scholarly journals Precommitted Investment Strategy versus Time-Consistent Investment Strategy for a General Risk Model with Diffusion

2014 ◽  
Vol 2014 ◽  
pp. 1-15
Author(s):  
Lidong Zhang ◽  
Ximin Rong ◽  
Ziping Du
Keyword(s):  
Risk Model ◽  
Theoretical Perspective ◽  
Optimal Strategies ◽  
Investment Strategy ◽  
Value Functions ◽  
Lagrange Method ◽  
Bellman Equations ◽  
Strategy Versus ◽  
Hamilton Jacobi Bellman ◽  
Mean Variance

We mainly study a general risk model and investigate the precommitted strategy and the time-consistent strategy under mean-variance criterion, respectively. A lagrange method is proposed to derive the precommitted investment strategy. Meanwhile from the game theoretical perspective, we find the time-consistent investment strategy by solving the extended Hamilton-Jacobi-Bellman equations. By comparing the precommitted strategy with the time-consistent strategy, we find that the company under the time-consistent strategy has to give up the better current utility in order to keep a consistent satisfaction over the whole time horizon. Furthermore, we theoretically and numerically provide the effect of the parameters on these two optimal strategies and the corresponding value functions.

scholarly journals Precommitted Investment Strategy versus Time-Consistent Investment Strategy for a Dual Risk Model

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Lidong Zhang ◽  
Ximin Rong ◽  
Ziping Du
Keyword(s):  
Risk Model ◽  
Optimal Investment ◽  
Investment Strategy ◽  
Bellman Equations ◽  
Optimal Investment Strategy ◽  
Optimization Principle ◽  
Dual Risk Model ◽  
Strategy Versus ◽  
Hamilton Jacobi Bellman ◽  
Original Time

We are concerned with optimal investment strategy for a dual risk model. We assume that the company can invest into a risk-free asset and a risky asset. Short-selling and borrowing money are allowed. Due to lack of iterated-expectation property, the Bellman Optimization Principle does not hold. Thus we investigate the precommitted strategy and time-consistent strategy, respectively. We take three steps to derive the precommitted investment strategy. Furthermore, the time-consistent investment strategy is also obtained by solving the extended Hamilton-Jacobi-Bellman equations. We compare the precommitted strategy with time-consistent strategy and find that these different strategies have different advantages: the former can make value function maximized at the original timet=0and the latter strategy is time-consistent for the whole time horizon. Finally, numerical analysis is presented for our results.

scholarly journals Equilibrium Investment Strategy for DC Pension Plan with Inflation and Stochastic Income under Heston’s SV Model

2016 ◽  
Vol 2016 ◽  
pp. 1-18 ◽  
Author(s):  
Jingyun Sun ◽  
Zhongfei Li ◽  
Yongwu Li
Keyword(s):  
Pension Plan ◽  
Investment Strategy ◽  
Investment Strategies ◽  
Value Functions ◽  
Defined Contribution ◽  
Hjb Equations ◽  
Plan Member ◽  
The Mean ◽  
Hamilton Jacobi Bellman ◽  
Mean Variance

We consider a portfolio selection problem for a defined contribution (DC) pension plan under the mean-variance criteria. We take into account the inflation risk and assume that the salary income process of the pension plan member is stochastic. Furthermore, the financial market consists of a risk-free asset, an inflation-linked bond, and a risky asset with Heston’s stochastic volatility (SV). Under the framework of game theory, we derive two extended Hamilton-Jacobi-Bellman (HJB) equations systems and give the corresponding verification theorems in both the periods of accumulation and distribution of the DC pension plan. The explicit expressions of the equilibrium investment strategies, corresponding equilibrium value functions, and the efficient frontiers are also obtained. Finally, some numerical simulations and sensitivity analysis are presented to verify our theoretical results.

STOCHASTIC DIFFERENTIAL GAMES BETWEEN TWO INSURERS WITH GENERALIZED MEAN-VARIANCE PREMIUM PRINCIPLE

2018 ◽  
Vol 48 (1) ◽  
pp. 413-434 ◽  
Author(s):  
Shumin Chen ◽  
Hailiang Yang ◽  
Yan Zeng
Keyword(s):  
Differential Game ◽  
Game Problem ◽  
Stochastic Differential Game ◽  
Value Functions ◽  
Programming Technique ◽  
Bellman Equations ◽  
Equilibrium Strategies ◽  
Generalized Mean ◽  
Hamilton Jacobi Bellman ◽  
Mean Variance

AbstractWe study a stochastic differential game problem between two insurers, who invest in a financial market and adopt reinsurance to manage their claim risks. Supposing that their reinsurance premium rates are calculated according to the generalized mean-variance principle, we consider the competition between the two insurers as a non-zero sum stochastic differential game. Using dynamic programming technique, we derive a system of coupled Hamilton–Jacobi–Bellman equations and show the existence of equilibrium strategies. For an exponential utility maximizing game and a probability maximizing game, we obtain semi-explicit solutions for the equilibrium strategies and the equilibrium value functions, respectively. Finally, we provide some detailed comparative-static analyses on the equilibrium strategies and illustrate some economic insights.

scholarly journals Dynamic Proportional Reinsurance and Approximations for Ruin Probabilities in the Two-Dimensional Compound Poisson Risk Model

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Yan Li ◽  
Guoxin Liu
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Optimal Strategies ◽  
Proportional Reinsurance ◽  
Two Dimensional ◽  
Compound Poisson ◽  
Compound Poisson Risk Model ◽  
Optimal Value ◽  
Hamilton Jacobi Bellman ◽  
Poisson Risk Model

We consider the dynamic proportional reinsurance in a two-dimensional compound Poisson risk model. The optimization in the sense of minimizing the ruin probability which is defined by the sum of subportfolio is being ruined. Via the Hamilton-Jacobi-Bellman approach we find a candidate for the optimal value function and prove the verification theorem. In addition, we obtain the Lundberg bounds and the Cramér-Lundberg approximation for the ruin probability and show that as the capital tends to infinity, the optimal strategies converge to the asymptotically optimal constant strategies. The asymptotic value can be found by maximizing the adjustment coefficient.

On the structure of multifactor optimal portfolio strategies

2018 ◽  
Vol 24 (3) ◽  
pp. 1043-1058
Author(s):  
Nikolai Dokuchaev
Keyword(s):  
Mutual Funds ◽  
Portfolio Selection ◽  
Optimal Strategies ◽  
Optimal Portfolio ◽  
Performance Criteria ◽  
Value Functions ◽  
Portfolio Strategies ◽  
Optimal Portfolio Selection ◽  
Admissible Strategies ◽  
Hamilton Jacobi Bellman

The paper studies problem of optimal portfolio selection. It is shown that, under some mild conditions, near optimal strategies for investors with different performance criteria can be constructed using a limited number of fixed processes (mutual funds), for a market with a larger number of available risky stocks. This implies dimension reduction for the optimal portfolio selection problem: all rational investors may achieve optimality using the same mutual funds plus a saving account. This result is obtained under mild restrictions for the utility functions without any assumptions on regularity of the value function. The proof is based on the method of dynamic programming applied indirectly to some convenient approximations of the original problem that ensure certain regularity of the value functions. To overcome technical difficulties, we use special time dependent and random constraints for admissible strategies such that the corresponding HJB (Hamilton–Jacobi–Bellman) equation admits “almost explicit” solutions generating near optimal admissible strategies featuring sufficient regularity and integrability.

scholarly journals Optimal Execution considering Trading Signal and Execution Risk Simultaneously

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Yuan Cheng ◽  
Lan Wu
Keyword(s):  
Kernel Function ◽  
Price Impact ◽  
Value Functions ◽  
Feedback Form ◽  
Bellman Equations ◽  
Impact Function ◽  
Optimal Execution ◽  
Dirac Function ◽  
Hamilton Jacobi Bellman ◽  
Optimal Execution Problem

In this paper, we study the optimal execution problem by considering the trading signal and the transaction risk simultaneously. We propose an optimal execution problem by taking into account the trading signal and the execution risk with the associated decay kernel function and the transient price impact function being of generalized forms. In particular, we solve the stochastic optimal control problems under the assumptions that the decay kernel function is the Dirac function and the transient price function is a linear function. We give the optimal executing strategies in state-feedback form and the Hamilton‐Jacobi‐Bellman equations that the corresponding value functions satisfy in the cases of a constant execution risk and a linear execution risk. We also demonstrate that our results can recover previous results when the process of the trading signal degenerates.

Optimal investment-reinsurance problems with common shock dependent risks under two kinds of premium principles

2019 ◽  
Vol 53 (1) ◽  
pp. 179-206
Author(s):  
Junna Bi ◽  
Kailing Chen
Keyword(s):  
Risk Model ◽  
Optimal Strategies ◽  
Optimal Investment ◽  
Investment Strategy ◽  
Model Parameters ◽  
Compound Poisson ◽  
Compound Poisson Risk Model ◽  
Common Shock ◽  
The Impact ◽  
Poisson Risk Model

This paper considers the optimal investment-reinsurance strategy in a risk model with two dependent classes of insurance business under two kinds of premium principles, where the two claim number processes are correlated through a common shock component. Under the criterion of maximizing the expected exponential utility with the expected value premium principle and the variance premium principle, we use the stochastic optimal control theory to derive the optimal strategy and the value function for the compound Poisson risk model as well as for the Brownian motion diffusion risk model. In particular, we find that the optimal investment strategy on the risky asset is independent to the reinsurance strategy and the reinsurance strategy for the compound Poisson risk model are very different from those for the diffusion model under both two kinds of premium principles, but the investment strategies are the same in this two risk models. Finally, numerical examples are presented to show the impact of model parameters in the optimal strategies.

scholarly journals Alpha-robust mean-variance investment strategy for DC pension plan with uncertainty about jump-diffusion risk

2020 ◽  
Author(s):  
Danping Li ◽  
Junna BI ◽  
Mengcong Hu
Keyword(s):  
Ambiguity Aversion ◽  
Pension Plan ◽  
Optimal Investment ◽  
Jump Diffusion ◽  
Investment Strategy ◽  
Investment Strategies ◽  
Value Functions ◽  
Defined Contribution ◽  
Optimal Investment Problem ◽  
Mean Variance

This paper considers an alpha-robust optimal investment problem for a defined contribution (DC) pension plan with uncertainty about jump and diffusion risks in a mean-variance framework. Our model allows the pension manager to have different levels of ambiguity aversion, rather than only consider the extremely ambiguity-averse attitude. Moreover, in the DC pension plan, contributions are supposed to be a predetermined amount of money as premiums and the pension funds are allowed to be invested in a financial market which consists of a risk-free asset, and a risky asset satisfying a jump-diffusion process. Notice that a part of pension members could die during the accumulation phase, and their premiums should be withdrawn. Thus, we consider the return of premiums clauses by an actuarial method and assume that the surviving members will share the difference between the return and the accumulation equally. Taking account of the pension fund size and the volatility of the accumulation, a mean-variance criterion as the investment objective for the DC plan can be formulated. By applying a game theoretic framework, the equilibrium investment strategies and the corresponding equilibrium value functions can be obtained explicitly. Economic interpretations are given in the numerical simulation, which is presented to illustrate our results.

scholarly journals Utility Indifference Option Pricing Model with a Non-Constant Risk-Aversion under Transaction Costs and Its Numerical Approximation

2021 ◽  
Vol 14 (9) ◽  
pp. 399
Author(s):  
Pedro Pólvora ◽  
Daniel Ševčovič
Keyword(s):  
Risk Aversion ◽  
Nonlinear Partial Differential Equation ◽  
Option Price ◽  
Transformation Method ◽  
Value Functions ◽  
Hjb Equations ◽  
Bellman Equations ◽  
Option Pricing Model ◽  
Utility Indifference ◽  
Hamilton Jacobi Bellman

Our goal is to analyze the system of Hamilton-Jacobi-Bellman equations arising in derivative securities pricing models. The European style of an option price is constructed as a difference of the certainty equivalents to the value functions solving the system of HJB equations. We introduce the transformation method for solving the penalized nonlinear partial differential equation. The transformed equation involves possibly non-constant the risk aversion function containing the negative ratio between the second and first derivatives of the utility function. Using comparison principles we derive useful bounds on the option price. We also propose a finite difference numerical discretization scheme with some computational examples.

WORST-CASE PORTFOLIO OPTIMIZATION IN A MARKET WITH BUBBLES

2016 ◽  
Vol 19 (02) ◽  
pp. 1650009 ◽  
Author(s):  
CHRISTOPH BELAK ◽  
SÖREN CHRISTENSEN ◽  
OLAF MENKENS
Keyword(s):  
Regime Switching ◽  
Asset Price ◽  
Optimal Strategies ◽  
Coupled System ◽  
Maximization Problem ◽  
Worst Case ◽  
Bellman Equations ◽  
Asset Price Bubbles ◽  
Price Bubbles ◽  
Hamilton Jacobi Bellman

We investigate a utility maximization problem in the presence of asset price bubbles. At random times, the investor receives warnings that a bubble has formed in the market which may lead to a crash in the risky asset. We propose a regime-switching model for the warnings and we make no assumptions about the distribution of the timing and the size of the crashes. Instead, we assume that the investor takes a worst-case perspective towards their impacts, i.e. the investor maximizes her expected utility under the worst-case crash scenario. We characterize the value function by a system of Hamilton–Jacobi–Bellman equations and derive a coupled system of ordinary differential equations for the optimal strategies. Numerical examples are provided.

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