On the Hamilton--Jacobi--Bellman Equation for an Optimal Consumption Problem: II. Verification Theorem

2012 ◽  
Vol 50 (4) ◽  
pp. 2401-2430 ◽  
Author(s):  
Hiroaki Hata ◽  
Shuenn-Jyi Sheu
Keyword(s):  
Bellman Equation ◽  
Optimal Consumption ◽  
Verification Theorem ◽  
Hamilton Jacobi Bellman Equation ◽  
Hamilton Jacobi Bellman

scholarly journals Optimal Dynamic XL Reinsurance

2003 ◽  
Vol 33 (2) ◽  
pp. 193-207 ◽  
Author(s):  
Christian Hipp ◽  
Michael Vogt
Keyword(s):  
Ruin Probability ◽  
Bellman Equation ◽  
Compound Poisson Process ◽  
Risk Process ◽  
Infinite Time ◽  
Verification Theorem ◽  
Hamilton Jacobi Bellman Equation ◽  
Optimal Dynamic ◽  
Xl Reinsurance ◽  
Hamilton Jacobi Bellman

We consider a risk process modelled as a compound Poisson process. We find the optimal dynamic unlimited excess of loss reinsurance strategy to minimize infinite time ruin probability, and prove the existence of a smooth solution of the corresponding Hamilton-Jacobi-Bellman equation as well as a verification theorem. Numerical examples with exponential, shifted exponential, and Pareto claims are given.

scholarly journals Optimal Dynamic XL Reinsurance

2003 ◽  
Vol 33 (02) ◽  
pp. 193-207 ◽  
Author(s):  
Christian Hipp ◽  
Michael Vogt
Keyword(s):  
Ruin Probability ◽  
Bellman Equation ◽  
Compound Poisson Process ◽  
Risk Process ◽  
Infinite Time ◽  
Verification Theorem ◽  
Hamilton Jacobi Bellman Equation ◽  
Optimal Dynamic ◽  
Xl Reinsurance ◽  
Hamilton Jacobi Bellman

We consider a risk process modelled as a compound Poisson process. We find the optimal dynamic unlimited excess of loss reinsurance strategy to minimize infinite time ruin probability, and prove the existence of a smooth solution of the corresponding Hamilton-Jacobi-Bellman equation as well as a verification theorem. Numerical examples with exponential, shifted exponential, and Pareto claims are given.

Interception of automated adversarial drone swarms in partially observed environments

2021 ◽  
pp. 1-14
Author(s):  
Daniel Saranovic ◽  
Martin Pavlovski ◽  
William Power ◽  
Ivan Stojkovic ◽  
Zoran Obradovic
Keyword(s):  
Bellman Equation ◽  
Pid Controllers ◽  
Partially Observed ◽  
Hamilton Jacobi Bellman Equation ◽  
Partial Feedback ◽  
Point Solution ◽  
Physical Limitations ◽  
Defensive Mechanisms ◽  
Hamilton Jacobi Bellman ◽  
New Framework

As the prevalence of drones increases, understanding and preparing for possible adversarial uses of drones and drone swarms is of paramount importance. Correspondingly, developing defensive mechanisms in which swarms can be used to protect against adversarial Unmanned Aerial Vehicles (UAVs) is a problem that requires further attention. Prior work on intercepting UAVs relies mostly on utilizing additional sensors or uses the Hamilton-Jacobi-Bellman equation, for which strong conditions need to be met to guarantee the existence of a saddle-point solution. To that end, this work proposes a novel interception method that utilizes the swarm’s onboard PID controllers for setting the drones’ states during interception. The drone’s states are constrained only by their physical limitations, and only partial feedback of the adversarial drone’s positions is assumed. The new framework is evaluated in a virtual environment under different environmental and model settings, using random simulations of more than 165,000 swarm flights. For certain environmental settings, our results indicate that the interception performance of larger swarms under partial observation is comparable to that of a one-drone swarm under full observation of the adversarial drone.

scholarly journals Optimal Investment and Consumption for Multidimensional Spread Financial Markets with Logarithmic Utility

Stats ◽  
2021 ◽  
Vol 4 (4) ◽  
pp. 1012-1026
Author(s):  
Sahar Albosaily ◽  
Serguei Pergamenchtchikov
Keyword(s):  
Financial Markets ◽  
Numerical Simulations ◽  
Financial Market ◽  
Optimal Investment ◽  
Optimal Consumption ◽  
Dynamical Programming ◽  
Verification Theorem ◽  
Logarithmic Utility ◽  
Hamilton Jacobi Bellman ◽  
Optimal Investment And Consumption

We consider a spread financial market defined by the multidimensional Ornstein–Uhlenbeck (OU) process. We study the optimal consumption/investment problem for logarithmic utility functions using a stochastic dynamical programming method. We show a special verification theorem for this case. We find the solution to the Hamilton–Jacobi–Bellman (HJB) equation in explicit form and as a consequence we construct optimal financial strategies. Moreover, we study the constructed strategies with numerical simulations.

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