Insurance: Mathematics and Economics 40 (2007) 77–84www.elsevier.com/locate/ime

Optimal investment for an insurer with exponential utility preference

Nan Wang

Cass Business School, City University, EC1Y 8TZ London, United Kingdom

Received October 2005; received in revised form February 2006; accepted 28 February 2006

Abstract

This paper considers the optimal investment choice for a general insurer in the sense of maximizing the exponential utility of hisor her reserve at a future time. The claim process is supposed to be a pure jump process (not necessarily compound Poisson) andthe insurer has the option of investing in multiple risky assets whose price processes are described by the Black–Scholes marketmodel. It is shown in this paper that the optimal strategy is to put a fixed amount of money in each risky asset if there is no risk-freeasset. If there is a risk-free asset, the discounted amount held in each risky asset is fixed. In the case where the claim process iscompound Poisson, the optimal strategy with respect to a properly selected utility function can result in a reserve process which issafer than that without risky investment.c© 2006 Elsevier B.V. All rights reserved.

Keywords: Exponential utility; Admissible strategy; Ito’s formula; Exponential martingale; Adjustment coefficient

1. Introduction

The problem of optimal investment for a general insurer has attracted more and more attention since the work ofHipp and Plum (2000). In Hipp and Plum (2000), the classical Cramer–Lundberg model is adopted for the risk reserve(i.e., the constant premium rate and compound Poisson claim process) and the insurer is supposed to have an optionof investing part of his or her reserve in a risky asset (there is no risk-free asset) whose price process is a geometricBrownian motion with the target of minimizing the probability of eventual ruin. This set-up is adopted by most of theworks on this subject since 2000.

An important earlier work is Browne (1995), where the author uses a different model (or say, approximation) forthe claim process, which is a drifted Brownian motion. Browne (1995) shows that the target of minimizing the ruinprobability and the target of maximizing the exponential utility of the reserve (at a future time) produce the same typeof strategy, that is to invest a fixed amount of money in the risky asset.

When a compound Poisson process is considered, the result becomes different because the strategy whichminimizes the ruin probability is not the investment of a fixed amount money in the risky asset. However, as shownin Hipp and Schmidli (2004), if the initial reserve is large enough, the amount in the risky asset is almost a constant.In Gaier et al. (2003), the authors show that the strategy which produces the optimal asymptotic decay of the ruinprobability is to invest a fixed amount of money in the risky asset regardless of the reserve level.

E-mail address: nan.wang1@btopenworld.com.

0167-6687/$ - see front matter c© 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.insmatheco.2006.02.008

78 N. Wang / Insurance: Mathematics and Economics 40 (2007) 77–84

What is the strategy that maximizes the exponential utility of the reserve in the case of the compound Poisson claimprocess then? Will this strategy differ much from (or be similar to) the strategy which minimizes the ruin probability?

In this paper we adopt the Black–Scholes asset model and suppose there are multiple risky assets available to beinvested in. The claim process in this paper is an increasing pure jump process. With the target of maximizing theexponential utility of the reserve at a future time, we show that the optimal investment strategy is indeed to invest afixed amount of money in each risky asset if there is no risk-free asset. In the case where there is a risk-free asset, thediscounted amount held in each risky asset is fixed.

A similar question of exponential utility maximizing is also considered in Yang and Zhang (2005) through theapproach of stochastic dynamic programming (the HJB equation) as used in Browne (1995). In Yang and Zhang(2005), the claim process is compound Poisson; the risk reserve is also disturbed by a Brownian motion which iscorrelated with the risky asset process. The optimal strategy in Yang and Zhang (2005) is free of the compoundPoisson claim process. The approach in this paper shows that the type of claim process in fact has nothing to do withthe optimal strategy (for exponential utility) as long as it is an increasing process.

The organization of the paper is as follows. In Section 2, we give the mathematical description of the investmentproblem with unhedgable claim outflows. In Section 3, we first prove the assertions made above and then considerthe influence of risky investment on the safety of the insurer when the claim process is compound Poisson. Finally,in Section 4, we illustrate the limitation of the HJB equation for the problem and we also mention an approach in thecase of a more general asset price model.

2. Mathematical descriptions of the problem

All stochastic processes introduced below are supposed to be adapted processes in a filtered probability space(Ω ,F(Ft ), P), where Ft , t ≥ 0 is a filtration satisfying the usual conditions. Suppose the initial reserve is x , thepremium rate is constant c, the cumulative claim process, denoted as Ct , is an increasing pure jump process (right-continuous) and there are d + 1 assets with price processes, X t = (X (0)

t , X (1)t , . . . , X (d)

t )∗, given by1

dX (0)t = r X (0)

t dt, dX (i)t = X (i)

t[(r + µi )dt + σ ∗

i dWt], t ≥ 0, 1 ≤ i ≤ d (1)

where r is the interest rate, µi , 1 ≤ i ≤ d , are positive constants, Wt = (W (1)t , . . . , W (m)

t )∗ is a standardm-dimensional Brownian motion, and σ ∗

i = (σi1, . . . , σim) is the i-th row of matrix σ = σi j 1≤i≤d,1≤ j≤m . X (0)t

represents the price of a risk-free asset. Without any loss of generality, we assume X (0)0 = 1.

Throughout this paper, Ct is assumed independent of Wt . The probability space therefore can be taken as thecomplete product space

(Ω ,FT (Ft )0≤t≤T , P) := (Ω1 ⊗ Ω2,F (1)T ⊗ F (2)

T (F (1)t ⊗ F (2)

t )0≤t≤T , P1 ⊗ P2)

where (Ω1,F (1)T (F (1)

t )0≤t≤T , P1) is the probability space containing process Ct with F (1)t being the natural filtration

generated by Ct (suppose F (1)0 contains all P1-null sets) and (Ω2,F (2)

T (F (2)t )0≤t≤T , P2) is the probability space

containing the m-dimensional Brownian motion Wt with F (2)t being the argumented filtration generated by Wt .

An investment plan can be expressed in terms of the number of units invested in each asset. Suppose all the moneyis invested either in those risky assets or in the risk-free asset. Let θ

(i)t be the number of units of asset X (i)

t in theinvestment portfolio at time t . The portfolio value is then θ∗

t X t , where θt = (θ(0)t , θ

(1)t , . . . , θ

(d)t )∗. The portfolio

value at a time t is the result of premium inflow, claim outflow and the earning from investment in time period [0, t];therefore (see for example Bjork (1998) for details of portfolio dynamics)

θ∗t X t = x +

∫ t

0θ∗

s dXs + ct − Ct . (2)

To be mathematically rigorous, for each 0 ≤ i ≤ d , θ(i)t should be a predictable process (with respect to (Ft )t≥0)

which is integrable with respect to X (i)t .

1 The sign ∗ means transposition (so, X t is a column vector). In this paper, the product operation in all mathematical expressions is the productof matrices (vectors).

N. Wang / Insurance: Mathematics and Economics 40 (2007) 77–84 79

Define Vt = θ∗t X t , which is, from (2), a semimartingale, not continuous however. Applying Ito’s formula (for Ito’s

lemma with respect to a semimartingale with jumps, see pages 81–82 of Protter (2004) and the integration sign∫ t

0

used in this paper means∫ t

0+ used in Protter (2004)) to Vt (X (0)t )−1 (note that X (0)

t = er t ) produces

Vt (X (0)t )−1

− x =

∫ t

0e−rsdVs +

∫ t

0Vs−de−rs

+

∫ t

0θ∗

s d[e−rs, Xs] +

∑0<s≤t

(Vse−rs

− Vs−e−rs− e−rs1Vs

)

=

t∫0

e−rsdVs +

∫ t

0Vs−de−rs

where Vt− = limst Vs and 1Vt = Vt − Vt− . The integration∫ t

0 Vs−de−rs has no difference from∫ t

0 Vsde−rs since

Vs− = θ∗s Xs + 1Cs,

∫ t

01Csde−rs

= 0.

From (2), we have∫ t

0e−rsdVs =

∫ t

0e−rsθ∗

s dXs + c∫ t

0e−rsds −

∫ t

0e−rsdCs .

Bringing this into the above expression and noting that

d(Xse−rs) = e−rsdXs + Xsde−rs

gives (define X t = X t e−rs)

θ∗t X t = x +

∫ t

0θ∗

s dXs + c∫ t

0e−rsds −

∫ t

0e−rsdCs . (3)

An investment strategy θt is said to be admissible in this paper if θ (i) is integrable with respect to X (i) foreach 1 ≤ i ≤ d and (3) holds. The set of all admissible strategies is written as A. We will use the notationVt = Vt e−r t

= θ∗t X t from now on, which represents the discounted portfolio value. Clearly, V0 = V0 = x .

Suppose the time horizon is [0, T ]. The investment problem of this paper is the following optimization:

maxθ∈D

E(1 − e−αVT ) (4)

where α is a positive constant and D is a certain subset of A.

3. The optimal strategy and the influence of risky investment

Instead of number of units, an investment strategy can also be expressed by the amount of wealth invested in eachasset. Define

πt =

(π

(1)t , . . . , π

(d)t

)∗

:=

(θ

(1)t X (1)

t , . . . , θ(d)t X (d)

t

)∗

. (5)

The amount in the risk-free asset is clearly π(0)t := Vt −

∑di=1 π

(i)t . Denote µ = (µ1, . . . , µd)∗. Bringing (5) and

(1) into (3), we obtain

Vt = x + c∫ t

0e−rsds −

∫ t

0e−rsdCs +

∫ t

0e−rsπ∗

s µds +

∫ t

0e−rsπ∗

s σdWs . (6)

Since exponential utility is considered in this paper, we assume

E

[exp

(α

∫ T

0e−r t dCt

)]< ∞

80 N. Wang / Insurance: Mathematics and Economics 40 (2007) 77–84

and we also confine the range of admissible strategies to all of those such that

E

[exp

(α2

2

∫ T

0e−2r tπ∗

t σσ ∗πt dt

)]< ∞. (7)

Denote the set of all admissible strategies (use π rather than θ from now on) satisfying (7) as D. The optimizationproblem is rewritten as

maxπ∈D

E(1 − e−αVT ). (8)

Theorem 1. Suppose for any fixed 0 ≤ t ≤ T the quadratic optimization problem

minx∈Rd

(12αe−r t x∗σσ ∗x − x∗µ

)(9)

has solutions with finite Rd -norm. Let at be one of the solutions, which clearly has the form

at =er t

α(σσ ∗)−µ,

where (σσ ∗)− is a g-inverse of matrix σσ ∗. Then πt = at , 0 ≤ t ≤ T , is a solution of problem (8).

We give a lemma first, which will be used in the proof of the theorem. Recall that the probability space we areworking on is a product probability space as described in the introduction.

Lemma 2. For any strategy π ∈ D,

E[

e−12

∫ T0 α2e−2r t π∗

t σσ ∗πt dt−∫ T

0 αe−r t π∗t σdWt

∣∣∣F (1)T ⊗ Ω2

]= 1

almost surely with respect to probability measure P1.

Proof. Denote ω1, ω2 as general elements in Ω1 and Ω2 respectively. The stochastic processes πt and Wt are thenexpressed as πt (ω1, ω2) and Wt (ω2). An investment strategy must be a progressively measurable process, i.e., for anyfixed t , πs(ω1, ω2), as a function of (s, ω1, ω2) ∈ [0, t] ⊗ Ω1 ⊗ Ω2, is a measurable mapping of(

[0, t] ⊗ Ω1 ⊗ Ω2,B([0, t]) ⊗ F (1)t ⊗ F (2)

t

)→

(Rd ,B(Rd)

).

If an element ω1 ∈ Ω1 is fixed (almost surely w.r.t. P1), by Fubini’s theorem, πs(, ω2), as a function of (s, ω2) ∈

[0, t] ⊗ Ω2, is a measurable mapping from([0, t] ⊗ Ω2,B([0, t]) ⊗ F (2)

t

)→

(Rd ,B(Rd)

)which means πt (, ω2) is progressively measurable in (Ω2,F (2)

T (F (2)t )0≤t≤T , P2). Noting at the condition (7), still by

Fubini’s theorem, we see that πt (, ω2) satisfies the Novikov condition (see corollary 5.13 on page 199 of Karatzas andShreve (1991)). Therefore, for any fixed ω1 ∈ Ω1 (almost surely w.r.t P1),

e−12

∫ t0 α2e−2rsπ∗

s (ω1,ω2)σσ ∗πs (ω1,ω2)ds−∫ t

0 αe−rsπ∗s (ω1,ω2)σdWs (ω2),F (2)

t , 0 ≤ t ≤ T

is an exponential martingale in (Ω2,F (2)T (F (2)

t )0≤t≤T , P2), and hence

E2

[e−

12

∫ t0 α2e−2rsπ∗

s (ω1,ω2)σσ ∗πs (ω1,ω2)ds−∫ t

0 αe−rsπ∗s (ω1,ω2)σdWs (ω2)

]= 1.

It is easy to check that

E[

e−12

∫ T0 α2e−2r t π∗

t (ω1,ω2)σσ ∗πt (ω1,ω2)dt−∫ T

0 αe−r t π∗t (ω1,ω2)σdWt (ω2)

∣∣∣F (1)T ⊗ Ω2

]= E2

[e−

12

∫ t0 α2e−2rsπ∗

s (ω1,ω2)σσ ∗πs (ω1,ω2)ds−∫ t

0 αe−rsπ∗s (ω1,ω2)σdWs (ω2)

].

The proof is thus completed.

N. Wang / Insurance: Mathematics and Economics 40 (2007) 77–84 81

Proof of Theorem 1. Clearly, optimization problem (8) is the same as minπ∈D E(e−αVT ). It is seen from (6) that

E(

e−αVT)

= E

e−α

(x+c

∫ T0 e−r t dt−

∫ T0 e−r t dCt

)· E

[e−α

(∫ T0 e−r t π∗

t µdt+∫ T

0 e−r t π∗t σdWt

)∣∣∣∣F (1)T ⊗ Ω2

].

For the conditional expectation part, we have

E

[e−α

(∫ T0 e−r t π∗

t µdt+∫ T

0 e−r t π∗t σdWt

)∣∣∣∣F (1)T ⊗ Ω2

]= E

[(e−α

∫ T0 e−r t π∗

t µdt+ α22

∫ T0 e−2r t π∗

t σσ ∗πt dt) (

e−α22

∫ T0 e−2r t π∗

t σσ ∗πt dt−α∫ T

0 e−r t π∗t σdWt

)∣∣∣∣F (1)T ⊗ Ω2

].

Note that

−α

∫ T

0e−r tπ∗

t µdt +α2

2

∫ T

0e−2r tπ∗

t σσ ∗πt dt ≥ −

∫ T

0αe−r t a∗

t µdt +12

∫ T

0α2e−2r t a∗

t σσ ∗at dt

since at is a solution of quadratic optimization (9). So, by Lemma 2,

E

[e−α

(∫ T0 e−r t π∗

t µdt+∫ T

0 e−r t π∗t σdWt

)∣∣∣∣F (1)T ⊗ Ω2

]≥ e−

∫ T0 αe−r t a∗

t µdt+ 12

∫ T0 α2e−2r t a∗

t σσ ∗at dt E[

e−12

∫ T0 α2e−2r t π∗

t σσ ∗πt dt−∫ T

0 αe−r t π∗t σdWt

∣∣∣F (1)T ⊗ Ω2

]= e−α

∫ T0 e−r t a∗

t µdt+ α22

∫ T0 e−2r t a∗

t σσ ∗at dt .

Therefore, for any π ∈ B,

E[e−αVT

]≥ e−α

∫ T0 e−r t a∗

t µdt+ α22

∫ T0 e−2r t a∗

t σσ ∗at dt· Ee

−α(

x+c∫ T

0 e−r t dt−∫ T

0 e−r t dCt

). (10)

On the other hand, if we choose π = a (this strategy is in D since at has finite Rd -norm), then it can be checked withthe same arguments as above that equality holds in (10). This ends the proof.

Corollary 3. If σ is an invertible square matrix, then the optimal strategy is given by πt = α−1er t (σσ ∗)−1µ, 0 ≤

t ≤ T .

Proof. It is obvious from Theorem 1.

Risky investment could be very dangerous for an insurance business as shown in Frovola et al. (2002). In thefollowing part of this section, we set r = 0, which corresponds to the situation of no risk-free asset, and considerwhether the strategy given in Corollary 3 (which is not the type in Frovola et al. (2002)) contributes to the insurer’ssafety when the claim process is a compound Poisson process.

Note that the strategy in Corollary 3 is free of time parameter t since r = 0. So, we may omit the time horizon[0, T ] and consider the problem over the whole time range [0, ∞).

From (6), the portfolio value Vt (note that Vt and Vt are the same in this case) corresponding to the strategy givenin Corollary 3 is

Vt = x + ct − Ct +1α

µ∗(σσ ∗)−1µt +1α

µ∗(σσ ∗)−1σ Wt

= x + ct −

Nt∑i=1

Yi +1α

µ∗(σσ ∗)−1µt +1α

µ∗(σσ ∗)−1σ Wt (11)

where Nt is a Poisson process with intensity λ and Yi , i ≥ 1 are i.i.d. random variables such that c > λE(Y1) (i.e., werequest the positive loading condition).

We measure the risk of ruin by the so-called adjustment coefficient of process Vt , which is a positive constant γ

such that eγ (x−Vt ), t ≥ 0, is a martingale. If this γ exists, the probability of ruin then satisfies

P

(mint≥0

Vt < 0)

≤ e−γ x .

82 N. Wang / Insurance: Mathematics and Economics 40 (2007) 77–84

See, for example, Proposition 1.1 of Asmussen (2000) for details. Since both the compound Poisson process andBrownian motion have stationary and independent increments, it is checked for Vt that

E[

eγ (x−Vt+s )∣∣∣Fs

]= eγ (x−Vs )E

eγ

Nt+s∑i=Ns+1

Yi −γ(

c+ 1αµ∗(σσ ∗)−1µ

)t−γ

(1αµ∗(σσ ∗)−1σ

)(Wt+s−Ws )

= eγ (x−Vs )E

eγ

Nt∑i=1

Yi −γ(

c+ 1αµ∗(σσ ∗)−1µ

)t−γ

(1αµ∗(σσ ∗)−1σ

)Wt

= eγ (x−Vs )e

λ[M(γ )−1]t−γ(

c+ 1αµ∗(σσ ∗)−1µ

)t+γ 2

(1

2α2 µ∗(σσ ∗)−1µ)

t

where M(γ ) = E(eγ Y1). The adjustment coefficient is thus the solution of the equation

λ[M(γ ) − 1] −

[c +

1α

µ∗(σσ ∗)−1µ

]γ +

[1

2α2 µ∗(σσ ∗)−1µ

]γ 2

= 0. (12)

The adjustment coefficient for the reserve process without investment (the classical Cramer–Lundberg model),Rt := x + ct −

∑Nti=1 Yi , is well known to be the solution of the equation

λ[M(γ ) − 1] − cγ = 0. (13)

The left sides of Eqs. (12) and (13) are both convex about γ and, with the positive loading requirement, both equationsmust have solutions. To distinguish them, define γ1 and γ0 as the solutions of (12) and (13). It is not difficult to checkthat

γ1 ≥ γ0, if 2α ≥ γ0. (14)

This shows that a properly selected α will increase the degree of safety of the insurer compared with the case withoutinvestment. Whatever α is, the average growth rate of (11), c + α−1µ∗(σσ ∗)−1µ − λE(Y1), is always greater thanc − λE(Y1), the average growth rate without investment.

In the case where there is only one risky asset, i.e., σ is a constant rather than a matrix, we are back to the settingof Hipp and Plum (2000). The optimal strategy in Corollary 3 reduces to

πt =µ

ασ 2 . (15)

An investment strategy of this type is also obtained in Gaier et al. (2003) and Hipp and Schmidli (2004). In Gaier et al.(2003), the investment strategy yielding the optimal asymptotic decay of the probability of eventual ruin has form (15)and in Hipp and Schmidli (2004), the strategy that minimizes the probability of ruin has (15) as its limit.

4. Some final remarks

(1) Yang and Zhang (2005) solve their problem using the HJB equation. This approach however is not suitable herefor the general claim process Ct . The problem in Yang and Zhang (2005) can be expressed according to the notationof this paper as

maxπ

E(1 − e−α′VT )

where α′ is some constant. They consider the optimization problem with respect to the portfolio value Vt instead of thediscounted portfolio value Vt . This optimization problem goes back to problem (4) if we take α′

= αe−rT . Therefore,the two problems are not different in nature when the interest rate is a constant.

By (1), (2) and (5), it is easy to see that

Vt − V0 =

∫ t

0[c + r Vs + π∗

s µ]ds +

∫ t

0π∗

s σdWs − Ct (V0 = x).

N. Wang / Insurance: Mathematics and Economics 40 (2007) 77–84 83

Suppose Vt is a Markov process (so, there should be a requirement on Ct , for example, a process with independentincrements) and define

U (t, v) = supπs ,t≤s≤T

E[1 − e−α′VT |Vt = v].

Suppose U (t, v) is smooth enough to allow all of the differential operations in the due course. Then, from Ito’sformula,

U (t + h, Vt+h) − U (t, v) =

∫ t+h

t

∂U (s, Vs)

∂sds +

∫ t+h

t

∂U (s, Vs)

∂v[c + π∗

s µ + r Vs]ds

+

∫ t+h

t

∂U (s, Vs)

∂vπ∗

s ΣdWs +12

∫ t+h

t

∂2U (s, Vs)

∂v2 π∗s σσ ∗πsds

+

∑t<s≤t+h

[U (s, Vs) − U (s, Vs−)

].

By the law of dynamic programming,

U (t, v) = supπs ,t≤s≤t+h

E[U (t + h, Vt+h)].

The HJB equation is the result of the above two expressions on setting h → 0. To write out the HJB equation explicitly,one has to specify the probability law of the claim process Ct in order to deal with

∑t<s≤t+h E[U (s, Vs)−U (s, Vs−)].

For example, when claim process Ct is compound Poisson, the HJB equation is

∂U (t, v)

∂t+ sup

πt

[∂U (t, v)

∂v(c + π∗

t µ + rv) +12

∂2U (t, v)

∂v2 π∗t σσ ∗πt

]+ λE[U (t, v − Z) − U (t, v)] = 0

with boundary condition U (T, v) = 1 − e−α′v . Here λ is the intensity parameter of the Poisson process and Z is theclaim amount random variable. Yang and Zhang (2005) find the optimal investment strategy by solving this equation.

(2) If the interest rate is a stochastic process having correlation with other asset price processes, the situation willbecome very complicated, because the discounted outflow will also be correlated with the asset price processes andthe arguments in the proof of Theorem 1 are no longer valid.

The optimal investment problem with general semimartingale assets is usually tackled through a static dualityprogram and the problem is turned into a problem of finding a minimum martingale measure; see for exampleKramkov and Schachermayer (1999) and the references there. For problem (4) of this paper where there areunhedgable wealth outflows (the claim process), the idea in Delbaen et al. (2002) may be borrowed to turn the probleminto a static alternative. The solving of the alternative program depends on concrete modelling of the asset process,interest rate process and claim process, which is a difficult task in general.

(3) Comparing the study in this paper and those in Gaier et al. (2003) and Hipp and Schmidli (2004), one canfeel that maximizing exponential utility does have some inherent relation with minimizing ruin probability. At least,properly selected utility would not compromise on the safety of an insurer (in the sense of the adjustment coefficientof the risk process Vt ) as analyzed in Section 3.

We point out finally that the assumption of constant premium rate can be waived without affecting most of theresults in the paper. For example, we can let the premium rate be a stochastic process, say ct . As long as ct isindependent of the asset price processes X t , Theorem 1 still holds.

References

Asmussen, S., 2000. Ruin Probabilities. World Scientific.Bjork, T., 1998. Arbitrage Theory in Continuous Time. Oxford.Browne, S., 1995. Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin.

Mathematics of Operations Research 20, 937–957.Delbaen, F., Grandits, P., Rheinlander, T., Samperi, D., Schweizer, M., Stricker, C., 2002. Exponential hedging and entropic penalties. Mathematical

Finance 12, 99–123.Frovola, A.G., Kabanov, Yu.M., Pergamenshchikov, S.M., 2002. In the insurance business risky investments are dangerous. Finance and Stochastics

6, 227–235.

84 N. Wang / Insurance: Mathematics and Economics 40 (2007) 77–84

Gaier, J., Grandits, P., Schachermayer, W., 2003. Asymptotic ruin probabilities and optimal investment. Annals of Applied Probability 13,1054–1076.

Hipp, C., Plum, M., 2000. Optimal investment for insurers. Insurance: Mathematics and Economics 27, 215–228.Hipp, C., Schmidli, H., 2004. Asymptotics of ruin probabilities for controlled risk processes in the small claims case. Scandinavian Actuarial

Journal.Karatzas, S., Shreve, S., 1991. Brownian Motion and Stochastic Calculus. Springer.Kramkov, M., Schachermayer, W., 1999. The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Annals of

Applied Probability 9, 904–950.Protter, P., 2004. Stochastic Integration and Differential Equations. Springer.Yang, H., Zhang, L., 2005. Optimal investment for insurer with jump-diffusion risk process. Insurance: Mathematics and Economics 37, 615–634.