# Optimal Investment with Multiple Risky Assets for an Insurer in an Incomplete Market

Authors: Hui Zhao; Ximin Rong; Jiling Cao
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

This paper studies the optimal investment problem for an insurer in an incomplete market. The insurer's risk process is modeled by a Lévy process and the insurer is supposed to have the option of investing in multiple risky assets whose price processes are described by the standard Black-Scholes model. The insurer aims to maximize the
expected utility of terminal wealth. After the market is completed, we obtain the optimal strategies for quadratic utility and constant absolute risk aversion (CARA) utility explicitly via the martingale approach. Finally, computational results are presented for given raw market data.


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## Body

## 1. Introduction

Recently, the problem of optimal investment for an insurer has attracted a lot of attention, due to the fact that the insurer is allowed to invest in financial markets in practice. In the meantime, this is also a very interesting portfolio selection problem in the finance theory. Some important early work is done by Browne [1], where the risk process is approximated by a Brownian motion with drift and the stock price is modeled by a geometric Brownian motion. Under these assumptions, when the interest rate of a risk-free bond is zero, Browne [1] shows that minimizing the ruin probability and maximizing the expected constant absolute risk aversion (CARA) utility of the terminal wealth produce the same type of strategy. In Hipp and Plum [2], the classical Cramér-Lundberg model is adopted for the risk reserve and the insurer can invest in a risky asset to minimize the ruin probability. However, the interest rate of the bond in their model is implicitly assumed to be zero. Liu and Yang [3] extend the model of Hipp and Plum [2] to incorporate a nonzero interest rate. But in this case, a closed-form solution cannot be obtained. Azcue and Muler [4] consider the borrowing constraints for an optimal investment problem of an insurer, and some numerical solutions to minimize the ruin probability are obtained. In the literature, the optimal investment problem for an insurer is usually studied via the stochastic control theory; see [5] for details. By this approach, Yang and Zhang [6] use a jump-diffusion process to model the risk process and obtain the optimal investment strategy for CARA utility maximization. The martingale approach, which has been widely used in mathematical finance, is applied to investigate the optimal investment problem for an insurer by Wang et al. [7]. In their paper, the risk process is modeled by a Lévy process, the security market containing a bond and a stock is described by the standard Black-Scholes model, and the optimal strategies are worked out explicitly for the mean-variance criteria and the CARA utility. Zhou [8] studies a model similar to that in [7], where the prices of assets are described by the Lévy processes, and the optimal strategy is obtained explicitly for the CARA utility.In most of these papers, the optimal investment for an insurer is considered in a complete market. But the real market is always incomplete; that is, the number of risky assets (stocks) is strictly less than the dimension of the underlying Brownian motion. Hence, it is necessary to find an optimal strategy in an incomplete market. However, the traditional martingale method cannot be used directly in an incomplete market. To overcome the problem of incompleteness, many researchers have developed different ways to handle the problem. For instance, Karatzas et al. [9] complete the market by introducing additional fictitious stocks and then making them untradable. Zhang [10] provides an easier method by directly making the dimension of the underlying Brownian motion equal to the number of available stocks.The optimal investment problem for an insurer in an incomplete market is studied in the present paper. For this case, the traditional martingale method is problematic. Thus, we first complete the market via the approach proposed by Zhang [10] so that the martingale method can be used. In addition, only one stock is considered in [7, 8]. But an insurer will invest in multiple stocks to decrease risk. Thus in order to make our model more practical, we consider a financial market of one bond and m stocks. Moreover, the insurer’s objective in this paper is to maximize the utility of the discounted value of his/her terminal wealth, which is different from that in most works. When the risk process is modeled by a Lévy process and the security market is described by the standard Black-Scholes model, the optimal strategies are obtained explicitly for the quadratic and CARA utilities.An optimal strategy of CARA utility maximizing in an incomplete market is also obtained in Wang [11]. The result is general but not unique and cannot be expressed concretely in [11]. Whereas our corresponding optimal strategy is explicit and concrete.This paper is structured as follows. In Section2 the problem is formulated. The incomplete market is transformed into a complete one in Section 3. The optimal portfolios for quadratic and CARA utility functions are worked out explicitly in Section 4. Section 5 presents some remarks and computational results for optimal strategies in a complete market and an incomplete market. Section 6 contains some conclusions.


## 2. Formulation of the Problem

The insurer can invest in a financial market consisting of one riskless bond with priceSt(0) given by
(1)dSt(0)=rSt(0)dt,S0(0)=1,
and m stocks with prices St(i) satisfying
(2)dSt(i)=St(i)[bidt+∑j=1dσijdWt(j)],i=1,…,m,
where r is the interest rate and Wt:=(Wt(1),…,Wt(d))T is a d-dimensional Brownian motion defined on a filtered complete probability space (Ω,ℱ,(ℱt),ℙ) with the component Brownian motions Wt(j),j=1,…,d being independent. b:=(b1,…,bm)T denotes the appreciation rate vector, σ:=(σij)m×d is the volatility matrix, and define T>0 as a fixed and finite time horizon with 0≤t≤T. In addition, the matrix σ is assumed to have the full row rank. As in [10], under these assumptions, we restrict ourselves to the situation where m≤d that is, the number of stocks in the market is smaller than or equal to the dimension of the underlying Brownian motion. This financial market is complete when m=d, while incompleteness arises if  m<d. In this paper, we consider the insurer’s investment problem in an incomplete market.Suppose the initial reserve isx0 and the premium rate is a constant α; the risk process (Rt) of the insurer is modeled by
(3)Rt=x0+αt-∑i=1N(t)‍Yi,
where ∑i=1N(t)Yi is a compound Poisson process defined on (Ω,ℱ,ℙ), N(t) is a homogeneous Poisson process with intensity λ and represents the number of claims occurring in the time interval [0,t], Yi is the size of the ith claim. Thus the compound Poisson process ∑i=1N(t)Yi represents the cumulative amount of claims in the time interval [0,t]. The claims’ sizes Y={Yi,i≥1} are assumed to be an i.i.d. sequence with a common cumulative distribution function F satisfying F(0)=0 and ∫0∞‍x2dF(x)<∞. Furthermore, it is usually assumed that N,{Yi,i≥1}, and the d-dimensional Brownian motion W are mutually independent. If Lt denotes the compensated compound Poisson process ∑i=1N(t)Yi-λmFt, where mF is the mean of F, then the risk process (3) can be rewritten as
(4)Rt=x0+(α-λmF)t-Lt.
As in [7], we generalize the above risk process model to a stochastic cash flow, which is still denoted by (Rt) and satisfies
(5)Rt=x0+ct-Lt,
where L is a 1-dimensional compensated pure jump Lévy process defined on (Ω,ℱ,(ℱt),ℙ), c is a constant, (ℱt) is the usual augmentation of the natural filtration of (W,L) with ℱ=ℱT, and W and L are mutually independent. Let μ denote the jump measure of L. Its dual predictable projection ν has the form ν(dt,dx)=dt×m(dx) with m({0})=0 and ∫ℝ‍(x2∧1)m(dx)<∞. To avoid some tedious technical arguments, we assume ∫ℝ‍x2m(dx)<∞ throughout this paper. Under this assumption, it is well known that L is square integrable and has the following Lévy decomposition property:
(6)Lt=∫0t‍∫ℝ‍x(μ(ds,dx)-ν(ds,dx)).
For details, refer to Cont and Tankov [12]. Further, note that for the risk process model (3), we have m(dx)=λF(dx).The insurer is allowed to invest in thosem stocks as well as in the bond. A trading strategy can be expressed in terms of the dollar amount invested in each asset. Let πt(i),i=1,…,m, be the dollar amount of the ith stock in the investment portfolio at time t, and let πt:=(πt(1),…,πt(m))T and π:=(πt). To be mathematically rigorous, for each 1≤i≤m,  πt(i) is an (ℱt)-predictable process.Definition 1.
A trading strategyπ is called admissible if 𝔼[∫0T‍πtTπtdt]<∞.The set of all admissible trading strategies is denoted byΠ. For each π∈Π and an initial capital x0, the wealth process Xx0,π of the insurer satisfies the following dynamics:
(7)dXtx0,π=πtT[bdt+σdWt]+(Xtx0,π-πtT1m)rdt+cdt-dLtX0x0,π=x0,
where 1m:=(1,…,1)T is an m×1 vector. Wang et al. [7] consider an optimal investment problem similar to (7) in the one stock case and solve it elegantly by the martingale method in a complete market. However, in an incomplete market, the martingale method will be problematic. Thus, we will use the method proposed by Zhang [10] to transform our incomplete market into a complete one.


## 3. Transformation from an Incomplete Market to a Complete One

In this section, we “complete” the incomplete market described in (1) and (2) in order to use the martingale method in the sequel.Step 1.
Reducing the dimension of the Brownian motion.
For eachi=1,…,m,   let σi:=(σi1,σi2,…,σid). Then the volatility matrix in the incomplete market can be written as σ=(σ1,σ2,…,σm)T. Define
(8)Bt(i):=∑j=1d‍σij∥σi∥Wt(j),i=1,…,m,
where ∥·∥ denotes the Euclidean norm. Expressing the stock prices in terms of Bt(i), it gives
(9)dSt(i)=St(i)(bidt+∥σi∥dBt(i)),i=1,…,m,
where (Bt(1),…,Bt(m))T is an m-dimensional Brownian motion. But the component Brownian motions are no longer independent. Specifically, one can consult Exercise  4.15 in [13] for the following result:
(10)dBt(i)dBt(k)=ρikdt,∀i≠k,
where
(11)ρik=1∥σi∥∥σk∥∑j=1d‍σijσkj=〈σi,σk〉∥σi∥∥σk∥,
and 〈σi,σk〉 represents the inner product of σi and σk. It follows from the Cauchy-Schwarz inequality that |ρik|≤1. Under the assumption that the volatility matrix has the full row rank, we have
(12)|ρik|<1fori≠k.Step 2.
Creating independent component Brownian motions from the correlated ones.
Let(13)Ψ:=[ρ11ρ12⋯ρ1mρ21ρ22⋯ρ2m⋮⋮⋱⋮ρm1ρm2⋯ρmm]
be the matrix generated by the correlation coefficients of the correlated m-dimensional Brownian motion (Bt(1),…,Bt(m))T with
(14)ρik{=1,ifi=k,<1,ifi≠k.
The matrix Ψ is nonsingular, symmetric, and positively semi-definite. So, there exists a nonsingular matrix A-:=(a-ij)m×m such that Ψ=A-A-T. It can be shown that there exist m independent Brownian motions W~t(1),…,W~t(m) such that
(15)Bt(i)=∑j=1m‍a-ijW~t(j),∀i=1,…,m.
For details, refer to Exercise  4.16 in [13]. So far, we have derived a complete market with m stocks and m-independent component Brownian motions. The m-stocks in the incomplete market can now be re-written as
(16)dSt(i)=St(i)(bidt+∥σi∥∑j=1m‍a-ijdW~t(j)),∀i=1,…,m.
The volatility matrix, denoted by σ~, under the completed market is given by
(17)σ~=[∥σ1∥a-11∥σ1∥a-12⋯∥σ1∥a-1m∥σ2∥a-21∥σ2∥a-22⋯∥σ2∥a-2m⋮⋮⋱⋮∥σm∥a-m1∥σm∥a-m2⋯∥σm∥a-mm]=[∥σ1∥0⋯00∥σ2∥⋯0⋮⋮⋱⋮00⋯∥σm∥]·A-,
Put
(18)Σ:=[∥σ1∥0⋯00∥σ2∥⋯0⋮⋮⋱⋮00⋯∥σm∥].
Then
(19)σ~=ΣA-.
With the help of σ~ and W~t=(W~t(1),…,W~t(m)), we transfer the model (7) to the following one in the completed market
(20)dXtx0,π=πtT[bdt+σ~dW~t]+(Xtx0,π-πtT1m)rdt+cdt-dLtX0x0,π=x0,
which is equivalent to
(21)Xtx0,π=ert(x0+∫0t‍e-rs(πsT(b-r1m)+c)ds+∫0t‍e-rsπsTσ~dW~s-∫0t‍e-rsdLs)=ertx0+c(ert-1)r+∫0t‍er(t-s)(πsT(b-r1m))ds+∫0t‍er(t-s)πsTσ~dW~s-∫0t‍er(t-s)dLs.
Define
(22)X^tx0,π=e-rtXtx0,π=x0+c(1-e-rt)r+∫0t‍e-rs(πsT(b-r1m))ds+∫0t‍e-rsπsTσ~dW~s-∫0t‍e-rsdLs,
which represents the discounted value of Xtx0,π.
Suppose that the insurer has a utility functionU of his wealth and he/she aims to maximize the expected utility of his/her discounted terminal wealth, that is,
(23)maximize𝔼[U(X^Tx0,π)]overπ∈Π.
The utility function U is strictly concave and continuously differentiable on (-∞,∞); thus there exists at most a unique optimal discounted terminal wealth for the company.
Furthermore, for anyπ*∈Π and π∈Π, we have
(24)𝔼[U(X^Tx0,π*)]-𝔼[U(X^Tx0,π)]≥𝔼[U′(X^Tx0,π*)(X^Tx0,π*-X^Tx0,π)],
from the concavity of U. according to (24) we claim the following proposition.Proposition 2.
If there exists a strategyπ*∈Π such that
(25)𝔼[U′(X^Tx0,π*)X^Tx0,π]isconstantoverπ∈Π,
then π* is the optimal trading strategy.Remark 3.
Such sufficient conditions to the optimal investment are well known in the martingale approach; refer to [9] and others.
By (22), (25) is equivalent to that
(26)𝔼[U′(X^Tx0,π*)∫0T‍(e-rsπsT(b-r1m)ds+e-rsπsTσ~dW~s)]isconstantoverπ∈Π.
Thus, problem (23) can be solved by applying (26) in terms of the concrete utility functions.
It is well known that maximizing the expected quadratic utility is equivalent to finding a mean-variance efficient strategy while CARA utility is commonly used. Thus, we will work out the optimal strategies explicitly for these two utilities in the next section.


## 4. Optimal Strategies for Different Utility Functions

First, we introduce some notations and a martingale representation theorem that will be used. Let𝒫 be the predictable σ-algebra on Ω×[0,T], which is generated by all left-continuous and (ℱt)-adapted processes, and 𝒫~:=𝒫⊗ℬ(ℝ). Let L(𝒫) (resp., L2(𝒫)) be the set of all (ℱt)-predictable, ℝ-valued processes θ such that ∫0T‍|θ(t)|2dt<∞ a.s. (resp.,𝔼[∫0T‍|θ(t)|2dt]<∞). Let L(𝒫~) be the set of all 𝒫~-measurable, ℝ-valued functions θ defined on Ω×[0,T]×ℝ such that
(27)∑0<s≤t‍|θ(s,ΔLs)|2I{ΔLs≠0}
is locally integrable and increasing, and also ∫ℝ‍|θ(t,x)|m(dx)<∞ a.s. for all t∈[0,T], where I{⋯} is the indicator function. Moreover, let L2(𝒫~) be the set of all 𝒫~-measurable, ℝ-valued functions θ defined on Ω×[0,T]×ℝ such that 𝔼[∫0T‍∫ℝ‍|θ(t,x)|2m(dx)dt]<∞. Finally, Lℱ2 denotes the set of all (ℱt)-adapted processes (Xt) with Càdlàg paths such that 𝔼[𝕤𝕦𝕡0≤t≤T|Xt|2]<∞. It is well known that every square-integrable martingale belongs to Lℱ2, and it is easy to see that if π∈Π then X^x0,π∈Lℱ2.One can consult Proposition  9.4 in [12] for the following result.Lemma 4 (Martingale Representation).
For any local (resp., square-integrable) martingale(Zt), there exist some θ1=(θ1(1),…,θ1(m))T with  θ1T∈L(𝒫)×⋯×L(𝒫)︸m, and θ2∈L(𝒫~)  (resp.,θ1T∈L2(𝒫)×⋯×L2(𝒫)︸m,andθ2∈L2(𝒫~)) such that
(28)Zt=Z0+∫0t‍[θ1(s)]TdW~s+∫0t‍∫ℝ‍θ2(s,x)(μ(ds,dx)-ν(ds,dx)),
for all t∈[0,T].


### 4.1. Quadratic Utility

In this subsection, we consider problem (23) for the quadratic utility function U(x)=x-(γ/2)x2, where γ>0 is a parameter. In case that there is only one stock and one bond in a complete market, a closed-form solution is obtained by Wang et al. [7]. Motivated by their work, we show that a closed-form solution can also be obtained when there are several stocks in an incomplete market. Since U′(x)=1-γx, condition (26) can be written as
(29)𝔼[(1-γX^Tx0,π*)∫0T‍(e-rsπsT(b-r1m)ds+e-rsπsTσ~dW~s)]isconstantoverπ∈Π.
Put
(30)Zt*=𝔼[1-γX^Tx0,π*∣ℱt],t∈[0,T].
Then ZT*=1-γX^Tx0,π* and Zτ*=𝔼[ZT*∣ℱτ] a.s. for any stopping time τ≤T a.s.Lemma 5.
Letπ*∈Π. Then π* satisfies condition (29) if and only if there exists a θ2*∈L2(𝒫~) such that (X^x0,π*,π*,Z*,θ2*) solves the following forward-backward stochastic differential equation (FBSDE)
(31)dX^t=e-rtπtT(b-r1m)dt+e-rtπtTσ~dW~t+ce-rtdt-e-rtdLt,X^0=x0,dZt=-Zt-(b-r1m)T(σ~-1)TdW~t+d(∫0t‍∫ℝ‍θ2(s,x)(μ(ds,dx)-ν(ds,dx))),ZT=1-γX^T,
for (X^,π,Z,θ2)∈Lℱ2×Π×Lℱ2×L2(𝒫~).Proof.
Supposeπ* satisfies condition (29). From (22) we have (X^x0,π*,π*)∈Lℱ2×Π solving the forward SDE in (31) for (X^,π) and it is clear that (Zt*) is a square-integrable martingale. For any stopping time τ≤T, let πtτ,i=(0,…,0,I[t≤τ],0,…,0,)T,i=1,…,m, whose ith component is I[t≤τ] and the other components are all 0. Then πτ,i∈Π,i=1,…,m. Denote by σ~i the ith row vector of σ~ for i=1,…,m; that is, σ~i:=(σ~i1,…,σ~im). For any positive integer 1≤i≤m, substituting πτ,i into (29), we have that
(32)𝔼[ZT*∫0τ‍(e-rs(bi-r)ds+e-rsσ~idW~s)]=𝔼[𝔼[ZT*∣ℱτ]∫0τ‍(e-rs(bi-r)ds+e-rsσ~idW~s)]=𝔼[Zτ*∫0τ‍(e-rs(bi-r)ds+e-rsσ~idW~s)]
is constant over all stopping times τ≤T a.s., which implies that
(33)Zt*∫0t‍(e-rs(bi-r)ds+e-rsσ~idW~s)isamartingale,i=1,…,m.
Since (Zt*) is a square-integrable martingale, by Lemma 4, there are θ1*=(θ1(1),*,…,θ1(m),*)T with (θ1*)T∈L2(𝒫)×⋯×L2(𝒫)︸m and θ2*∈L2(𝒫~) such that
(34)dZt*=[θ1*(t)]TdW~t+d(∫0t‍∫ℝ‍θ2*(s,x)(μ(ds,dx)-ν(ds,dx))).
By Itô’s formula,
(35)d(Zt*∫0t‍(e-rs(bi-r)ds+e-rsσ~idW~s))=((bi-r)Zt-*+σ~iθ1*(t))e-rtdt+alocalmartingale,
which together with (33) implies (bi-r)Zt-*+σ~iθ1*(t)=0 for each i=1,…,m. Thus, it is easy to see that
(36)σ~θ1*(t)=-(b-r1m)Zt-*,
that is,
(37)θ1*(t)=-σ~-1(b-r1m)Zt-*.
By (34), (Z*,θ2*) solves the backward SDE in (31) for (Z,θ2), and hence (X^x0,π*,π*,Z*,θ2*) solves FBSDE (31).
Conversely, suppose that there exists(Z*,θ2*)∈Lℱ2×L2(𝒫~) such that (X^x0,π*,π*,Z*,θ2*) solves FBSDE (31). It is easy to verify that for any π∈Π, by Itô’s formula, (Zt*Mtπ) is a local martingale, where
(38)Mtπ:=∫0t‍e-rsπsT(b-r1m)ds+e-rsπsTσ~dW~s.
Furthermore, for any π∈Π, we have Mπ∈Lℱ2. Hence,
(39)𝔼[sup0≤t≤T|Zt*Mtπ|]≤𝔼[sup0≤t≤T|Zt*|2]·𝔼[sup0≤t≤T|Mtπ|2]<∞,
which implies that the family
(40){Zτ*Mτπ:τisastoppingtimeandτ≤T}
is uniformly integrable and hence Z*Mπ is a martingale. Thus we have 𝔼[ZT*MTπ]=0 for any π∈Π, implying that π* satisfies condition (29).In what follows, we will solve FBSDE (31) by two steps.Step 1.
Conjecturing the form of solution.
Put(41)At=exp{∫0t‍asds},t∈[0,T],
where (at) is a nonrandom Lebesgue-integrable function to be determined. If (X^,π,Z,θ2) solves FBSDE (31), then by Itô’s formula,
(42)ATZT=Z0+∫0T‍AsdZs+∫0T‍Zs-dAs=Z0-∫0T‍AsZs-(b-r1m)T(σ~-1)TdW~s+∫0T‍∫ℝ‍Asθ2(s,x)(μ(ds,dx)-ν(ds,dx))+∫0T‍Zs-Asasds,
which implies
(43)1-ZTγ=1γ-1γATATZT=1γ-Z0γAT+1γAT∫0T‍AsZs-(b-r1m)T(σ~-1)TdW~s-1γAT∫0T‍∫ℝ‍Asθ2(s,x)(μ(ds,dx)-ν(ds,dx))-1γAT∫0T‍Zs-Asasds.
Let(X^,π,Z,θ2) solve FBSDE (31). Then X^T=(1-ZT)/γ. Comparing dW~t-term d(μ-ν)-term, respectively, in (43) with those in (22) and noting (6), it is reasonable to conjecture
(44)AtZt-γATσ~-1(b-r1m)=e-rtσ~Tπt,AtγATθ2(t,x)=e-rtx,
that is,
(45)πt=AtZt-ertγAT(σ~T)-1σ~-1(b-r1m),θ2(t,x)=γe-rtxATAt.
Note that by (19), Σ and A are both nonsingular, (σ~T)-1σ~-1=(σ~σ~T)-1=(ΣA-A-TΣT)-1=(ΣΨΣ)-1. Hence, (45) can be re-written as
(46)πt=AtZt-ertγAT(ΣΨΣ)-1(b-r1m),θ2(t,x)=γe-rtxATAt.
Substituting (44) and (46) into (43), we have
(47)1-ZTγ=1γ-Z0γAT+∫0T‍e-rsπsTσ~dW~s-∫0T‍e-rsdLs-1γAT∫0T‍Zs-Asasds=1γ-Z0γAT+X^T-x0-c(1-e-rT)r-∫0T‍e-rsπsT(b-r1m)ds-1γAT∫0T‍Zs-Asasds=X^T+1γ-Z0γAT-x0-c(1-e-rT)r-∫0T‍AsZs-γAT(b-r1m)T(ΣΨΣ)-1(b-r1m)ds-1γAT∫0T‍Zs-Asasds,
where the second equality follows from the forward SDE in (31) and the third equality holds owing to [(ΣΨΣ)-1]T=(ΣΨΣ)-1. If we take
(48)at=-(b-r1m)T(ΣΨΣ)-1(b-r1m),Z0=AT-γATx0-γATc(1-e-rT)r=(1-γx0-γc(1-e-rT)r)e-(b-r1m)T(ΣΨΣ)-1(b-r1m)T,
then (47) is reduced as
(49)1-ZTγ=X^T
and FBSDE (31) is solved.Step 2.
Verifying the conjecture in Step1.
Let(50)θ2*(t,x)=γexp{-rt-(b-r1m)T(ΣΨΣ)-1(b-r1m)(T-t)}x,(51)Z0*=(1-γx0-γc(1-e-rT)r)e-(b-r1m)T(ΣΨΣ)-1(b-r1m)T.
Then the following SDE
(52)dZt=-Zt-(b-r1m)T(σ~-1)TdW~t+d(∫0t‍∫ℝ‍θ2*(s,x)(μ(ds,dx)-ν(ds,dx)))
has a solution of the form
(53)Zt*=Mt(Z0*+∫0t‍∫ℝ‍Ms-1θ2*(s,x)(μ(ds,dx)-ν(ds,dx)))=Mt(Z0*+γ∫0t‍Ms-1exp{-rs-(b-r1m)T(ΣΨΣ)-1×(b-r1m)(T-s)(b-r1m)T}dLs∫0t‍),
where
(54)Mt=exp{12-(b-r1m)T(σ~-1)TW~t-12(b-r1m)T(ΣΨΣ)-1(b-r1m)t}.
Furthermore, let
(55)πt*=1γexp{(b-r1m)T(ΣΨΣ)-1(b-r1m)(T-t)+rt}×Zt-*(ΣΨΣ)-1(b-r1m),
and X^Tx0,π* is obtained through the forward SDE in (31). Then by the procedure same as that in (47), it is easy to verify that
(56)1-ZT*γ=X^Tx0,π*,thatis,1-γX^Tx0,π*=ZT*.
Thus, (X^x0,π*,π*,Z*,θ2*) solves FBSDE (31).
Finally, by Proposition2, Lemma 5, and arguments in Steps 1−2, we have the following theorem.Theorem 6.
Letπ* be defined as in (51)–(55). Then π* is the optimal trading strategy of problem (23) for the quadratic utility function U(x)=x-(γ/2)x2.


### 4.2. CARA Utility

In this subsection, we consider problem (23) for the CARA utility function U(x)=-(1/γ)e-γx, where γ>0. The procedure is similar to what is done in [7]. Since U′(x)=e-γx, condition (26) can be written as
(57)𝔼[e-γX^Tx0,π*∫0T‍(e-rsπsT(b-r1m)ds+e-rsπsTσ~dW~s)]isconstantoverπ∈Π.
Similar to the case of quadratic utility, in what follows, we split the procedure into three steps.Step 1.
Conjecturing the form ofπ* that satisfies condition (57).
Put(58)ZT*:=e-γX^Tx0,π*𝔼[e-γX^Tx0,π*],
and Zt*:=𝔼[ZT*∣ℱt] for all t∈[0,T]. Then Zτ*:=𝔼[ZT*∣ℱt] a.s. for any stopping time τ≤T a.s. Let ℚ be a probability measure on (Ω,ℱ) such that dQ/dP=ZT*.
For any stopping timeτ≤T a.s., let πtτ,i=(0,…,0,I[t≤τ],0,…,0,)T,i=1,…,m, whose ith component is I[t≤τ] and the other components are all 0. Then πτ,i∈Π,i=1,…,m. As in Section 4.1, denote by σ~i the ith row vector of σ~ for i=1,…,m; that is, σ~i:=(σ~i1,…,σ~im). For any positive integer 1≤i≤m, substituting πτ,i into (57), we have
(59)𝔼[ZT*∫0τ‍(e-rs(bi-r)ds+e-rsσ~idW~s)]=𝔼ℚ[∫0τ‍e-rs(bi-r)ds+e-rsσ~idW~s]
being constant over all stopping times τ≤T a.s., which implies that
(60)∫0t‍e-rs(bi-r)ds+e-rsσ~idW~sisamartingaleunderℚ,i=1,…,m.
Since (Zt*) is a martingale, then
(61)Kt:=∫0t‍1Zs-*dZs*,t∈[0,T],
is a local martingale. By Lemma 4, there exist some θ1=(θ1(1),…,θ1(m))T satisfying θ1T∈L(𝒫)×⋯×L(𝒫)︸m and θ2∈L(𝒫~) such that
(62)dKt=(θ1(t))TdW~t+d(∫0t‍∫ℝ‍θ2(s,x)(μ(ds,dx)-ν(ds,dx))),
that is,
(63)dZt*=Zt-*[∫0t‍(θ1(t))TdW~t+d(∫0t‍∫ℝ‍θ2(s,x)(μ(ds,dx)-ν(ds,dx)))].
Furthermore, by the Doléans-Dade exponential formula, we have
(64)Zt*=exp{∫0t‍(θ1(s))TdW~s-12∫0t‍(θ1(s))Tθ1(s)ds+∫0t‍∫ℝ‍θ2(s,x)(μ(ds,dx)-ν(ds,dx))+∫0t‍∫ℝ‍(log(1+θ2(s,x))-θ2(s,x))μ(ds,dx)}.
By Girsanov’s Theorem, we know that W~t-∫0t‍θ1(s)ds is an m-dimensional Brownian motion under ℚ, which together with (60) implies σ~iθ1(t)=-(bi-r),  i=1,…,m. Thus, it is easy to see that
(65)σ~θ1(t)=-(b-r1m),
that is,
(66)θ1(t)=-σ~-1(b-r1m).
On the other hand, by (22), we have
(67)e-γX^Tx0,π*=exp{-γx0-γc(1-e-rT)r-γ∫0T‍e-rs(πs*)T(b-r1m)ds-γ∫0T‍e-rs(πs*)Tσ~dW~s+γ∫0T‍e-rsdLsγc(1-e-rT)r}.
In order to have (58), comparing dW~t-term and dμ-term, respectively, in (64) with those in (67) and taking (66) into account, it is reasonable to conjecture
(68)-σ~-1(b-r1m)=-γe-rtσ~Tπt*,log(1+θ2(t,x))=γe-rtx,
that is,
(69)πt*=ert(σ~T)-1σ~-1(b-r1m)γ=ert(ΣΨΣ)-1(b-r1m)γθ2(t,x)=eγxe-rt-1.Step 2.
Verifying thatZT* in (64) satisfies (58), where π* and θ2 are defined in (69), and θ1 is defined in (66).
Substituting (69) into (67), we have
(70)e-γX^Tx0,π*=ITHT,
where
(71)IT=exp{-γx0-γc(1-e-rT)r-(b-r1m)T(ΣΨΣ)-1(b-r1m)Tγc(1-e-rT)r},HT=exp{-(b-r1m)T(σ~T)-1W~T+γ∫0T‍e-rsdLs}.
For θ1 and θ2 defined as above, it is easy to see that Z* is a martingale. Substituting (66) and (69) into (64), we have
(72)ZT*=JTHT,
where
(73)JT=exp{-(b-r1m)T(ΣΨΣ)-1(b-r1m)2+∫0T‍∫ℝ‍(γe-rsx-eγxe-rs+1)ν(ds,dxT)(b-r1m)T(ΣΨΣ)-1(b-r1m)2}
is a constant. Since Z* is a martingale, we have 𝔼[ZT*]=1 and hence
(74)𝔼[HT]=JT-1.
Finally, by (70)–(74), we have
(75)e-γX^Tx0,π*𝔼[e-γX^Tx0,π*]=ITHTIT𝔼[HT]=JTHT=ZT*,
which is just what we desired.Step 3.
Verifying thatπ* in (69) satisfies condition (57).
It is clear that𝔼[(ZT*)2]<∞. For any π∈Π, by Girsanov’s Theorem, we can see that
(76)Mtπ:=∫0t‍e-rsπsT(b-r1m)ds+e-rsπsTσ~dW~s,t∈[0,T],
is a local martingale under ℚ. Furthermore, for any π∈Π, we have Mπ∈Lℱ2. Hence
(77)𝔼ℚ[sup0≤t≤T|Mtπ|]=𝔼[ZT*sup0≤t≤T|Mtπ|]≤𝔼[(ZT*)2]·𝔼[sup0≤t≤T|Mtπ|2]<∞,
which implies that the family
(78){Mτπ:τisastoppingtimeandτ≤T}
is uniformly integrable under ℚ. Hence Mπ is a martingale under ℚ. Thus we have 𝔼ℚ[MTπ]=0 for any π∈Π, which implies (57).
By Proposition2 and arguments in steps 5–7, we can claim the following theorem.Theorem 7.
Letπ* be defined as in (69). Then π* is the optimal trading strategy of problem (23) for the CARA utility function U(x)=-(1/γ)e-γx.


## 4.1. Quadratic Utility

In this subsection, we consider problem (23) for the quadratic utility function U(x)=x-(γ/2)x2, where γ>0 is a parameter. In case that there is only one stock and one bond in a complete market, a closed-form solution is obtained by Wang et al. [7]. Motivated by their work, we show that a closed-form solution can also be obtained when there are several stocks in an incomplete market. Since U′(x)=1-γx, condition (26) can be written as
(29)𝔼[(1-γX^Tx0,π*)∫0T‍(e-rsπsT(b-r1m)ds+e-rsπsTσ~dW~s)]isconstantoverπ∈Π.
Put
(30)Zt*=𝔼[1-γX^Tx0,π*∣ℱt],t∈[0,T].
Then ZT*=1-γX^Tx0,π* and Zτ*=𝔼[ZT*∣ℱτ] a.s. for any stopping time τ≤T a.s.Lemma 5.
Letπ*∈Π. Then π* satisfies condition (29) if and only if there exists a θ2*∈L2(𝒫~) such that (X^x0,π*,π*,Z*,θ2*) solves the following forward-backward stochastic differential equation (FBSDE)
(31)dX^t=e-rtπtT(b-r1m)dt+e-rtπtTσ~dW~t+ce-rtdt-e-rtdLt,X^0=x0,dZt=-Zt-(b-r1m)T(σ~-1)TdW~t+d(∫0t‍∫ℝ‍θ2(s,x)(μ(ds,dx)-ν(ds,dx))),ZT=1-γX^T,
for (X^,π,Z,θ2)∈Lℱ2×Π×Lℱ2×L2(𝒫~).Proof.
Supposeπ* satisfies condition (29). From (22) we have (X^x0,π*,π*)∈Lℱ2×Π solving the forward SDE in (31) for (X^,π) and it is clear that (Zt*) is a square-integrable martingale. For any stopping time τ≤T, let πtτ,i=(0,…,0,I[t≤τ],0,…,0,)T,i=1,…,m, whose ith component is I[t≤τ] and the other components are all 0. Then πτ,i∈Π,i=1,…,m. Denote by σ~i the ith row vector of σ~ for i=1,…,m; that is, σ~i:=(σ~i1,…,σ~im). For any positive integer 1≤i≤m, substituting πτ,i into (29), we have that
(32)𝔼[ZT*∫0τ‍(e-rs(bi-r)ds+e-rsσ~idW~s)]=𝔼[𝔼[ZT*∣ℱτ]∫0τ‍(e-rs(bi-r)ds+e-rsσ~idW~s)]=𝔼[Zτ*∫0τ‍(e-rs(bi-r)ds+e-rsσ~idW~s)]
is constant over all stopping times τ≤T a.s., which implies that
(33)Zt*∫0t‍(e-rs(bi-r)ds+e-rsσ~idW~s)isamartingale,i=1,…,m.
Since (Zt*) is a square-integrable martingale, by Lemma 4, there are θ1*=(θ1(1),*,…,θ1(m),*)T with (θ1*)T∈L2(𝒫)×⋯×L2(𝒫)︸m and θ2*∈L2(𝒫~) such that
(34)dZt*=[θ1*(t)]TdW~t+d(∫0t‍∫ℝ‍θ2*(s,x)(μ(ds,dx)-ν(ds,dx))).
By Itô’s formula,
(35)d(Zt*∫0t‍(e-rs(bi-r)ds+e-rsσ~idW~s))=((bi-r)Zt-*+σ~iθ1*(t))e-rtdt+alocalmartingale,
which together with (33) implies (bi-r)Zt-*+σ~iθ1*(t)=0 for each i=1,…,m. Thus, it is easy to see that
(36)σ~θ1*(t)=-(b-r1m)Zt-*,
that is,
(37)θ1*(t)=-σ~-1(b-r1m)Zt-*.
By (34), (Z*,θ2*) solves the backward SDE in (31) for (Z,θ2), and hence (X^x0,π*,π*,Z*,θ2*) solves FBSDE (31).
Conversely, suppose that there exists(Z*,θ2*)∈Lℱ2×L2(𝒫~) such that (X^x0,π*,π*,Z*,θ2*) solves FBSDE (31). It is easy to verify that for any π∈Π, by Itô’s formula, (Zt*Mtπ) is a local martingale, where
(38)Mtπ:=∫0t‍e-rsπsT(b-r1m)ds+e-rsπsTσ~dW~s.
Furthermore, for any π∈Π, we have Mπ∈Lℱ2. Hence,
(39)𝔼[sup0≤t≤T|Zt*Mtπ|]≤𝔼[sup0≤t≤T|Zt*|2]·𝔼[sup0≤t≤T|Mtπ|2]<∞,
which implies that the family
(40){Zτ*Mτπ:τisastoppingtimeandτ≤T}
is uniformly integrable and hence Z*Mπ is a martingale. Thus we have 𝔼[ZT*MTπ]=0 for any π∈Π, implying that π* satisfies condition (29).In what follows, we will solve FBSDE (31) by two steps.Step 1.
Conjecturing the form of solution.
Put(41)At=exp{∫0t‍asds},t∈[0,T],
where (at) is a nonrandom Lebesgue-integrable function to be determined. If (X^,π,Z,θ2) solves FBSDE (31), then by Itô’s formula,
(42)ATZT=Z0+∫0T‍AsdZs+∫0T‍Zs-dAs=Z0-∫0T‍AsZs-(b-r1m)T(σ~-1)TdW~s+∫0T‍∫ℝ‍Asθ2(s,x)(μ(ds,dx)-ν(ds,dx))+∫0T‍Zs-Asasds,
which implies
(43)1-ZTγ=1γ-1γATATZT=1γ-Z0γAT+1γAT∫0T‍AsZs-(b-r1m)T(σ~-1)TdW~s-1γAT∫0T‍∫ℝ‍Asθ2(s,x)(μ(ds,dx)-ν(ds,dx))-1γAT∫0T‍Zs-Asasds.
Let(X^,π,Z,θ2) solve FBSDE (31). Then X^T=(1-ZT)/γ. Comparing dW~t-term d(μ-ν)-term, respectively, in (43) with those in (22) and noting (6), it is reasonable to conjecture
(44)AtZt-γATσ~-1(b-r1m)=e-rtσ~Tπt,AtγATθ2(t,x)=e-rtx,
that is,
(45)πt=AtZt-ertγAT(σ~T)-1σ~-1(b-r1m),θ2(t,x)=γe-rtxATAt.
Note that by (19), Σ and A are both nonsingular, (σ~T)-1σ~-1=(σ~σ~T)-1=(ΣA-A-TΣT)-1=(ΣΨΣ)-1. Hence, (45) can be re-written as
(46)πt=AtZt-ertγAT(ΣΨΣ)-1(b-r1m),θ2(t,x)=γe-rtxATAt.
Substituting (44) and (46) into (43), we have
(47)1-ZTγ=1γ-Z0γAT+∫0T‍e-rsπsTσ~dW~s-∫0T‍e-rsdLs-1γAT∫0T‍Zs-Asasds=1γ-Z0γAT+X^T-x0-c(1-e-rT)r-∫0T‍e-rsπsT(b-r1m)ds-1γAT∫0T‍Zs-Asasds=X^T+1γ-Z0γAT-x0-c(1-e-rT)r-∫0T‍AsZs-γAT(b-r1m)T(ΣΨΣ)-1(b-r1m)ds-1γAT∫0T‍Zs-Asasds,
where the second equality follows from the forward SDE in (31) and the third equality holds owing to [(ΣΨΣ)-1]T=(ΣΨΣ)-1. If we take
(48)at=-(b-r1m)T(ΣΨΣ)-1(b-r1m),Z0=AT-γATx0-γATc(1-e-rT)r=(1-γx0-γc(1-e-rT)r)e-(b-r1m)T(ΣΨΣ)-1(b-r1m)T,
then (47) is reduced as
(49)1-ZTγ=X^T
and FBSDE (31) is solved.Step 2.
Verifying the conjecture in Step1.
Let(50)θ2*(t,x)=γexp{-rt-(b-r1m)T(ΣΨΣ)-1(b-r1m)(T-t)}x,(51)Z0*=(1-γx0-γc(1-e-rT)r)e-(b-r1m)T(ΣΨΣ)-1(b-r1m)T.
Then the following SDE
(52)dZt=-Zt-(b-r1m)T(σ~-1)TdW~t+d(∫0t‍∫ℝ‍θ2*(s,x)(μ(ds,dx)-ν(ds,dx)))
has a solution of the form
(53)Zt*=Mt(Z0*+∫0t‍∫ℝ‍Ms-1θ2*(s,x)(μ(ds,dx)-ν(ds,dx)))=Mt(Z0*+γ∫0t‍Ms-1exp{-rs-(b-r1m)T(ΣΨΣ)-1×(b-r1m)(T-s)(b-r1m)T}dLs∫0t‍),
where
(54)Mt=exp{12-(b-r1m)T(σ~-1)TW~t-12(b-r1m)T(ΣΨΣ)-1(b-r1m)t}.
Furthermore, let
(55)πt*=1γexp{(b-r1m)T(ΣΨΣ)-1(b-r1m)(T-t)+rt}×Zt-*(ΣΨΣ)-1(b-r1m),
and X^Tx0,π* is obtained through the forward SDE in (31). Then by the procedure same as that in (47), it is easy to verify that
(56)1-ZT*γ=X^Tx0,π*,thatis,1-γX^Tx0,π*=ZT*.
Thus, (X^x0,π*,π*,Z*,θ2*) solves FBSDE (31).
Finally, by Proposition2, Lemma 5, and arguments in Steps 1−2, we have the following theorem.Theorem 6.
Letπ* be defined as in (51)–(55). Then π* is the optimal trading strategy of problem (23) for the quadratic utility function U(x)=x-(γ/2)x2.


## 4.2. CARA Utility

In this subsection, we consider problem (23) for the CARA utility function U(x)=-(1/γ)e-γx, where γ>0. The procedure is similar to what is done in [7]. Since U′(x)=e-γx, condition (26) can be written as
(57)𝔼[e-γX^Tx0,π*∫0T‍(e-rsπsT(b-r1m)ds+e-rsπsTσ~dW~s)]isconstantoverπ∈Π.
Similar to the case of quadratic utility, in what follows, we split the procedure into three steps.Step 1.
Conjecturing the form ofπ* that satisfies condition (57).
Put(58)ZT*:=e-γX^Tx0,π*𝔼[e-γX^Tx0,π*],
and Zt*:=𝔼[ZT*∣ℱt] for all t∈[0,T]. Then Zτ*:=𝔼[ZT*∣ℱt] a.s. for any stopping time τ≤T a.s. Let ℚ be a probability measure on (Ω,ℱ) such that dQ/dP=ZT*.
For any stopping timeτ≤T a.s., let πtτ,i=(0,…,0,I[t≤τ],0,…,0,)T,i=1,…,m, whose ith component is I[t≤τ] and the other components are all 0. Then πτ,i∈Π,i=1,…,m. As in Section 4.1, denote by σ~i the ith row vector of σ~ for i=1,…,m; that is, σ~i:=(σ~i1,…,σ~im). For any positive integer 1≤i≤m, substituting πτ,i into (57), we have
(59)𝔼[ZT*∫0τ‍(e-rs(bi-r)ds+e-rsσ~idW~s)]=𝔼ℚ[∫0τ‍e-rs(bi-r)ds+e-rsσ~idW~s]
being constant over all stopping times τ≤T a.s., which implies that
(60)∫0t‍e-rs(bi-r)ds+e-rsσ~idW~sisamartingaleunderℚ,i=1,…,m.
Since (Zt*) is a martingale, then
(61)Kt:=∫0t‍1Zs-*dZs*,t∈[0,T],
is a local martingale. By Lemma 4, there exist some θ1=(θ1(1),…,θ1(m))T satisfying θ1T∈L(𝒫)×⋯×L(𝒫)︸m and θ2∈L(𝒫~) such that
(62)dKt=(θ1(t))TdW~t+d(∫0t‍∫ℝ‍θ2(s,x)(μ(ds,dx)-ν(ds,dx))),
that is,
(63)dZt*=Zt-*[∫0t‍(θ1(t))TdW~t+d(∫0t‍∫ℝ‍θ2(s,x)(μ(ds,dx)-ν(ds,dx)))].
Furthermore, by the Doléans-Dade exponential formula, we have
(64)Zt*=exp{∫0t‍(θ1(s))TdW~s-12∫0t‍(θ1(s))Tθ1(s)ds+∫0t‍∫ℝ‍θ2(s,x)(μ(ds,dx)-ν(ds,dx))+∫0t‍∫ℝ‍(log(1+θ2(s,x))-θ2(s,x))μ(ds,dx)}.
By Girsanov’s Theorem, we know that W~t-∫0t‍θ1(s)ds is an m-dimensional Brownian motion under ℚ, which together with (60) implies σ~iθ1(t)=-(bi-r),  i=1,…,m. Thus, it is easy to see that
(65)σ~θ1(t)=-(b-r1m),
that is,
(66)θ1(t)=-σ~-1(b-r1m).
On the other hand, by (22), we have
(67)e-γX^Tx0,π*=exp{-γx0-γc(1-e-rT)r-γ∫0T‍e-rs(πs*)T(b-r1m)ds-γ∫0T‍e-rs(πs*)Tσ~dW~s+γ∫0T‍e-rsdLsγc(1-e-rT)r}.
In order to have (58), comparing dW~t-term and dμ-term, respectively, in (64) with those in (67) and taking (66) into account, it is reasonable to conjecture
(68)-σ~-1(b-r1m)=-γe-rtσ~Tπt*,log(1+θ2(t,x))=γe-rtx,
that is,
(69)πt*=ert(σ~T)-1σ~-1(b-r1m)γ=ert(ΣΨΣ)-1(b-r1m)γθ2(t,x)=eγxe-rt-1.Step 2.
Verifying thatZT* in (64) satisfies (58), where π* and θ2 are defined in (69), and θ1 is defined in (66).
Substituting (69) into (67), we have
(70)e-γX^Tx0,π*=ITHT,
where
(71)IT=exp{-γx0-γc(1-e-rT)r-(b-r1m)T(ΣΨΣ)-1(b-r1m)Tγc(1-e-rT)r},HT=exp{-(b-r1m)T(σ~T)-1W~T+γ∫0T‍e-rsdLs}.
For θ1 and θ2 defined as above, it is easy to see that Z* is a martingale. Substituting (66) and (69) into (64), we have
(72)ZT*=JTHT,
where
(73)JT=exp{-(b-r1m)T(ΣΨΣ)-1(b-r1m)2+∫0T‍∫ℝ‍(γe-rsx-eγxe-rs+1)ν(ds,dxT)(b-r1m)T(ΣΨΣ)-1(b-r1m)2}
is a constant. Since Z* is a martingale, we have 𝔼[ZT*]=1 and hence
(74)𝔼[HT]=JT-1.
Finally, by (70)–(74), we have
(75)e-γX^Tx0,π*𝔼[e-γX^Tx0,π*]=ITHTIT𝔼[HT]=JTHT=ZT*,
which is just what we desired.Step 3.
Verifying thatπ* in (69) satisfies condition (57).
It is clear that𝔼[(ZT*)2]<∞. For any π∈Π, by Girsanov’s Theorem, we can see that
(76)Mtπ:=∫0t‍e-rsπsT(b-r1m)ds+e-rsπsTσ~dW~s,t∈[0,T],
is a local martingale under ℚ. Furthermore, for any π∈Π, we have Mπ∈Lℱ2. Hence
(77)𝔼ℚ[sup0≤t≤T|Mtπ|]=𝔼[ZT*sup0≤t≤T|Mtπ|]≤𝔼[(ZT*)2]·𝔼[sup0≤t≤T|Mtπ|2]<∞,
which implies that the family
(78){Mτπ:τisastoppingtimeandτ≤T}
is uniformly integrable under ℚ. Hence Mπ is a martingale under ℚ. Thus we have 𝔼ℚ[MTπ]=0 for any π∈Π, which implies (57).
By Proposition2 and arguments in steps 5–7, we can claim the following theorem.Theorem 7.
Letπ* be defined as in (69). Then π* is the optimal trading strategy of problem (23) for the CARA utility function U(x)=-(1/γ)e-γx.


## 5. Remarks and Computational Results

In this section, we present some remarks and conduct computational experiments for the optimal investment problems of this paper according to the real market data. The objective of these computational tests is to contrast the insurer’s optimal strategies in an incomplete market with those in a complete market.According to Theorems6 and 7, the optimal strategy of CARA utility is independent of the claim process, while that of quadratic utility is related to the claim process. This is due to the specific nature of the CARA utility.


### 5.1. Quadratic Utility

For quadratic utility, as shown in (51)–(55), π* is a left-continuous process with a jump
(79)πt+*-πt*=(ΣΨΣ)-1(b-r1m)ΔLt,
at any time t, where ΔLt is the claim size at time t. From (79) the effect of claims on the efficient strategy of quadratic utility can be interpreted as follows.(i)
When no claim occurs at timet, the wealth process is continuous, and so is πt*.(ii)
When a claim occurs at timet, the wealth process has a jump -ΔLt, which is just the corresponding claim size; the optimal strategy also has a jump and the jump size on each stock is the corresponding element of (ΣΨΣ)-1(b-r1m)ΔLt.In terms of (79), we can consider the transaction cost of realizing the optimal strategy of quadratic utility. According to [14], the transaction cost is quantified by the portfolio weight turnover for a portfolio π, that is,
(80)∥πt+-πt∥1,
where ∥·∥1 denotes the 1-norm. Since transaction cost is an important aspect of any investment strategy, we are interested in comparing the costs of implementing the optimal strategies in an incomplete market with those in a complete market.Suppose there are 12 underline Brownian motions, 12 stocks for investment in a complete market, and 10 stocks in an incomplete market. The universe of assets are chosen from the 12 industry categories of finance.cn.yahoo.com in 2009 (see Tables1 and 2). The appreciation rate vector b is estimated through the mean rate of return of each stock. The covariance matrix a of the rate of return on 12 stocks is estimated based on the real market data. In a complete market, the volatility matrix σ^ is calculated from a such that σ^σ^T=a. Next, the volatility matrix σ of the 10 stocks in an incomplete market is obtained by eliminating corresponding row of σ^. Let the risk-free rate r=0.02 and the claim size is distributed according to the exponential distribution with parameter 1; that is, F(y)=1-e-y. Let t=0,1/12,1/6,…,2. With exponential claim size ΔLt randomly generated and Ψ,Σ calculated from (13) and (18), Figure 1 plots the transaction cost in an incomplete market and in a complete market, respectively.Table 1
Stocks in a complete market.
Industry
Company
Code
Aerospace and defense
Honeywell Intl.
HON
Automanufactures
Toyota Motor Corp. ADR
TM
Biotechnology and drug manufacturers
Johnson & Johnson
JNJ
Chemicals
EI DuPont de Nemours & Co.
DD
Communication equipment
Qualcomm
QCOM
Computer software
Microsoft
MSFT
Discount
Wal-Mart Stores Inc.
WMT
Diversified computer systems
Hewlett-Packard Co.
HPQ
Major integrated oil and gas
BP Plc.
BP
Semiconductor-broad line
Intel Corp.
INTC
Telecom services
AT & T
T
Utilities (gas and electric)
Duke Energy Corporation
DUKTable 2
Stocks in an incomplete market.
Industry
Company
Code
Aerospace and defense
Honeywell Intl.
HON
Automanufactures
Toyota Motor Corp. ADR
TM
Biotechnology and drug manufacturers
Johnson & Johnson
JNJ
Chemicals
EI DuPont de Nemours & Co.
DD
Communication equipment
Qualcomm
QCOM
Computer software
Microsoft
MSFT
Diversified computer systems
Hewlett-Packard Co.
HPQ
Major integrated oil & gas
BP Plc.
BP
Semiconductor-broad line
Intel Corp.
INTC
Utilities (gas & electric)
Duke Energy Corporation
DUKFigure 1
Transaction cost for quadratic utility.We easily observe that the optimal strategies in an incomplete market incurs lower transaction cost than those in a complete market. Because there are fewer stocks in an incomplete market, than in a complete market, the number of stocks whose investment amount will be changed is smaller in an incomplete market. Thus the transaction cost for fewer stocks is low.


### 5.2. CARA Utility

For the optimal strategy of CARA utility, (69) shows that the discounted amount held in each stock is fixed. A similar investment strategy for CARA utility in an incomplete market is also obtained in Wang [11]. However, the expression of the optimal strategy in [11] contains a g-inverse of matrix σσT, which is not unique. In our results, (ΣΨΣ)-1 replaces the g-inverse matrix, which makes the optimal strategy explicit and deterministic. Due to the determinacy and nonrandomness of the optimal strategy for CARA utility, we can analyze the differences between complete market and incomplete market conveniently from the results of CARA utility.The experimental procedure is the same as in Section5.1 and all the parameters are set as in Section 5.1 except t. Let T=1 represent one year and the insurer rebalances his portfolio once a month. Then t=0,1/12,1/6,…,11/12 and there are 12 investment periods.In Figure2, we compare the average total dollar amount invested in stocks in an incomplete market with that in a complete market as the risk aversion coefficient γ ranges from 1 to 100. The average total dollar amount invested in stocks for a sequence of portfolios {πt}t=0n with m stocks is defined as
(81)π-=1n+1∑t=0n‍πtT1m.
From Figure 2 we find that in an incomplete market, the insurer will invest more money in stocks (risky assets). The practical number of stocks in an incomplete market is less than the ideal number of stocks. For example, in our experiments, there is only 10 stocks for investment while the ideal number of attainable stocks is 12 in the incomplete market. Thus an insurer will invest more in the only 10 stocks. An investor is venturesome in an incomplete market. Also we see that the total amount invested in stocks decreases as γ increases. Since γ is the absolute risk aversion coefficient, this result is consistent with intuition.Figure 2
Comparison of the average total dollar amount invested in stocks in a complete market and an incomplete market.Figure3 is a plot of the relative difference of the optimal strategy in an incomplete market with respect to that in a complete market; that is, it plots
(82)(π-in-π-co)π-co
as a function of γ. The above π-in,π-co denote the average total dollar amount invested in stocks in an incomplete market and in a complete market, respectively. It is obvious that this difference is independent of the risk aversion coefficient γ but only related to the difference of the market structure. No matter what the risk aversion coefficients are, the effects of the incomplete market on the corresponding optimal strategies are the same.Figure 3
Relative differences of the average total dollar amount invested in stocks between the two markets.Given a sequence of portfolios{πt}t=0n, we quantify its average transaction cost by
(83)1n∑t=0n-1‍∥πt+1-πt∥1.
Figure 4 plots the average transaction costs of 12 periods optimal strategies for CARA utility in the two different markets. Same as the quadratic utility case, the cost of implementing the optimal investment in an incomplete market is lower than that in a complete market.Figure 4
Average transaction cost for CARA utility in the two markets.


## 5.1. Quadratic Utility

For quadratic utility, as shown in (51)–(55), π* is a left-continuous process with a jump
(79)πt+*-πt*=(ΣΨΣ)-1(b-r1m)ΔLt,
at any time t, where ΔLt is the claim size at time t. From (79) the effect of claims on the efficient strategy of quadratic utility can be interpreted as follows.(i)
When no claim occurs at timet, the wealth process is continuous, and so is πt*.(ii)
When a claim occurs at timet, the wealth process has a jump -ΔLt, which is just the corresponding claim size; the optimal strategy also has a jump and the jump size on each stock is the corresponding element of (ΣΨΣ)-1(b-r1m)ΔLt.In terms of (79), we can consider the transaction cost of realizing the optimal strategy of quadratic utility. According to [14], the transaction cost is quantified by the portfolio weight turnover for a portfolio π, that is,
(80)∥πt+-πt∥1,
where ∥·∥1 denotes the 1-norm. Since transaction cost is an important aspect of any investment strategy, we are interested in comparing the costs of implementing the optimal strategies in an incomplete market with those in a complete market.Suppose there are 12 underline Brownian motions, 12 stocks for investment in a complete market, and 10 stocks in an incomplete market. The universe of assets are chosen from the 12 industry categories of finance.cn.yahoo.com in 2009 (see Tables1 and 2). The appreciation rate vector b is estimated through the mean rate of return of each stock. The covariance matrix a of the rate of return on 12 stocks is estimated based on the real market data. In a complete market, the volatility matrix σ^ is calculated from a such that σ^σ^T=a. Next, the volatility matrix σ of the 10 stocks in an incomplete market is obtained by eliminating corresponding row of σ^. Let the risk-free rate r=0.02 and the claim size is distributed according to the exponential distribution with parameter 1; that is, F(y)=1-e-y. Let t=0,1/12,1/6,…,2. With exponential claim size ΔLt randomly generated and Ψ,Σ calculated from (13) and (18), Figure 1 plots the transaction cost in an incomplete market and in a complete market, respectively.Table 1
Stocks in a complete market.
Industry
Company
Code
Aerospace and defense
Honeywell Intl.
HON
Automanufactures
Toyota Motor Corp. ADR
TM
Biotechnology and drug manufacturers
Johnson & Johnson
JNJ
Chemicals
EI DuPont de Nemours & Co.
DD
Communication equipment
Qualcomm
QCOM
Computer software
Microsoft
MSFT
Discount
Wal-Mart Stores Inc.
WMT
Diversified computer systems
Hewlett-Packard Co.
HPQ
Major integrated oil and gas
BP Plc.
BP
Semiconductor-broad line
Intel Corp.
INTC
Telecom services
AT & T
T
Utilities (gas and electric)
Duke Energy Corporation
DUKTable 2
Stocks in an incomplete market.
Industry
Company
Code
Aerospace and defense
Honeywell Intl.
HON
Automanufactures
Toyota Motor Corp. ADR
TM
Biotechnology and drug manufacturers
Johnson & Johnson
JNJ
Chemicals
EI DuPont de Nemours & Co.
DD
Communication equipment
Qualcomm
QCOM
Computer software
Microsoft
MSFT
Diversified computer systems
Hewlett-Packard Co.
HPQ
Major integrated oil & gas
BP Plc.
BP
Semiconductor-broad line
Intel Corp.
INTC
Utilities (gas & electric)
Duke Energy Corporation
DUKFigure 1
Transaction cost for quadratic utility.We easily observe that the optimal strategies in an incomplete market incurs lower transaction cost than those in a complete market. Because there are fewer stocks in an incomplete market, than in a complete market, the number of stocks whose investment amount will be changed is smaller in an incomplete market. Thus the transaction cost for fewer stocks is low.


## 5.2. CARA Utility

For the optimal strategy of CARA utility, (69) shows that the discounted amount held in each stock is fixed. A similar investment strategy for CARA utility in an incomplete market is also obtained in Wang [11]. However, the expression of the optimal strategy in [11] contains a g-inverse of matrix σσT, which is not unique. In our results, (ΣΨΣ)-1 replaces the g-inverse matrix, which makes the optimal strategy explicit and deterministic. Due to the determinacy and nonrandomness of the optimal strategy for CARA utility, we can analyze the differences between complete market and incomplete market conveniently from the results of CARA utility.The experimental procedure is the same as in Section5.1 and all the parameters are set as in Section 5.1 except t. Let T=1 represent one year and the insurer rebalances his portfolio once a month. Then t=0,1/12,1/6,…,11/12 and there are 12 investment periods.In Figure2, we compare the average total dollar amount invested in stocks in an incomplete market with that in a complete market as the risk aversion coefficient γ ranges from 1 to 100. The average total dollar amount invested in stocks for a sequence of portfolios {πt}t=0n with m stocks is defined as
(81)π-=1n+1∑t=0n‍πtT1m.
From Figure 2 we find that in an incomplete market, the insurer will invest more money in stocks (risky assets). The practical number of stocks in an incomplete market is less than the ideal number of stocks. For example, in our experiments, there is only 10 stocks for investment while the ideal number of attainable stocks is 12 in the incomplete market. Thus an insurer will invest more in the only 10 stocks. An investor is venturesome in an incomplete market. Also we see that the total amount invested in stocks decreases as γ increases. Since γ is the absolute risk aversion coefficient, this result is consistent with intuition.Figure 2
Comparison of the average total dollar amount invested in stocks in a complete market and an incomplete market.Figure3 is a plot of the relative difference of the optimal strategy in an incomplete market with respect to that in a complete market; that is, it plots
(82)(π-in-π-co)π-co
as a function of γ. The above π-in,π-co denote the average total dollar amount invested in stocks in an incomplete market and in a complete market, respectively. It is obvious that this difference is independent of the risk aversion coefficient γ but only related to the difference of the market structure. No matter what the risk aversion coefficients are, the effects of the incomplete market on the corresponding optimal strategies are the same.Figure 3
Relative differences of the average total dollar amount invested in stocks between the two markets.Given a sequence of portfolios{πt}t=0n, we quantify its average transaction cost by
(83)1n∑t=0n-1‍∥πt+1-πt∥1.
Figure 4 plots the average transaction costs of 12 periods optimal strategies for CARA utility in the two different markets. Same as the quadratic utility case, the cost of implementing the optimal investment in an incomplete market is lower than that in a complete market.Figure 4
Average transaction cost for CARA utility in the two markets.


## 6. Conclusion

In this paper, we consider the optimal investment problem for an insurer in an incomplete market. After transforming the incomplete market into a complete one, we solve the problem via the martingale approach. From the computational experiments, we find that an insurer’s optimal investment strategy in an incomplete market incurs lower transaction cost than the one in a complete market. The amount invested in stocks in an incomplete market is more than that in a complete market. For CARA utility, the computational results also show that the incomplete market produces the same effects on the optimal strategies of insurers with different risk aversion coefficients.



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*Source: 751846-2013-03-31.xml*