB St+ U dqt .(4)+The payout on retirement is still assumed to follow that in 2, i.e. the larger of the two quantities:(a) an amount depending on the salary St at the retirement time t; (b) a dened amount which is independent of the salary St . Let T be the maturity
20、of the retirement pension plan and T0 (0, T ) be the time to allow early retirement. Then the payout function (S, t) is as follows(S, t) = (1 T t )T T0maxA, aS, (S, t) R+ 0, T (5)where A is a dened amount mentioned in (b) and aS is the amount mentioned in (a) with the positive constant a.Following 2
21、, also from the actuarial mathematics literature 7, the insurance benets are paid from the retirement funds due to death, or disability, at the amount Ad (S, t) with force of exit d . Benets are also paid if a member withdraws in order to transfer to another pension fund; we denote this amount by Aw
22、 (S, t) and intensity by w . Then the total benets are B(S, t) = d Ad (S, t) + w Aw (S, t). Throughout this paper we always assume B(S, t) = S. Denote the risk-free interest rate by r. Let = r + d + w .Let V (S, t) be the nancial value of the retirement benets during 0, T . Denote by t,T the set of
23、all stopping time of the process St , then the nancial value V (S, t) of the pensionplan can be characterized as the solution of the following optimal stopping time problemV (S, t) = sup E t,T e( t) (S , ) +t e(t) B(S )d St = S where St evolves according to (4).This optimal stopping time problem is
24、related to the following variational inequalityminMV B, V = 0 a.e(S, t) R+ 0, T ), V (S, T ) = (S, T ), S R+where M is a parabolic operator(6) V1 22 2 VVMV = + St 2S2 + ( k)S S ( + )V + EV (1 + U )S, t)It is also related to a parabolic free boundary problem. Let denote the boundary of the set V 0 t
25、T . Then the optimal strategy is to retire as soon as the path t (St , t) hits which means determining the free boundary is also very important in our valuation. If V (S, t) is a C 1 solution of (6), then we should impose the following two conditions (see 8):1. V = on ,2. There holds a.e.V1+t 22 S22
26、 VS2V+ ( k)S S ( + )V + EV (1 + U )S, t) S.From the nancial point of view, a rational member will choose to retire if and only if he gets the full value of the retirement benets. Thus he will not elect early retirement if V (S, t) (S, t) which also implies equation MV B = 0 holds.aLet x = log A . Ta
27、keS = ex+x , u(x, t) = V (S, T t), (x, t) = (S, T t), N (z) = P (ez 1).Then the operator M V becomesLu =u 2 2 u ( k 2 u)+ ( + )u +u(x + z, t)dN (z), (7)in whicht 2x22 x +N () = 0, N (+) = 1,Later on we shall need this operator L0 :ez dN (z) = 1 + k,u2 2 u2 uL0 u =t 2 x2 ( k ) + ( + )u.2 xThe payout
28、function (5) becomes+(x, t) = A(1 t)T T0maxex , 1, (x, t) QT.(8)aLet f (x) = B(S) = ex+x = A ex , then u(x, t) solves the nonlocal parabolic integro- dierential variational inequality (1).The rest of this paper is organized as follows. The existence and uniqueness of the solution are obtained throug
29、h solving a related penalty problem in section 1. Then sections 2,3 analyze the important properties of the free boundary, including continuity and dierentiability.1 Variational inequalityThroughout this paper, we always assume + |z|2 e|z|dN (z) 0), (9)u(x, t)|p D 0,(10)then u(x, t) 0 in D.Proof. Le
30、t DL = D |x| L and consider the following function in DLm(ex + 1)w(x, t) = u(x, t) +(x2 + t)etL2where L, , are positive constants to be chosen.Setk1 = +zdN (z), k2 = +z2 dN (z), k3 = +zez dN (z), k4 = +z2 ez dN (z).Simple computation yieldswhereLw m eL2x ( + A1 )x2+ B1 x + + C1 + ( + A2 )x2+ B2 x +
31、+ C2 andA1 = , B1 = 2 2 2(k + k3 ), C1 = 2 k4 ,A2 = , B2 = 2 2 2(k + k1 ), C2 = 2 k4 .Choose , large enough, we obtain Lw 0 in DL .It is clear that w 0 on p DL . We also claim w(x, t) 0 in D L . Otherwise, we assumew(x, t) attains its negative minimum at (x0 , t0 ) DL , which leads to +Lw(x0 , t0 )
32、( + )w(x0 , t0 ) w(x0 + z, t0 )dN (z) w(x0 , t0 ) 0, there exists a constant M 0 such thatu(x, t) M eB(xx0 ) in D. (11)Proof. Apply Lemma 1 to the function w = u + C eK t eB(xx0 ) with suciently large constantK and the assertion in Lemma 2 follows.Dene a function (x, t) C 2 ( 0) satisfying: (x, t) =
33、 (x, t), if |x| ,x 0,2x2 0,(12) (x, t) (x, t), if , 0.To get uniform estimates, we change the form of variational inequality (1). Dene h(x, t) =eK t (ex + 1) withK = max0, 1 (1 + k) +Let v = u , = and f = f . Then we have22 .(13)h handh0 (x, t) A, 0 x 2A,xx 15A, (14)(1) becomes0 f(x, t), fx , fxx A.
34、(15)awhereminL v f, v = 0, a.e. (x, t) QTv(x, 0) = (x, 0), x R,(16)2L v = v 2 v 2 2 hx v2 hxx +ex+z + 1t 2x2 (k 2 +) +(+Kh x 2)v hex + 1 v(x+z, t)dN (z).(17)We work with the following approximating penalty problem as in 2 and 5 to prove existence of a solution to the integro-dierential variational i
35、nequality (16) and then to (1).For 0, dene a penalty function (s) C such that there exists a positive constantC1 independent of A (s + 1) (s) 0, (0) C1 , (s) 0, (s) 0,0,s 0lim (s) =0, s 0.Let QT ,R = QT |x| 0,0 v,R (x, t) C2 .Proof. It is obviously that v,R (x, t) 0. Let w(x, t) = C2 . Choose K as i
36、n (13) and C2 large enough. Then for (x, t) QT ,R ,L w + (w ) (K + + (1 + k) 22 1 )C Cf2and w ), (x, t) p QT ,R . We have v,R (x, t) w = C2 by the comparison principle.Next we consider the bound of the function ,R (u ) which requires to be independent of , R.Lemma 4. There exists a positive constant
37、 C3 such that, for any , R 0,0 (v ) C3 .Proof. Let w(x, t) = (v ). Its easily seen that w 0. Let be the minimum of w(x, t) and (0) 0, take R R0 + 1. Noting that (14) and (15), we can apply the standard regularity theory of PDE and proceed as in 2 to prove the following estimate.Lemma 5. There exist positive constants C4 and D such that, for any smal l 0,|vx |C 0 (QT ,R0 ) C4 , |v,R |W 2,1+ Dp (QT ,R0 B (0)for any p 1. Furthermore+|vt |, |vxx | D , in QT ,R0 B (0). (19)We conclude this subsection with the following theoremTheorem 2. There exists a unique conti
