scholarly journals Analytical solution to an LQG homing problem in two dimensions

2020 ◽  
Vol 34 ◽  
pp. 01003
Author(s):  
Mario Lefebvre
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Value Function ◽  
Separation Of Variables ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Airy Functions ◽  
Dimensional Diffusion ◽  
Infinite Sum ◽  
The Value Function

An analytical solution is found to the problem of maximising the time spent in the first quadrant by the two-dimensional diffusion process (X(t), Y(t)), where Y(t) is a controlled Brownian motion and X(t) is proportional to its integral. Moreover, we force the process to exit the first quadrant through the y-axis. This type of problem is known as LQG homing and is very difficult to solve explicitly, especially in two or more dimensions. Here the partial differential equation satisfied by a transformation of the value function is solved by making use of the method of separation of variables. The exact solution is expressed as an infinite sum of Airy functions.

An analytical solution of a two-dimensional temperature field in a hollow cylinder under a time periodic boundary condition using Fourier series

2009 ◽  
Vol 223 (8) ◽  
pp. 1889-1901 ◽  
Author(s):  
G Atefi ◽  
M A Abdous ◽  
A Ganjehkaviri ◽  
N Moalemi
Keyword(s):  
Boundary Condition ◽  
Temperature Field ◽  
Fourier Series ◽  
Temperature Distribution ◽  
Analytical Solution ◽  
Hollow Cylinder ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Separation Of Variables ◽  
Two Dimensional

The objective of this article is to derive an analytical solution for a two-dimensional temperature field in a hollow cylinder, which is subjected to a periodic boundary condition at the outer surface, while the inner surface is insulated. The material is assumed to be homogeneous and isotropic with time-independent thermal properties. Because of the time-dependent term in the boundary condition, Duhamel's theorem is used to solve the problem for a periodic boundary condition. The periodic boundary condition is decomposed using the Fourier series. This condition is simulated with harmonic oscillation; however, there are some differences with the real situation. To solve this problem, first of all the boundary condition is assumed to be steady. By applying the method of separation of variables, the temperature distribution in a hollow cylinder can be obtained. Then, the boundary condition is assumed to be transient. In both these cases, the solutions are separately calculated. By using Duhamel's theorem, the temperature distribution field in a hollow cylinder is obtained. The final result is plotted with respect to the Biot and Fourier numbers. There is good agreement between the results of the proposed method and those reported by others for this geometry under a simple harmonic boundary condition.

scholarly journals Dynkin game under g-expectation in continuous time

2020 ◽  
Vol 9 (2) ◽  
pp. 459-470
Author(s):  
Helin Wu ◽  
Yong Ren ◽  
Feng Hu
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Continuous Time ◽  
Value Function ◽  
Backward Stochastic Differential Equation ◽  
Saddle Points ◽  
Value Functions ◽  
Dynkin Game ◽  
The Value Function

Abstract In this paper, we investigate some kind of Dynkin game under g-expectation induced by backward stochastic differential equation (short for BSDE). The lower and upper value functions $$\underline{V}_t=ess\sup \nolimits _{\tau \in {\mathcal {T}_t}} ess\inf \nolimits _{\sigma \in {\mathcal {T}_t}}\mathcal {E}^g_t[R(\tau ,\sigma )]$$ V ̲ t = e s s sup τ ∈ T t e s s inf σ ∈ T t E t g [ R ( τ , σ ) ] and $$\overline{V}_t=ess\inf \nolimits _{\sigma \in {\mathcal {T}_t}} ess\sup \nolimits _{\tau \in {\mathcal {T}_t}}\mathcal {E}^g_t[R(\tau ,\sigma )]$$ V ¯ t = e s s inf σ ∈ T t e s s sup τ ∈ T t E t g [ R ( τ , σ ) ] are defined, respectively. Under some suitable assumptions, a pair of saddle points is obtained and the value function of Dynkin game $$V(t)=\underline{V}_t=\overline{V}_t$$ V ( t ) = V ̲ t = V ¯ t follows. Furthermore, we also consider the constrained case of Dynkin game.

scholarly journals Similarity Solutions of Partial Differential Equations in Probability

2011 ◽  
Vol 2011 ◽  
pp. 1-13
Author(s):  
Mario Lefebvre
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Diffusion Processes ◽  
Separation Of Variables ◽  
Similarity Solutions ◽  
Two Dimensional ◽  
Partial Differential ◽  
Generalized Fourier Series ◽  
Dimensional Diffusion ◽  
Concentric Circles

Two-dimensional diffusion processes are considered between concentric circles and in angular sectors. The aim of the paper is to compute the probability that the process will hit a given part of the boundary of the stopping region first. The appropriate partial differential equations are solved explicitly by using the method of similarity solutions and the method of separation of variables. Some solutions are expressed as generalized Fourier series.

2D Analytical Model Investigating the Effect of Structural Heat Recirculation on the Temperature Profiles in a Parallel Plate Reactor With a Premixed Laminar Flame

2007 ◽  
Author(s):  
Ananthanarayanan Veeraragavan ◽  
Christopher P. Cadou
Keyword(s):  
Analytical Solution ◽  
Separation Of Variables ◽  
Outer Wall ◽  
Temperature Fields ◽  
Channel Height ◽  
Two Dimensional ◽  
Thermal Coupling ◽  
Power Sources ◽  
Wall Boundary ◽  
Heat Recirculation

A two-dimensional model for heat transfer in reacting channel flow is developed along with an analytical solution that relates the temperature field in the channel to the flow Pe number. The solution is derived from first principles by modeling the flame as a volumetric heat source and by applying “jump conditions” across the flame. The model explores the role of heat recirculation via the channel’s walls by accounting for the thermal coupling between the wall and the gas. The uniqueness of the model lies in that it is developed by simultaneously solving the two dimensional temperature fields in both the wall and structure analytically. The solution is obtained using separation of variables in the streamwise (x) and the transverse (y) direction. Thermal coupling between the wall and gas is achieved by requiring that the temperature and heat flux match at the interface. The outer wall boundary can be either adiabatic or have a convective heat loss based on Newton’s law of cooling. The resulting solution is a Fourier series (for both wall and gas temperature fields) which depends on the flow Pe and the outer wall boundary condition. This simple model and the resulting analytical solution provide an extremely computationally efficient tool for exploring the effects of varying channel height and gas velocity on the temperature distribution associated with reacting (combusting) flow a channel. Understanding these tradeoffs is important for developing miniaturized, combustion-based power sources.

ON THE CREDIT RISK OF SECURED LOANS WITH MAXIMUM LOAN-TO-VALUE COVENANTS

2014 ◽  
Vol 17 (08) ◽  
pp. 1450055
Author(s):  
Fabian Astic ◽  
Agnès Tourin
Keyword(s):  
Differential Equation ◽  
Credit Risk ◽  
Value Function ◽  
Expected Loss ◽  
Trade Off ◽  
Closed Form Solutions ◽  
Control Techniques ◽  
Lending Policy ◽  
Lending Decisions ◽  
The Value Function

We propose a framework for analyzing the credit risk of secured loans with maximum loan-to-value covenants. Here, we do not assume that the collateral can be liquidated as soon as the maximum loan-to-value is breached. Closed-form solutions for the expected loss are obtained for nonrevolving loans. In the revolving case, we introduce a minimization problem with an objective function parameterized by a risk reluctance coefficient, capturing the trade-off between minimizing the expected loss incurred in the event of liquidation and maximizing the interest gain. Using stochastic control techniques, we derive the partial integro-differential equation satisfied by the value function, and solve it numerically with a finite difference scheme. The experimental results and their comparison with a standard loan-to-value-based lending policy suggest that stricter lending decisions would benefit the lender.

Dynamic programming and feedback analysis of the two dimensional tidal dynamics system

2020 ◽  
Vol 26 ◽  
pp. 109
Author(s):  
Manil T. Mohan
Keyword(s):  
Optimal Control ◽  
Dynamic Programming ◽  
Jacobi Equation ◽  
Value Function ◽  
Two Dimensional ◽  
Tidal Dynamics ◽  
Infinite Dimensional ◽  
Bellman Principle ◽  
The Value Function ◽  
Hamilton Jacobi Equation

In this work, we consider the controlled two dimensional tidal dynamics equations in bounded domains. A distributed optimal control problem is formulated as the minimization of a suitable cost functional subject to the controlled 2D tidal dynamics equations. The existence of an optimal control is shown and the dynamic programming method for the optimal control of 2D tidal dynamics system is also described. We show that the feedback control can be obtained from the solution of an infinite dimensional Hamilton-Jacobi equation. The non-differentiability and lack of smoothness of the value function forced us to use the method of viscosity solutions to obtain a solution of the infinite dimensional Hamilton-Jacobi equation. The Bellman principle of optimality for the value function is also obtained. We show that a viscosity solution to the Hamilton-Jacobi equation can be used to derive the Pontryagin maximum principle, which give us the first order necessary conditions of optimality. Finally, we characterize the optimal control using the adjoint variable.

scholarly journals Viscosity Solution of Mean-Variance Portfolio Selection of a Jump Markov Process with No-Shorting Constraints

2016 ◽  
Vol 2016 ◽  
pp. 1-14 ◽  
Author(s):  
Moussa Kounta
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Portfolio Selection ◽  
Viscosity Solution ◽  
Value Function ◽  
Auxiliary Problem ◽  
Partial Differential ◽  
Mean Variance Portfolio ◽  
Mean Variance ◽  
The Value Function

We consider the so-called mean-variance portfolio selection problem in continuous time under the constraint that the short-selling of stocks is prohibited where all the market coefficients are random processes. In this situation the Hamilton-Jacobi-Bellman (HJB) equation of the value function of the auxiliary problem becomes a coupled system of backward stochastic partial differential equation. In fact, the value functionVoften does not have the smoothness properties needed to interpret it as a solution to the dynamic programming partial differential equation in the usual (classical) sense; however, in such casesVcan be interpreted as a viscosity solution. Here we show the unicity of the viscosity solution and we see that the optimal and the value functions are piecewise linear functions based on some Riccati differential equations. In particular we solve the open problem posed by Li and Zhou and Zhou and Yin.

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