In this paper, we consider an AAI with two types of insurance business with p-thinning dependent
claims risk, diversify claims risk by purchasing proportional reinsurance, and invest in a stock with Heston
model price process, a risk-free bond, and a credit bond in the financial market with the objective of maximizing
the expectation of the terminal wealth index effect, and construct the wealth process of AAI as well as the the
model of robust optimal reinsurance-investment problem is obtained, using dynamic programming, the HJB
equation to obtain the pre-default and post-default reinsurance-investment strategies and the display expression
of the value function, respectively, and the sensitivity of the model parameters is analyzed through numerical
experiments to obtain a realistic economic interpretation. The model as well as the results in this paper are a
generalization and extension of the results of existing studies.
The power trading and ancillary services provision comprise technical and financial risks and therefore require a structured risk management. Focus in this paper is on financial risk management that is important for the system operator faces when providing and using ancillary services for balancing of power system. Risk on ancillary services portfolio is modeled through value at risk and conditional value at risk measures. The application of these risk measures in power system is given in detail to show how to using the risk concept in practice. Conditional value at risk optimization is analysed in the context of portfolio selection and how to apply this optimization for hedging a portfolio consisting of different types of ancillary services.
Fair valuation of participating life insurance contracts with jump riskAlex Kouam
A C++ based program which prices the fair value of a participating life insurance whereby the underlying follows a Kou process and the insurer's default occurs only at contract's maturity.
:In this paper, we consider the equity premium puzzle under a general utility function. We derive that
the optimal strategy under a general utility function approximate the optimal strategy under the special utility
function. This result posed in the present paper can be regarded as a generalization of the work by Gong and
Zou [13]
Optimal Portfolio Selection for a Defined Contribution Pension Fund with Retu...BRNSS Publication Hub
This paper investigates the optimal investment strategies for a defined contribution pension fund with return clauses of premiums with interest under the mean-variance criterion. Using the actuarial symbol, we formalize the problem as a continuous time mean-variance stochastic optimal control. The pension fund manager considers investments in risk and risk-free assets to increase the remaining accumulated funds to meet the retirement needs of the remaining members. Using the variational inequalities methods, we established an optimized problem from the extended Hamilton–Jacobi–Bellman Equations and solved the optimized problem to obtain the optimal investment strategies for both risk-free and risky assets and also the efficient frontier of the pension member. Furthermore, we evaluated analytically and numerically the effect of various parameters of the optimal investment strategies on it. We observed that the optimal investment strategy for the risky asset decreases with an increase in the risk-averse level, initial wealth, and the predetermined interest rate.
The power trading and ancillary services provision comprise technical and financial risks and therefore require a structured risk management. Focus in this paper is on financial risk management that is important for the system operator faces when providing and using ancillary services for balancing of power system. Risk on ancillary services portfolio is modeled through value at risk and conditional value at risk measures. The application of these risk measures in power system is given in detail to show how to using the risk concept in practice. Conditional value at risk optimization is analysed in the context of portfolio selection and how to apply this optimization for hedging a portfolio consisting of different types of ancillary services.
Fair valuation of participating life insurance contracts with jump riskAlex Kouam
A C++ based program which prices the fair value of a participating life insurance whereby the underlying follows a Kou process and the insurer's default occurs only at contract's maturity.
:In this paper, we consider the equity premium puzzle under a general utility function. We derive that
the optimal strategy under a general utility function approximate the optimal strategy under the special utility
function. This result posed in the present paper can be regarded as a generalization of the work by Gong and
Zou [13]
Optimal Portfolio Selection for a Defined Contribution Pension Fund with Retu...BRNSS Publication Hub
This paper investigates the optimal investment strategies for a defined contribution pension fund with return clauses of premiums with interest under the mean-variance criterion. Using the actuarial symbol, we formalize the problem as a continuous time mean-variance stochastic optimal control. The pension fund manager considers investments in risk and risk-free assets to increase the remaining accumulated funds to meet the retirement needs of the remaining members. Using the variational inequalities methods, we established an optimized problem from the extended Hamilton–Jacobi–Bellman Equations and solved the optimized problem to obtain the optimal investment strategies for both risk-free and risky assets and also the efficient frontier of the pension member. Furthermore, we evaluated analytically and numerically the effect of various parameters of the optimal investment strategies on it. We observed that the optimal investment strategy for the risky asset decreases with an increase in the risk-averse level, initial wealth, and the predetermined interest rate.
Measuring the behavioral component of financial fluctuation: an analysis bas...SYRTO Project
Measuring the behavioral component of financial fluctuation: an analysis based on the S&P500 - Caporin M., Corazzini L., Costola M. June, 27 2013. IFABS 2013 - Posters session.
Abstract. Regulations of the market require disclosure of information about the nature and extent of risks arising from the trades of the market instruments. There are several significant drawbacks in fixed income pricing modeling. In this paper we interpret a corporate bond price as a random variable. In this case the spot price does not a complete characteristic of the price. The price should be specified by the spot price as well as its value of market risk. This interpretation is similar to a random variable in Probability Theory where an estimate of the random variable completely defined by its cumulative distribution function. The buyer market risk is the value of the chance that the spot price is higher than it is implied by the market scenarios. First we quantify credit risk of the corporate bonds and then consider marked-to-market pricing adjustment to bond price. In the case when issuer of the corporate bond is the counterparty of the bond buyer counterparty and credit risks are coincide.
Bid and Ask Prices Tailored to Traders' Risk Aversion and Gain Propension: a ...Waqas Tariq
Risky asset bid and ask prices “tailored” to the risk-aversion and the gain-propension of the traders are set up. They are calculated through the principle of the Extended Gini premium, a standard method used in non-life insurance. Explicit formulae for the most common stochastic distributions of risky returns, are calculated. Sufficient and necessary conditions for successful trading are also discussed.
These slides introduce the lifecontingencies R package functionalities. Pricing, reserving and simulating life contingent insurance will be shown. Similarly, joining Lee Carter mortality projections with demography R package and annuities evaluation with lifecontingencies R package is shown. The work has been all done with R markdown.
I conti economici trimestrali: avanzamenti metodologici e prospettive di innovazione
Seminario
Roma, 21 aprile 2016
Istat, Aula Magna
Via Cesare Balbo, 14
PROBLEMS IN SELECTION OF SECURITY PORTFOLIOS THE PERFORMANDaliaCulbertson719
PROBLEMS IN SELECTION OF SECURITY PORTFOLIOS
THE PERFORMANCE OF MUTUAL FUNDS IN THE PERIOD 1945-1964
MICHAEL C. JENSEN*
I. INTRODUCTION
A CENTRAL PROBLEM IN FINANCE (and especially portfolio management) has
been that of evaluating the "performance" of portfolios of risky investments.
The concept of portfolio "performance" has at least two distinct dimensions:
1) The ability of the portfolio manager or security analyst to increase re-
turns on the portfolio through successful prediction of future security
prices, and
2) The ability of the portfolio manager to minimize (through "efficient"
diversification) the amount of "insurable risk" born by the holders of
the portfolio.
The major difficulty encountered in attempting to evaluate the performance
of a portfolio in these two dimensions has been the lack of a thorough under-
standing of the nature and measurement of "risk." Evidence seems to indicate
a predominance of risk aversion in the capital markets, and as long as in-
vestors correctly perceive the "riskiness" of various assets this implies that
"risky" assets must on average yield higher returns than less "risky" assets.'
Hence in evaluating the "performance" of portfolios the effects of differential
degrees of risk on the returns of those portfolios must be taken into account.
Recent developments in the theory of the pricing of capital assets by
Sharpe [20], Lintner [15] and Treynor [25] allow us to formulate explicit
measures of a portfolio's performance in each of the dimensions outlined
above. These measures are derived and discussed in detail in Jensen [11].
However, we shall confine our attention here only to the problem of evaluating
a portfolio manager's predictive ability-that is his ability to earn returns
through successful prediction of security prices which are higher than those
which we could expect given the level of riskiness of his portfolio. The founda-
tions of the model and the properties of the performance measure suggested
here (which is somewhat different than that proposed in [11]) are discussed
in Section II. The model is illustrated in Section III by an application of it
to the evaluation of the performance of 115 open end mutual funds in the
period 1945-1964.
A number of people in the past have attempted to evaluate the performance
of portfolios2 (primarily mutual funds), but almost all of these authors have
* University of Rochester College of Business. This paper has benefited from comments and
criticisms by G. Benston, E. Fama, J. Keilson, H. Weingartner, and especially M. Scholes.
1. Assuming, of course, that investors' expectations are on average correct.
2. See for example [2, 3, 7, 8, 9, 10, 21, 24].
389
390 The Journal of Finance
relied heavily on relative measures of performance when what we really need
is an absolute measure of performance. That is, they have relied mainly on
procedures for ranking portfolios. For example, if there are two por ...
There are several significant drawbacks in derivative price modeling which relate to global regulations of the derivatives market. Here we present a unified approach which in stochastic market interprets option price as a random variable. Therefore spot price does not complete characteristic of the price in stochastic environment. Complete derivatives price includes the spot price as well as thevalue of market risk implied by the use of the spot price. This interpretation is similar to the notion of therandom variable in Probability Theory in which an estimate of the random variable completely defined by its cumulative distribution function
Many countries have seen the importance of financial education by making financial
education a national strategy. In Vietnam, although the National Strategies for Inclusive Financial
Education has been proposed since 2017 and officially included in the National Financial Inclusion
Strategy in 2020, however, financial education is still quite new, and many people are not aware of
the necessity of financial l
Today, in the rapidly emerging globalization process, increasing the competitiveness of enterprises
depends on increasing of their firm performance. Although there are many methods and techniques affecting
firm performance, Information technology (IT) capabilities has become one of the most widely used method,
especially in dealing with supply chain matters of a firm. The aim of our study is to express whether innovation
and organization learning is effective as intermediate variable to the effects of IT capabilities at firm’s
performance. The opinion which claim
More Related Content
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Measuring the behavioral component of financial fluctuation: an analysis bas...SYRTO Project
Measuring the behavioral component of financial fluctuation: an analysis based on the S&P500 - Caporin M., Corazzini L., Costola M. June, 27 2013. IFABS 2013 - Posters session.
Abstract. Regulations of the market require disclosure of information about the nature and extent of risks arising from the trades of the market instruments. There are several significant drawbacks in fixed income pricing modeling. In this paper we interpret a corporate bond price as a random variable. In this case the spot price does not a complete characteristic of the price. The price should be specified by the spot price as well as its value of market risk. This interpretation is similar to a random variable in Probability Theory where an estimate of the random variable completely defined by its cumulative distribution function. The buyer market risk is the value of the chance that the spot price is higher than it is implied by the market scenarios. First we quantify credit risk of the corporate bonds and then consider marked-to-market pricing adjustment to bond price. In the case when issuer of the corporate bond is the counterparty of the bond buyer counterparty and credit risks are coincide.
Bid and Ask Prices Tailored to Traders' Risk Aversion and Gain Propension: a ...Waqas Tariq
Risky asset bid and ask prices “tailored” to the risk-aversion and the gain-propension of the traders are set up. They are calculated through the principle of the Extended Gini premium, a standard method used in non-life insurance. Explicit formulae for the most common stochastic distributions of risky returns, are calculated. Sufficient and necessary conditions for successful trading are also discussed.
These slides introduce the lifecontingencies R package functionalities. Pricing, reserving and simulating life contingent insurance will be shown. Similarly, joining Lee Carter mortality projections with demography R package and annuities evaluation with lifecontingencies R package is shown. The work has been all done with R markdown.
I conti economici trimestrali: avanzamenti metodologici e prospettive di innovazione
Seminario
Roma, 21 aprile 2016
Istat, Aula Magna
Via Cesare Balbo, 14
PROBLEMS IN SELECTION OF SECURITY PORTFOLIOS THE PERFORMANDaliaCulbertson719
PROBLEMS IN SELECTION OF SECURITY PORTFOLIOS
THE PERFORMANCE OF MUTUAL FUNDS IN THE PERIOD 1945-1964
MICHAEL C. JENSEN*
I. INTRODUCTION
A CENTRAL PROBLEM IN FINANCE (and especially portfolio management) has
been that of evaluating the "performance" of portfolios of risky investments.
The concept of portfolio "performance" has at least two distinct dimensions:
1) The ability of the portfolio manager or security analyst to increase re-
turns on the portfolio through successful prediction of future security
prices, and
2) The ability of the portfolio manager to minimize (through "efficient"
diversification) the amount of "insurable risk" born by the holders of
the portfolio.
The major difficulty encountered in attempting to evaluate the performance
of a portfolio in these two dimensions has been the lack of a thorough under-
standing of the nature and measurement of "risk." Evidence seems to indicate
a predominance of risk aversion in the capital markets, and as long as in-
vestors correctly perceive the "riskiness" of various assets this implies that
"risky" assets must on average yield higher returns than less "risky" assets.'
Hence in evaluating the "performance" of portfolios the effects of differential
degrees of risk on the returns of those portfolios must be taken into account.
Recent developments in the theory of the pricing of capital assets by
Sharpe [20], Lintner [15] and Treynor [25] allow us to formulate explicit
measures of a portfolio's performance in each of the dimensions outlined
above. These measures are derived and discussed in detail in Jensen [11].
However, we shall confine our attention here only to the problem of evaluating
a portfolio manager's predictive ability-that is his ability to earn returns
through successful prediction of security prices which are higher than those
which we could expect given the level of riskiness of his portfolio. The founda-
tions of the model and the properties of the performance measure suggested
here (which is somewhat different than that proposed in [11]) are discussed
in Section II. The model is illustrated in Section III by an application of it
to the evaluation of the performance of 115 open end mutual funds in the
period 1945-1964.
A number of people in the past have attempted to evaluate the performance
of portfolios2 (primarily mutual funds), but almost all of these authors have
* University of Rochester College of Business. This paper has benefited from comments and
criticisms by G. Benston, E. Fama, J. Keilson, H. Weingartner, and especially M. Scholes.
1. Assuming, of course, that investors' expectations are on average correct.
2. See for example [2, 3, 7, 8, 9, 10, 21, 24].
389
390 The Journal of Finance
relied heavily on relative measures of performance when what we really need
is an absolute measure of performance. That is, they have relied mainly on
procedures for ranking portfolios. For example, if there are two por ...
There are several significant drawbacks in derivative price modeling which relate to global regulations of the derivatives market. Here we present a unified approach which in stochastic market interprets option price as a random variable. Therefore spot price does not complete characteristic of the price in stochastic environment. Complete derivatives price includes the spot price as well as thevalue of market risk implied by the use of the spot price. This interpretation is similar to the notion of therandom variable in Probability Theory in which an estimate of the random variable completely defined by its cumulative distribution function
Similar to Robust Optimal Reinsurance and Investment Problem with p-Thinning Dependent and Default risks (20)
Many countries have seen the importance of financial education by making financial
education a national strategy. In Vietnam, although the National Strategies for Inclusive Financial
Education has been proposed since 2017 and officially included in the National Financial Inclusion
Strategy in 2020, however, financial education is still quite new, and many people are not aware of
the necessity of financial l
Today, in the rapidly emerging globalization process, increasing the competitiveness of enterprises
depends on increasing of their firm performance. Although there are many methods and techniques affecting
firm performance, Information technology (IT) capabilities has become one of the most widely used method,
especially in dealing with supply chain matters of a firm. The aim of our study is to express whether innovation
and organization learning is effective as intermediate variable to the effects of IT capabilities at firm’s
performance. The opinion which claim
Globally, the number of startup companies has rapidly expanded during the last 5-8 years. Offering
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Robust Optimal Reinsurance and Investment Problem with p-Thinning Dependent and Default risks
1. International Journal of Business Marketing and Management (IJBMM)
Volume 8 Issue 2 Mar-Apr 2023, P.P. 91-101
ISSN: 2456-4559
www.ijbmm.com
International Journal of Business Marketing and Management (IJBMM) Page 91
Robust Optimal Reinsurance and Investment Problem
with p-Thinning Dependent and Default risks
Yingqi Liang1
, Peibiao Zhao1
1
School of Mathematics and Statistics,Nanjing University of Science and Technology, Nanjing Jiangsu
210094,China
843186823@qq.com
Abstract: In this paper, we consider an AAI with two types of insurance business with p-thinning dependent
claims risk, diversify claims risk by purchasing proportional reinsurance, and invest in a stock with Heston
model price process, a risk-free bond, and a credit bond in the financial market with the objective of maximizing
the expectation of the terminal wealth index effect, and construct the wealth process of AAI as well as the the
model of robust optimal reinsurance-investment problem is obtained, using dynamic programming, the HJB
equation to obtain the pre-default and post-default reinsurance-investment strategies and the display expression
of the value function, respectively, and the sensitivity of the model parameters is analyzed through numerical
experiments to obtain a realistic economic interpretation. The model as well as the results in this paper are a
generalization and extension of the results of existing studies.
Key words: p-Thinning dependent risk, defaultable bonds, dynamic programming, HJB equation, robust
optimization.
I. Introduction
The sound operation and solid claiming ability of insurance companies are important to maintain social
stability and promote social development, and the purchase of reinsurance can well control the risk of insurance
companies. The current research on reinsurance in the actuarial field of insurance mainly focuses on the insurer's
objective function, investment type, claim process, and price process of assets. [1]-[3]studied the most available
reinsurance investment problems under different utility function. [4]-[7]investigate the optimal reinsurance
investment problem under different risky asset price process.
Most of the current literature considers only single-risk claims, while in reality the risks are usually
correlated with each other. The current literature on claim dependency risk is divided into two main types of
dependency, one is co-impact and the other is p-thinning dependency. P-thinning dependency means that when
one type of claim occurs, there is a certain probability p that another type of claim will occur, for example, a car
accident or fire will not only cause property damage, but also loss of life if the situation is serious. The problem
under diffusion approximation model, jump diffusion risk model is studied in[8][9],[10] studied the type of
dependent risk for common shocks, and [11] considered the risk process under p-thinning dependence. Most of
the current studies consider only two assets for considering investments in credit bonds. [12]-[14] take into
account defaulted bonds among the types of investments for insurers. Since models with diffusion and jump
terms usually introduce uncertainty in the model for insurers, insurers usually want to seek a more robust model.
[15]-[17] study the AAI most reinsurance-investment problem.
Based on this, this paper studies the optimal proportional reinsurance-investment problem of AAI under the
Heston model by considering both sparse dependence and claims risk based on [11][14][17]. The paper is
organized as follows: a robust optimal reinsurance-investment problem model under p-thinning dependence and
default risk is developed in Section 2. The value function is divided into pre-default and post-default in Section
3, and the optimal reinsurance-investment strategies are solved for pre-default and post-default using dynamic
programming, optimal control theory, and the HJB equation, respectively. Parameter sensitivities are analyzed,
and economic explanations are given in Section 4. Section 5 concludes the paper.
II. Model Formulation
Suppose *Ω, F, *𝐹𝑡+t∈,0,T -, P+ is a complete probability space, the positive number T denotes the final
value moment, [0, T ] is a fixed time interval, 𝐹𝑡 denotes the information in the market up to time t, and 𝔽 ∶=
*𝐹𝑡+t∈,0,T -denotes the standard Brownian motion 𝑊0(𝑡), 𝑊1(𝑡), 𝑊2(𝑡), 𝑊3(𝑡). Poisson process N(t), and the
right-continuous P-complete information flow generated by the sequence of random variables *𝑋𝑖, 𝑖 ≥ 1+,
*𝑌𝑖, 𝑖 ≥ 1+. ℍ ∶= (ℋ𝑡)t⩾0 is the information flow generated by the violation process H(t), let 𝔾: = (𝒢𝑡)t⩾0, be
the information flow generated by 𝔽, ℍ the expanded information flow, i.e., 𝔾: = ℱ𝑡 ∨ ℋ𝑡. By definition, each
𝔽 -harness is also the 𝔾 -harness. The probability measure P is a realistic probability measure and Q is a risk-
neutral measure. In addition, it is assumed that all transactions in the financial market are continuous, and no
2. Robust Optimal Reinsurance and Investment Problem with p-Thinning Dependent and Default ris
International Journal of Business Marketing and Management (IJBMM) Page 92
taxes do not incur transaction costs and all property is infinitely divisible.
1.1 Surplus Process
Assuming that the insurer operates two different lines of business and that there is a sparse dependency between
these two lines of business, the surplus process is as follows:
R(t) = 𝑥0 + 𝑐𝑡 − (∑ 𝑋𝑖
𝑁(𝑡)
𝑖=1
+ ∑ 𝑌𝑖
𝑁𝑝(𝑡)
𝑖=1
) . #(1)
where{𝑋𝑖,𝑖 ≥ 1} is independently and identically distributed in 𝐹𝑋(∙),𝐸(𝑋) = 𝜇𝑋 > 0,𝐸(𝑋2
) = 𝜍𝑋
2
, as the
claim amount of the first class of business,{𝑌𝑖,𝑖 ≥ 1} is independently and identically distributed in𝐹𝑌(∙
),𝐸(𝑌) = 𝜇𝑌 > 0,𝐸(𝑌2
) = 𝜍𝑌
2
, as the claim amount of the second class of business. the claim amount of the
second type of business. N(t)denotes the conforming Poisson process with parameter c denotes the premium of
the insurance company by the expected value premium there are c = (1 + θ1)λ𝜇𝑋 + (1 + θ2)λp𝜇𝑌.θ1 > 0,θ2 >
0.
1.2 Proportional Reinsurance
Assuming that the insurer diversifies the claim risk by purchasing proportional reinsurance, and let 𝑞1(𝑡),𝑞2(𝑡)
be the insurer's retention ratio, the claim after the insurer purchases reinsurance is:
𝑞1(𝑡)𝑋𝑖 , 𝑞2(𝑡)𝑌𝑖 then the reinsurance fee is δ(𝑞1(𝑡), 𝑞2(𝑡)) = (1 + η1)(1 − 𝑞1(𝑡))λ𝜇𝑋 + (1 + η2)(1 −
𝑞2(𝑡))λp𝜇𝑌.According to Grandell(1991)[8]
,the claims process can be diffusely approximated as:
d ∑ Xi
N(t)
i=1
= λE(Xi)dt − γ1dWX(t), γ1 = √λE(XI
2
)
d ∑ Yi
Np(t)
i=1
= λE(Yi)dt − γ2dWY(t), γ2 = √λE(YI
2
)
The correlation coefficient of 𝑊𝑋(𝑡)𝑊𝑌(𝑡 )is ρ
̂ =
λp
𝛾1𝛾2
E(𝑋𝑖)E(𝑌𝑖).Then the wealth process of the insurer after
joining the reinsurance is:
dXq1,q2 = ,λμX(θ1 − η1 + η1𝑞1(𝑡)) + λpμY(θ2 − η2
+η2𝑞2(𝑡))𝑑𝑡 + √(𝑞1𝛾1)2 + (𝑞2𝛾2)2 + 2ρ
̂𝑞1𝑞2𝛾1𝛾2dW0
1.3 Financial Market
Suppose the financial market consists of three assets: risk-free assets, stocks, and corporate bonds, and the price
processes of the three assets are as follows: The price process of risk-free bonds is given by:𝑑R(t) =
rR(t)dt.The stock price process S(t) obeys the Heston stochastic volatility model:
{
𝑑𝑆(𝑡) = 𝑆(𝑡),𝑟 + 𝛼𝐿(𝑡)𝑑𝑡 + √𝐿(𝑡)𝑑𝑊1(𝑡)-, 𝑆(0) = 𝑠0
𝑑𝐿(𝑡) = 𝑘(𝜔 − 𝐿(𝑡))𝑑𝑡 + 𝜍√𝐿(𝑡)𝑑𝑊2(𝑡), 𝐿(0) = 𝑙0
R is the risk-free rate,α, 𝑘, ς, are positive constant.𝐸,𝑊1𝑊2- = ρt, 2𝑘𝜔 ≥ ς2
Following Bielecki (2007)[18]
, the credit bond price process p(t, T1), under a realistic measure P, using an
approximate model to portray default risk is as follows:
dp(t, T1) = p(t−, T1),(r + (1 − H(t))δ(1 − Δ)dt
−(1 − H(t−))ζdMp
(t)-
Where Mp
(t) = H(t) − hQ
∫ Δ(1 − H(u))du
t
0
is aℊ − harness, δ = 𝑄
ζ is the credit spread.
1.4 Robust optimization problem
Assuming the insurer adopts a reinsurance investment strategyε(t) = (𝑞1(𝑡), 𝑞2(𝑡), π(t), β(𝑡)),𝑞1(𝑡), 𝑞2(𝑡)are
the reinsurance strategies adopted by the insurer at moment t in the first and second asset classes,
respectively.π(t) for the insurance company's investment in equities at time t.,β(𝑡)for the insurance company′s
investment in credit bonds at time t. Let 𝔸 denotes the set of all feasible strategies, then the dynamic process of
wealth of the insurance company at this moment 𝑋𝜀
(t) is:
d𝑋𝜀(𝑡) = 𝜋(𝑡)
𝑑𝑆(𝑡)
𝑆(𝑡)
+ β(𝑡)
𝑑𝑝(𝑡)
𝑝(𝑡)
+ (𝑋𝜀(𝑡) − 𝜋(𝑡) − β(𝑡))
𝑑𝐵(𝑡)
𝐵(𝑡)
3. Robust Optimal Reinsurance and Investment Problem with p-Thinning Dependent and Default ris
International Journal of Business Marketing and Management (IJBMM) Page 93
+𝑑𝑋𝑚(𝑡),Xε(0) = x0
Assume that the insurer maximizes the terminal T moment expected utility in the financial market, the portfolio
index utility, which takes the form of:
V(X(T)) = −
1
𝛾
𝑒−𝛾𝑋(𝑇)
Where 𝛾 > 0 is the ambiguity aversion coefficient of the insurer. The insurer's goal is to find the optimal
reinsurance-investment strategy𝜀∗
(𝑡) = (𝑞1
∗
(𝑡), 𝑞2
∗
(𝑡), 𝜋∗
(𝑡), β∗
(𝑡))to maximizing the expectation of end-use
wealth utility for insurers. The objective function of the insurer is:
Vε
(t, x, l, h) = E(u(Xε
(T))|Xε
(t) = x, L(t) = l)
The value function of the optimization problem is:
V(t, x, l, h) = Vε
(t, x, l, h)
ε∈Π
sup
, V(T,x,l,h)=v(x),
Assume that the ambiguity information is described by the probability P and the reference model is measured by
the probability 𝒫Φ
which is equivalent to P:𝒫: = *𝒫Φ
|𝒫Φ
~𝑃+.Next, the optional measure set is constructed,
defining the procedure :Φ(𝑡) = (Φ0(𝑡), Φ1(𝑡), Φ2(𝑡), Φ3(𝑡))s.t.:
1. Φ(𝑡) is a ℊ(𝑡)- measurable for any t[0,T]
2. Φi(t) = Φi(t, ω), i = 0,1,2,3 andΦi(t) ≥ 0 for all (t, ω) ∈ ,0, T- × Ω.
3.∫ ||Φ(t)||2
𝑑𝑡 < ∞;
𝑇
0
Let Σ be all processes shaped as Φ,for all Φ ∈ Σ define a G-adaptation process under a real measure
*ΛΦ
(𝑡)|𝑡 ∈ ,0, T-+.From Ito differentiation we have:
𝑑ΛΦ
(𝑡) = ΛΦ
(𝑡−)(−Φ0(𝑡)dW0 − Φ1(𝑡)𝑑𝑊1 − Φ2(𝑡)𝑑𝑊2 − (1 − Φ3(𝑡))𝑑Mp
)
Where ΛΦ
(0) = 1,P-a.s. ΛΦ
(𝑡) is a (P,G)- martingale,E[ΛΦ
(𝑇)] =1, for each Φ ∈ Σ,A new optional measure
is absolutely continuous to P, defined as
𝑑𝑃Φ
𝑑𝑃
|𝒢𝑇
= ΛΦ
(𝑇).
So far, we have constructed a class of real-world probability measures 𝑃Φ
,where Φ ∈ Σ,From Gisanova's
theorem[21]
,it follows that:
𝑑Wi
𝑃Φ
(𝑡) = dWi(𝑡) + Φ𝑖(𝑡)𝑑𝑡, 𝑖 = 0,1,2
Therefore the wealth process in the 𝑃Φ
is:
d𝑋𝜀
(𝑡) = ,xr + 𝜋(𝛼𝑙 − Φ1√𝑙) + β(1 − H(t))δ(1 − Δ)
+λμX(θ1 − η1) + λpμY((θ2 − η2) + η1𝑞1(𝑡)) + η2𝑞2(𝑡))
+√(𝑞1𝛾1)2 + (𝑞2𝛾2)2 + 2ρ
̂𝑞1𝑞2𝛾1𝛾2Φ0-dt + 𝜋√𝑙𝑑W1
𝑃Φ
+β(1 − H(t−))ζdMp
+ √(𝑞1𝛾1)2 + (𝑞2𝛾2)2 + 2ρ
̂𝑞1𝑞2𝛾1𝛾2𝑑W0
𝑃Φ
(2)
We assume that the insurer determines a robust portfolio strategy such that the worst-case scenario is the best
option. The insurer penalizes any deviation from the sub-reference model with a penalty that increases with this
deviation, using relative entropy to measure the deviation between the reference measure and the optional
measure. Inspired by Maenhout[19]
Branger[20]
the problem is modified to define the value function as:
V(t, x, l. h) = 𝐸𝑡,𝑥,𝑙,ℎ
𝑃Φ
0−
1
𝛾
𝑒−𝛾𝑋(𝑇)
+ ∫ 𝐺(𝑢, 𝑋𝜀(𝑢), 𝜙(𝑢))𝑑𝑢
𝑇
𝑡
1
Φ∈Σ
𝑖𝑛𝑓
𝜀∈𝛱
𝑠𝑢𝑝
𝐸𝑡,𝑥,𝑙,ℎ
𝑃Φ
Calculated under the optional measure , the initial value of the stochastic process is 𝑋𝜀(𝑡)=x, S(t)=s,
H(t)=h,
G(u, Xε(u), ϕ(u)) =
Φ0
2
2Ψ0(t, Xε(t), ϕ(t))
+
Φ1
2
2Ψ1(t, Xε(t), ϕ(t))
+
Φ2
2
2Ψ2(t, Xε(t), ϕ(t))
+
(Φ3 ln Φ3 − Φ3 + 1)hp
(1 − h)
2Ψ3(t, Xε(t), ϕ(t))
Where Ψ0 ≥ 0,Ψ1 ≥ 0,Ψ2 ≥ 0,Ψ3 ≥ 0 is state-dependent,let Ψ0 = −
𝑣0
𝛾V(t,x,l,h)
, Ψ1 = −
𝑣1
𝛾V(t,x,l,h)
, Ψ2 =
−
𝑣2
𝛾V(t,x,l,h)
, Ψ3 = −
𝑣3
𝛾V(t,x,l,h)
𝑣𝑖,i=0,1,2,3 is the risk aversion factor, the lager Ψi. According to the dynamic planning principle, the HJB
equation is as follows:
𝒜𝜀,𝜙
𝑉 + 𝐺(𝑢, 𝑋𝜀(𝑢), 𝜙(𝑢)) = 0
Φ𝜖Σ
𝑖𝑛𝑓
𝜀𝜖∏
𝑠𝑢𝑝
#(3)
𝒜𝜀,𝜙
is the infinitesimal operator under the measure 𝑃Φ
.
4. Robust Optimal Reinsurance and Investment Problem with p-Thinning Dependent and Default ris
International Journal of Business Marketing and Management (IJBMM) Page 94
III. Robust optimal reinsurance-investment strategy solving
This section will solve the robust optimal problem constructed in the previous section. This paper
divides the value function into pre-default and post-default components according to the time of default of the
credit bond:
V(T, x, l, h) = {
V(T, x, l, 0),h = 0(before default)
V(T, x, l, 1),h = 1(after default)
By decomposing the value function into two sub-functions, denoted as the value function before the zero-
coupon bond default and the value function after the zero-coupon bond default, the two sub-HJB equations are
obtained and solved successively to obtain the reinsurance and risky asset investment strategies and value
function expressions after default, and the reinsurance, risky asset and credit bond investment strategies and
value function expressions before default.
1.5 Optimal reinsurance and investment decisions after default
When H (t)=1,τ ∧ T ≤ t ≤ T,the insurer has constituted a default at or before time t, the HJB equation
degenerates to:
𝑉𝑡 +
[𝑟𝑥 + 𝜋(𝛼𝑙 − Φ1√𝑙) + 𝜆μX(θ1 − η1) + λpμY(θ2 − η2) + 𝜆μXη1𝑞1(𝑡) + λpμYη2𝑞2(𝑡) +
√(𝑞1𝛾1)2 + (𝑞2𝛾2)2 + 2ρ
̂𝑞1𝑞2𝛾1𝛾2Φ0]𝑉
𝑥 +
1
2
(π2
l + λ(q1ςX)2
+ λp(q2ςY)2
+ 2λpμXμY)Vxx + πlςρVxl +
(k(ω − l) − Φ1𝜍𝜌√𝑙 − Φ2𝜍√1 − 𝜌2√𝑙)Vl +
1
2
ς2
lVll −
Φ0
2
2v0
𝛾 −
Φ1
2
2v1
𝛾 −
Φ2
2
2v2
𝛾 = 0 (4)
Satisfied :V(T, x, l, 1) = −
1
γ
e−γX
The solution can be assumed to be of the form:
V(t, x, y, l, 1) = −
1
γ
exp*−γ𝑒𝑟(𝑇−𝑡)
x + G(t, l)+, G(T, l) = 0#(5)
Taking each partial derivative of V:
{
𝑉𝑡 = [γrx𝑒𝑟(𝑇−𝑡)
+ Gt]V, 𝑉
𝑥 = −γ𝑒𝑟(𝑇−𝑡)
V
𝑉
𝑥𝑥 = γ2
𝑒2𝑟(𝑇−𝑡)
𝑉,𝑉𝑥𝑙 = −γ𝐺𝑙𝑉
𝑉𝑙 = 𝐺𝑙𝑉, 𝑉𝑙𝑙 = 𝐺𝑙𝑙𝑉 + 𝐺𝑙
2
𝑉
The minimum value point of Φ∗
is obtained from the first order condition as:
{
Φ0
∗
= v0er(T−t)
√(q1γ1)2 + (q2γ2)2 + 2ρ
̂q1q2γ1γ2
Φ1
∗
= v1er(T−t)
π√l, Φ2
∗
= −ςρ√lGlv2
1
γ
Bringing in the HJB equation yields:
γrx𝑒𝑟(𝑇−𝑡)
+ Gt − ,λμX(θ1 − η1) + λpμY(θ2 − η2)-
γ𝑒𝑟(𝑇−𝑡)
+ k(ω − l)𝐺𝑙 +
(𝜍𝜌√𝑙𝐺𝑙)
2
v2
2
+
1
2
𝜍2
𝑙(𝐺𝑙𝑙 + 𝐺𝑙
2
)
+ *𝑓2(𝜋, 𝑡)+
π
inf
+ {λγ𝑒𝑟(𝑇−𝑡)
𝑓1(𝑞1, 𝑞2, 𝑡)}
𝑞1,𝑞2
inf
= 0#(6)
Where
𝑓1(𝑞1, 𝑞2, 𝑡) = −,μXη1𝑞1(𝑡) + pμYη2𝑞2(𝑡)-
+
(γ + v0)𝑒𝑟(𝑇−𝑡)
2
,(𝑞1𝜍𝑋)2
+ p(𝑞2𝜍𝑌)2
+ 2pςXςY𝑞1𝑞2-
𝑓2(𝜋, 𝑡) = 𝜋𝑙𝑒𝑟(𝑇−𝑡)(ςρ𝐺𝑙(γ + v1) − 𝛼) +
1
2
(γ + v1)lπ2
𝑒2𝑟(𝑇−𝑡)
Theorem III.1 Let m=
μX(η1𝜎𝑌
2−pη2μY
2)
𝜎𝑥
2𝜎𝑌
2−𝑝μX
2μY
2 , n=
μY(η2𝜎𝑋
2−η1μX
2)
𝜎𝑥
2𝜎𝑌
2−𝑝μX
2μY
2 ,then there exist 𝑡1, 𝑡2𝑡1
̂, 𝑡2
̂ and the following values
can be obtained:
5. Robust Optimal Reinsurance and Investment Problem with p-Thinning Dependent and Default ris
International Journal of Business Marketing and Management (IJBMM) Page 95
𝑡1 = 𝑇 −
1
𝑟
ln
𝑚
γ + v0
, 𝑡2 = 𝑇 −
1
𝑟
ln
𝑛
γ + v0
𝑡1
̂ = 𝑇 −
1
𝑟
ln
μXη1
(γ + v0)p(𝜍𝑋
2 + μXμY)
𝑡2
̂ = 𝑇 −
1
𝑟
ln
μyη2
(γ + v0)(𝜍𝑌
2 + μXμY)
When m ≤ γ(n ≤ γ),let 𝑡2 = 𝑇(𝑡1 = 𝑇).When m > γ(n > γ),let 𝑡2 = 0(𝑡1 = 0)(1)If 𝑚 ≤ n,then 𝑞1
∗
≤ 𝑞2
∗
,for
all tϵ,0, T-,The reinsurance strategy corresponding to the problem is(𝑞1
∗
, 𝑞2
∗
) = {
(𝑞1
̂ , 𝑞2
̂),0 ≤ 𝑡 ≤ 𝑡2
(𝑞1
̃, 1
̃),𝑡2 ≤ 𝑡 ≤ 𝑡1
̂
(1,1),𝑡1 ≤ 𝑡 ≤ 𝑇
(2)If
n > m,then for alltϵ,0, T-,The reinsurance strategy corresponding to the problem
is(𝑞1
∗
, 𝑞2
∗
) = {
(𝑞1
̂ , 𝑞2
̂),0 ≤ 𝑡 ≤ 𝑡1
(1, 𝑞2
̃), 𝑡1 ≤ 𝑡 ≤ 𝑡2
̂
(1,1), 𝑡2
̂ ≤ 𝑡 ≤ 𝑇
. Where
𝑞1
̂ =
μX(η1𝜍𝑌
2
− pη2μY
2)
(γ + v0)𝑒𝑟(𝑇−𝑡)(𝜍𝑥
2𝜍𝑌
2 − 𝑝μX
2μY
2)
#(7)
𝑞2
̂ =
μY(η2𝜍𝑋
2
− η1μX
2)
(γ + v0)𝑒𝑟(𝑇−𝑡)(𝜍𝑥
2𝜍𝑌
2 − 𝑝μX
2μY
2)
#(8)
𝑞1
̃ =
μXη1 − pμXμY(γ + v0)𝑒𝑟(𝑇−𝑡)
𝜍𝑋
2(γ + v0)𝑒𝑟(𝑇−𝑡)p
#(9)
𝑞2
̃ =
μyη2 − μXμY(γ + v0)𝑒𝑟(𝑇−𝑡)
𝜍𝑌
2(γ + v0)𝑒𝑟(𝑇−𝑡)
#(10)
Proof:By finding the first-order partial derivatives, second-order partial derivatives and second-order mixed
partial derivatives for𝑓1, the following system of equations and the Hessian array are obtained:
{
𝜕𝑓1
𝜕𝑞1
= −μXη1 + (γ + v0)𝑒𝑟(𝑇−𝑡)
,𝜍𝑋
2
𝑞1 + pςXςY𝑞2- = 0
𝜕𝑓1
𝜕𝑞2
= −μYη2 + (γ + v0)𝑒𝑟(𝑇−𝑡)
,𝜍𝑌
2
𝑞2 + ςXςY𝑞1- = 0
(11)
|
|
𝜕2
𝑓1
𝜕𝑞1𝜕𝑞1
𝜕2
𝑓1
𝜕𝑞1𝜕𝑞2
𝜕2
𝑓1
𝜕𝑞2𝜕𝑞1
𝜕2
𝑓1
𝜕𝑞2𝜕𝑞2
|
| = 𝑒4𝑟(𝑇−𝑡)
(γ + v0)2
|
−𝜍𝑥
2
(γ + v0) −pςXςY
−pςXςY −𝑝𝜍𝑦
2
(γ + v0)
|
From the Hessian array positive definite it is known that 𝑓1(𝑞1, 𝑞2, 𝑡) is a convex function and there exist
extreme value points; solving the system of equations (11) yields (7). (8).
Obviously,𝑞1
̂ 、𝑞2
̂ are both increasing functions with respect to t, when 𝑚 ≤ n,0 ≤ 𝑡 ≤ 𝑡1 or n > m,0 ≤
𝑡 ≤ 𝑡2the solution as (7)(8),Also the values of 𝑡1 and 𝑡2 can be found. When 𝑚 ≤ n,𝑡2 ≤ 𝑡 ≤ 𝑡1
̂ ,then
(q1
∗
, q2
∗
) = (𝑞1
̃, 1),q1 ∈ ,0,1-.When 𝑛 > m,𝑡1 ≤ 𝑡 ≤ 𝑡2
̂ then (q1
∗
, q2
∗
) = (1, 𝑞1
̃), q2 ∈ ,0,1-,Separate solve
df1(t,q1,1)
dq1
= 0,
df1(t,1,q2)
dq2
= 0 , then can get 𝑞1
̃, 𝑞2
̃ as (9), (10) End.
Theorem III.2 The post-default insurer's optimal risky asset investment strategy is to
π∗
=
α − ςρ(γ + v1)𝐺1
(γ + v1)𝑒𝑟(𝑇−𝑡)
#(12)
The expression of the optimal value function is:
V(t, x, y, l, 1) = −
1
γ
exp*−γ𝑒𝑟(𝑇−𝑡)(t)x + G1(t)l + G2(t)+#(13)
𝑤𝑒𝑟𝑒 ρ ≠ ±1, 𝐺1 =
𝑙1𝑙2 − 𝑙1𝑙2𝑒−
1
2
ς2(𝑙1−𝑙2)(γ+v1)(1−ρ2)(𝑇−𝑡)
𝑙2 − 𝑙1𝑒−
1
2
ς2(𝑙1−𝑙2)(γ+v1)(1−ρ2)(𝑇−𝑡)
, #(14)
ρ = 1, 𝐺1 =
α2
2(γ + v1)(ας + k)
(1 − 𝑒−(ας+k)(𝑇−𝑡)
), #(15)
7. Robust Optimal Reinsurance and Investment Problem with p-Thinning Dependent and Default ris
International Journal of Business Marketing and Management (IJBMM) Page 97
∗ 𝑙𝑛|
𝑙1 − 𝑙2
𝑙1 − 𝑙2𝑒0.5𝜎2(𝑙1−𝑙2)(1−𝜌2)(γ+v1)(𝑇−𝑡)
|),ρ ≠ ±1
α2
𝑘𝜔
2(𝑘 + ας)(γ + v1)
(T − t) +
1
2
(
α
(k + ας)(γ + v1)
)2
𝑘𝜔(1 − 𝑒(𝑘+ας)(𝑇−𝑡)
),ρ = 1
α2
𝑘𝜔
2(𝑘 − ας)(γ + v1)
(T − t) +
1
2
(
α
(k − ας)(γ + v1)
)2
𝑘𝜔(1 − 𝑒(𝑘−ας)(𝑇−𝑡)
),ρ = −1, k ≠ ας
𝑘𝜔
,(T−t)α-2
4(γ+v1)
,ρ = −1,k = ας. End.
1.6 Optimal reinsurance and investment decisions before default
This section considers the optimal pre-default reinsurance-investment strategy and the value function expression
based on the previous section, when H(t)=0, 0 ≤ t ≤ τ ∧ T.Let the solution of the default prior value function
have the following form:
V(T, x, l, 0) = −
1
𝛾
𝑒𝑥𝑝{−𝛾𝑒𝑟(𝑇−𝑡)(𝑡)𝑥 + 𝐾(𝑡, 𝑙)}#(22)
Satisfying the boundary conditions V(T, x, l, 0)=V(x), K(T, l)=0, the HJB equation is transformed into:
𝑉𝑡 + ,𝜋(𝛼𝑙 − Φ1√𝑙) + βδ(1 − Δ) + λμX(θ1 + η1𝑞1(𝑡) − 1))
+λpμY .θ2 + η2(𝑞2(𝑡) − 1)√(𝑞1𝛾1)2 + (𝑞2𝛾2)2 + 2ρ
̂𝑞1𝑞2𝛾1𝛾2Φ01 𝑉
𝑥
+
1
2
((𝜋)2
𝑙 + λ(𝑞1𝜍𝑋)2
+ λp(𝑞2𝜍𝑌)2
+ 2λp𝜍𝑋𝜍𝑌)𝑉
𝑥𝑥
+𝜋𝑙𝜍ρ𝑉𝑥𝑙 + (k(ω − l) − Φ1𝜍𝜌√𝑙 − Φ2𝜍√1 − 𝜌2√𝑙))Vl +
1
2
ς2
lVll
+𝑉(𝑒𝛾β(t)ζ𝐻1(𝑡)+𝐾(𝑡,𝑙)−G(𝑡,𝑙)
− 1)𝑝
−
Φ0
2
2v0
𝛾 −
Φ1
2
2v1
𝛾 −
Φ2
2
2v2
𝛾 −
(Φ3 ln Φ3 − Φ3 + 1)hp
(1 − h)
2v2
𝛾 = 0
Similarly find the partial derivative of V:
{
𝑉𝑡 = [γ𝑒𝑟(𝑇−𝑡)
+ Kt’]V, 𝑉
𝑥 = −γ𝑒𝑟(𝑇−𝑡)
V
𝑉
𝑥𝑥 = γ2
𝑒2𝑟(𝑇−𝑡)
𝑉,𝑉𝑥𝑙 = −γ𝑒𝑟(𝑇−𝑡)
𝐾𝑙𝑉
𝑉𝑙 = 𝐾𝑙𝑉, 𝑉𝑙𝑙 = 𝐾𝑙𝑙𝑉 + 𝐾𝑙
2
𝑉
Bringing the above expression into the HJB equation and fixing the reinsurance-investment strategy, the
minimum point of Φ is obtained according to the first-order condition as:
{
Φ0
∗
= v0𝑒𝑟(𝑇−𝑡)
√(𝑞1𝛾1)2 + (𝑞2𝛾2)2 + 2ρ
̂𝑞1𝑞2𝛾1𝛾2
Φ1
∗
= v1𝑒𝑟(𝑇−𝑡)
𝜋√𝑙, Φ2
∗
= −𝜍𝜌√𝑙K𝑙v2
1
𝛾
Φ3
∗
= 𝑒𝑥𝑝*
v3
𝛾
.𝑒−𝛾β(t)ζ𝑒𝑟(𝑇−𝑡)+𝐺1(𝑡,𝑙)−G(𝑡,𝑙)
− 1/+
#(23)
After bringing (23) into the HJB equation, by inverting the investment strategy we can obtain:
π∗
=
α − ςρ(γ + v1)Kl
(γ + v1)𝑒𝑟(𝑇−𝑡)
, β∗
=
ln
1
∆Φ3
+ K(t, l) − G(t, l)
ζγ𝑒𝑟(𝑇−𝑡)
and obviously the expressions for the reinsurance strategy before and after the bond default are the same, so that
𝑔1(𝑡, q1, q2) = 𝑓1(𝑡, q1, q2) = −,μXη1q1 + pμYη2q2-
+
1
2
,(q1ςX)2
+ p(q2ςY)2
+ pςXςYμXμYγ2
-(γ + v0)𝑒𝑟(𝑇−𝑡)
Bringing ε∗
= (π∗
, β∗
, 𝑞1
∗
, 𝑞2
∗
) into the equation, after finishing, we get:
,−γ𝑟𝑒𝑟(𝑇−𝑡)
+ Kt’- − ,λμX(θ1 − η1)
+λpμY(θ2 − η2)-γ𝑒𝑟(𝑇−𝑡)
+ k(ω − l)Kl +
1
2
ς2
l(Kll + Kl
2
)
−λγer(T−t)
𝑔1(q1
∗
, q2
∗
, t) − g2(π∗
, t) − g3(β∗
, t) = 0#(24)
Where
8. Robust Optimal Reinsurance and Investment Problem with p-Thinning Dependent and Default ris
International Journal of Business Marketing and Management (IJBMM) Page 98
𝑔2(π∗
, t) = π∗
𝑙𝑒𝑟(𝑇−𝑡)(ςρ𝐾𝑙(γ + v1) − 𝛼) +
1
2
(γ + v1)lπ∗2
𝑒2𝑟(𝑇−𝑡) (25)
𝑔3(β∗
, t) = −
(Φ3 − 1)γhp
v3
− δβγer(T−t)
#(26)
The derivative of equation (26) with respect to β :
β∗
=
ln
1
∆Φ3
+ K(t, l) − G(t, l)
ζγ𝑒𝑟(𝑇−𝑡)
#(27)
Bringing it into Φ3
∗
get :
hp
v3
Φ3(𝑡) ln Φ3(𝑡) + hp
Φ3(𝑡) −
𝛿
𝜁
= 0. The equation has dimension one positive roots
Φ3.
Let K(t, l)=K1(t)l + K2(t),satisfy the boundary conditions K1(T)=0,K2(T) = 0,bring it to HJB equation(24),
eliminating the effect of l on the equation yields two equations:
K1
’
− k𝐾1 +
1
2
ς2
𝐾1
2
−
𝐾1 − 𝐺1
ζ
δ − (ςρ𝐾1 + α(δ − r))α(δ − r)
+
1
2
(ςρ𝐾1 + α(δ − r))
2
− (ςρ𝐾1 + α(δ − r))ςρ𝐾1 = 0#(28)
K2
’
− ,λμX(θ1 − η1) + λpμY(θ2 − η2)-γ𝑒𝑟(𝑇−𝑡)
+ λpςXςYγ2
(𝑒𝑟(𝑇−𝑡)
)2
+kω𝐾1 −
ln
1
∆Φ3
+ 𝐾2 − 𝐺2
ζ
δ + (1 −
1
∆Φ3
) hp
− f(t, 𝑞1
∗
, 𝑞2
∗) = 0#(29)
When ρ ≠ ±1,equation (28) is the first-order RICCATI equation that,from the existence uniqueness of the
solution 𝐾1 = 𝐺1.Then we solve equation(29):let I(t)=𝐾2 − 𝐺2, Satisfying the boundary condition I(T) = 0, then
using (29) and (21) we get:I’ = K2
’
− G2
’
=
δ
ζ
I +
δ
ζ
ln
1
∆Φ3
+
γ(Φ3−1)
v3
hp
,
I = (ln
1
∆Φ3
+ ∆ − 1)𝑒
−
δ
ζ
(T−t)
− ln
1
∆Φ3
− ∆ + 1.Thus :
𝐾2 = (ln
1
∆Φ3
+
γ∆(Φ3 − 1)
v3
) 𝑒
−
δ
ζ
(T−t)
− ln
1
∆Φ3
+
γ∆(Φ3 − 1)
v3
+ 𝐺2(30)
Theorem III.3 The optimal pre-default investment and bond investment strategies are:
π∗
=
α − ςρ(γ + v1)Kl
(γ + v1)𝑒𝑟(𝑇−𝑡)
, 𝛽∗
=
𝑣3ln
1
ΔΦ3
+ Δ𝛾(Φ3 − 1)(𝑒
−
𝛿
𝜁
(T−t)
− 1)
𝑣3𝜁𝛾𝑒𝑟(𝑇−𝑡)
The expression of the value function before default is:
𝑉(𝑡, 𝑥, 𝑙, 0) = −
1
𝛾
exp{−𝛾𝑒𝑟(𝑇−𝑡)
𝑥 + K1(𝑡)𝑙+K2(𝑡)+
Where K1(t) = G1(t),as the formula (14)-(17),K2(t) as the formula (30).
IV. Parameter Sensitivity analysis
In this chapter, we give several numerical examples to test the effect of model parameters on the
optimal strategy. First, to make the parameters more realistic, the model parameters for credit bonds are set as
follows, based on the estimates from Berndt (2008)[22]
and Collin-Dufrensn & Solnik (2001)[23],
which are
estimated from market data:1/Δ = 2.53, hQ
= 0.013, 𝜁 = 0.52.The claim amounts for the first and second
categories of insurance respectively meet the parameters of λ𝑋 = 1.5, λ𝑌 = 1exponential distribution . If no
special statement is made, other model parameters are assumed as follows:: t = 3, T = 10, 𝑟 = 0.06, 𝛾 =
4,η1 = 2, η2 = 3, p = 0.5, 𝜈0 = 𝜈1 = 𝜈3 = 1, ρ = 1, ς = 0.16, 𝑘 = 2, α = 1.5.
1.7 Influence of parameters on optimal reinsurance strategy
Figure.1-2 represents the variation of the optimal reinsurance strategy with the parameters. where Figure.1
shows the variation of reinsurance strategy with probability p. When p increases, the insurer will reduce the
amount of reinsurance retention for class I reinsurance and increase the amount of reinsurance retention for class
II reinsurance. figure.2 represents the effect of risk aversion coefficient 𝛾 on reinsurance strategy, when the risk
aversion coefficient is larger, the insurer will reduce the reinsurance strategy and purchase more reinsurance to
diversify Claims risk.
9. Robust Optimal Reinsurance and Investment Problem with p-Thinning Dependent and Default ris
International Journal of Business Marketing and Management (IJBMM) Page 99
Figure. 1 Figure. 2
1.8 Influence of parameters on optimal investment strategy
Where Figure.3 represents the effect of the risk-free rate r on the optimal investment strategyπ∗
, when the risk-
free rate increases, the insurer will invest less in risky assets and more assets in risk-free assets. Figure.4
represents the effect of the volatile reversion rate k on the optimal investment strategyπ∗
. When the reversion
rate increases, the insurer will increase its investment in risky assets.
Figure. 3 Figure. 4
Figure.5 shows the impact of risk premium
1
Δ
on credit bonds, when the risk premium increases, investment in
credit bonds is increased. Figure.6 indicates that when the default loss rate 𝜁 increases, insurers will invest less
in credit bond.
Figure. 5 Figure. 6
1.9 Influence of ambiguity aversion coefficient on strategy
Figure.7 to Figure.9 represent the effect of optimal reinsurance-investment strategy subject to ambiguity
aversion sparsity. It can be seen that as the ambiguity aversion coefficient increases, the insurer's uncertainty
about the model increases and therefore reduces its optimal reinsurance-investment strategy.
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
p
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
q1
q2
2 2.5 3 3.5 4 4.5 5 5.5 6
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
q1
q2
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
r
0.05
0.1
0.15
0.2
0.25
0.3
1 1.5 2 2.5 3 3.5 4 4.5 5
k
0.222
0.2225
0.223
0.2235
0.224
0.2245
0.225
0.2255
0.226
1 2 3 4 5 6 7 8
1/
0
0.5
1
1.5
2
2.5
3
3.5
4
1
, =1.5
2
, =2.53
3
, =5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.5
1
1.5
2
2.5
3
3.5
1
,1/ =1.5
2
,1/ =2.53
3
,1/ =5
10. Robust Optimal Reinsurance and Investment Problem with p-Thinning Dependent and Default ris
International Journal of Business Marketing and Management (IJBMM) Page 100
Figure. 7 Figure. 8
Figure. 9
V. Conclusion
This paper assumes that the insurer owns two types of insurance business with sparse dependence risk,
and the claim process is described by a diffusion approximation model, and secondly, the insurer expands its
investment types by investing in the financial market with a stock, a risk-free asset and a credit bond, with the
stock price described by a Heston model and the credit bond price described by an approximate model. The
credit bond price is described by the approximate model, and considering that the insurer is ambiguous averse,
the robust optimal reinsurance-investment problem is established, and the explicit expressions of the robust
optimal reinsurance-investment and optimal value function are obtained by using stochastic control theory,
dynamic programming principle, and HJB equation, and sensitivity analysis is performed on the model
parameters. Based on this paper, further discussions can be made: 1) other situations of the claim process can be
considered: such as jump diffusion or common shock. 2) time lag effects can be considered. 3) game problems
can be considered.
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