A Stochastic Differential Game with Safe and Risky Choices

1988 ◽  
Vol 2 (1) ◽  
pp. 31-39
Author(s):  
J. M. McNamara
Keyword(s):  
Differential Equation ◽  
Diffusion Coefficient ◽  
Stochastic Differential Equation ◽  
Differential Game ◽  
The State ◽  
Stochastic Differential Game ◽  
Final Time ◽  
One Dimensional ◽  
Zero Sum ◽  
Drift Coefficient

This paper considers a two-person zero-sum stochastic differential game. The dynamics of the game are given by a one-dimensional stochastic differential equation whose diffusion coefficient may be controlled by the players. The drift coefficient is held constant and cannot be controlled. Player l's objective is to maximize the probability that the state at final time, T, is positive, while Player 2's objective is to maximize the probability that the state is negative.

Smooth first-passage densities for one-dimensional diffusions

1987 ◽  
Vol 24 (02) ◽  
pp. 370-377 ◽  
Author(s):  
E. J. Pauwels
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Time Scale ◽  
First Passage ◽  
One Dimensional ◽  
Drift Coefficient

The purpose of this paper is to show that smoothness conditions on the diffusion and drift coefficient of a one-dimensional stochastic differential equation imply the existence and smoothness of a first-passage density. In order to be able to prove this, we shall show that Brownian motion conditioned to first hit a point at a specified time has the same distribution as a Bessel (3)-process with changed time scale.

Nested variational inequalities and related optimal starting–stopping problems

1992 ◽  
Vol 29 (01) ◽  
pp. 104-115 ◽  
Author(s):  
M. Sun
Keyword(s):  
Differential Equation ◽  
Variational Inequalities ◽  
Stochastic Differential Equation ◽  
Differential Game ◽  
Diffusion Model ◽  
Value Function ◽  
Optimal Strategies ◽  
Stochastic Differential Game ◽  
The Value Function

This paper introduces several versions of starting-stopping problem for the diffusion model defined in terms of a stochastic differential equation. The problem could be regarded as a stochastic differential game in which the player can only decide when to start the game and when to quit the game in order to maximize his fortune. Nested variational inequalities arise in studying such a problem, with which we are able to characterize the value function and to obtain optimal strategies.

Nested variational inequalities and related optimal starting–stopping problems

1992 ◽  
Vol 29 (1) ◽  
pp. 104-115 ◽  
Author(s):  
M. Sun
Keyword(s):  
Differential Equation ◽  
Variational Inequalities ◽  
Stochastic Differential Equation ◽  
Differential Game ◽  
Diffusion Model ◽  
Value Function ◽  
Optimal Strategies ◽  
Stochastic Differential Game ◽  
The Value Function

This paper introduces several versions of starting-stopping problem for the diffusion model defined in terms of a stochastic differential equation. The problem could be regarded as a stochastic differential game in which the player can only decide when to start the game and when to quit the game in order to maximize his fortune. Nested variational inequalities arise in studying such a problem, with which we are able to characterize the value function and to obtain optimal strategies.

Smooth first-passage densities for one-dimensional diffusions

1987 ◽  
Vol 24 (2) ◽  
pp. 370-377 ◽  
Author(s):  
E. J. Pauwels
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Time Scale ◽  
First Passage ◽  
One Dimensional ◽  
Drift Coefficient

The purpose of this paper is to show that smoothness conditions on the diffusion and drift coefficient of a one-dimensional stochastic differential equation imply the existence and smoothness of a first-passage density.In order to be able to prove this, we shall show that Brownian motion conditioned to first hit a point at a specified time has the same distribution as a Bessel (3)-process with changed time scale.

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