scholarly journals A numerical algorithm for fully nonlinear HJB equations: An approach by control randomization

2014 ◽  
Vol 20 (2) ◽  
Author(s):  
Idris Kharroubi ◽  
Nicolas Langrené ◽  
Huyên Pham
Keyword(s):  
Numerical Algorithm ◽  
Detailed Comparison ◽  
Feedback Form ◽  
Hjb Equations ◽  
Alternative Scheme ◽  
Fully Nonlinear ◽  
Numerical Resolution ◽  
Partial Analysis ◽  
Hamilton Jacobi Bellman ◽  
Parametric Estimate

Abstract.We propose a probabilistic numerical algorithm to solve Backward Stochastic Differential Equations (BSDEs) with nonnegative jumps, a class of BSDEs introduced in [`Feynman–Kac representation for Hamilton–Jacobi–Bellman IPDE', Ann. Probab., to appear] for representing fully nonlinear HJB equations. This includes in particular numerical resolution for stochastic control problems with controlled volatility, possibly degenerate. Our backward scheme, based on least-squares regressions, takes advantage of high-dimensional properties of Monte Carlo methods, and also provides a parametric estimate in feedback form for the optimal control. A partial analysis of the algorithm error is presented, as well as numerical tests on the problem of option superreplication with uncertain volatilities and/or correlations, including a detailed comparison with the numerical results from the alternative scheme proposed in [J. Comput. Finance 14 (2011), 37–71].

scholarly journals Policy iteration for Hamilton–Jacobi–Bellman equations with control constraints

2021 ◽  
Author(s):  
Sudeep Kundu ◽  
Karl Kunisch
Keyword(s):  
Linear Equations ◽  
Hjb Equation ◽  
Policy Iteration ◽  
Optimal Feedback Control ◽  
Iteration Step ◽  
Control Constraints ◽  
Hjb Equations ◽  
Bellman Equations ◽  
Hamilton Jacobi Bellman ◽  
Feedback Control Theory

AbstractPolicy iteration is a widely used technique to solve the Hamilton Jacobi Bellman (HJB) equation, which arises from nonlinear optimal feedback control theory. Its convergence analysis has attracted much attention in the unconstrained case. Here we analyze the case with control constraints both for the HJB equations which arise in deterministic and in stochastic control cases. The linear equations in each iteration step are solved by an implicit upwind scheme. Numerical examples are conducted to solve the HJB equation with control constraints and comparisons are shown with the unconstrained cases.

Do Time Preferences Matter in Intertemporal Consumption and Portfolio Decisions?

2019 ◽  
Vol 19 (2) ◽  
Author(s):  
Shou Chen ◽  
Shengpeng Xiang ◽  
Hongbo He
Keyword(s):  
Absolute Risk ◽  
Hyperbolic Discounting ◽  
Time Preferences ◽  
Hjb Equations ◽  
Portfolio Decisions ◽  
Special Cases ◽  
Intertemporal Consumption ◽  
Hamilton Jacobi Bellman ◽  
Hara Utility ◽  
Exponential Discounting

Abstract We study the intertemporal consumption and portfolio rules in the model with the general hyperbolic absolute risk aversion (HARA) utility. The equivalent approximation approach is employed to obtain the Hamilton-Jacobi-Bellman (HJB) equations, and a remarkable property is shown: portfolio rules are independent of the discount function. Moreover, both the consumption and portfolio rates are non-increasing functions of wealth. Particularly illustrative cases examined in detail are the models with the most adopted discount functions, including exponential discounting and hyperbolic discounting. Explicit solutions for intertemporal decisions are found for these special cases, revealing that individual’s time preferences affect the consumption rules only. Moreover, the time-consistent consumption rate under hyperbolic discounting is larger than its counterpart under exponential discounting.

NATURAL GAS-FIRED POWER PLANTS VALUATION AND OPTIMIZATION UNDER LÉVY COPULAS AND REGIME SWITCHING

2017 ◽  
Vol 20 (01) ◽  
pp. 1750004 ◽  
Author(s):  
NEMAT SAFAROV ◽  
COLIN ATKINSON
Keyword(s):  
Natural Gas ◽  
Regime Switching ◽  
Power Plants ◽  
Control Strategies ◽  
Numerical Approach ◽  
Spot Price ◽  
Operating Characteristics ◽  
Hjb Equations ◽  
Explicit Finite Difference ◽  
Hamilton Jacobi Bellman

In this work, we analyze a stochastic control problem for the valuation of a natural gas power station while taking into account operating characteristics. Both electricity and gas spot price processes exhibit mean-reverting spikes and Markov regime-switches. The Lévy regime-switching model incorporates the effects of demand-supply fluctuations in energy markets and abrupt economic disruptions or business cycles. We make use of skewed Lévy copulas to model the dependence risk of electricity and gas jumps. The corresponding coupled Hamilton–Jacobi–Bellman (HJB) equations are solved by an explicit finite difference method. The numerical approach gives us both the value of the plant and its optimal operating strategy depending on the gas and electricity prices, current temperature of the boiler and time. The surfaces of control strategies and contract values are obtained by implementing the numerical method for a particular example.

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 Insurance-Package and Investment Problem for an Insurer

2018 ◽  
Vol 6 (1) ◽  
pp. 85-96
Author(s):  
Delei Sheng ◽  
Linfang Xing
Keyword(s):  
Value Function ◽  
Investment Strategy ◽  
Optimal Combination ◽  
Hjb Equations ◽  
Terminal Wealth ◽  
Optimal Insurance ◽  
Yield Rate ◽  
Investment Problem ◽  
Hamilton Jacobi Bellman ◽  
The Value Function

AbstractAn insurance-package is a combination being tie-in at least two different categories of insurances with different underwriting-yield-rate. In this paper, the optimal insurance-package and investment problem is investigated by maximizing the insurer’s exponential utility of terminal wealth to find the optimal combination-share and investment strategy. Using the methods of stochastic analysis and stochastic optimal control, the Hamilton-Jacobi-Bellman (HJB) equations are established, the optimal strategy and the value function are obtained in closed form. By comparing with classical results, it shows that the insurance-package can enhance the utility of terminal wealth, meanwhile, reduce the insurer’s claim risk.

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.

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