scholarly journals Construction of the Value Function and Optimal Rules in Optimal Stopping of One-Dimensional Diffusions

2010 ◽  
Vol 42 (1) ◽  
pp. 158-182 ◽  
Author(s):  
Kurt Helmes ◽  
Richard H. Stockbridge
Keyword(s):  
Optimal Stopping ◽  
Value Function ◽  
Optimization Problems ◽  
Strong Duality ◽  
Stopping Rules ◽  
One Dimensional ◽  
Reward Function ◽  
Restricted Form ◽  
The Family ◽  
The Value Function

A new approach to the solution of optimal stopping problems for one-dimensional diffusions is developed. It arises by imbedding the stochastic problem in a linear programming problem over a space of measures. Optimizing over a smaller class of stopping rules provides a lower bound on the value of the original problem. Then the weak duality of a restricted form of the dual linear program provides an upper bound on the value. An explicit formula for the reward earned using a two-point hitting time stopping rule allows us to prove strong duality between these problems and, therefore, allows us to either optimize over these simpler stopping rules or to solve the restricted dual program. Each optimization problem is parameterized by the initial value of the diffusion and, thus, we are able to construct the value function by solving the family of optimization problems. This methodology requires little regularity of the terminal reward function. When the reward function is smooth, the optimal stopping locations are shown to satisfy the smooth pasting principle. The procedure is illustrated using two examples.

scholarly journals Construction of the Value Function and Optimal Rules in Optimal Stopping of One-Dimensional Diffusions

2010 ◽  
Vol 42 (01) ◽  
pp. 158-182 ◽  
Author(s):  
Kurt Helmes ◽  
Richard H. Stockbridge
Keyword(s):  
Optimal Stopping ◽  
Value Function ◽  
Optimization Problems ◽  
Strong Duality ◽  
Stopping Rules ◽  
One Dimensional ◽  
Reward Function ◽  
Restricted Form ◽  
The Family ◽  
The Value Function

A new approach to the solution of optimal stopping problems for one-dimensional diffusions is developed. It arises by imbedding the stochastic problem in a linear programming problem over a space of measures. Optimizing over a smaller class of stopping rules provides a lower bound on the value of the original problem. Then the weak duality of a restricted form of the dual linear program provides an upper bound on the value. An explicit formula for the reward earned using a two-point hitting time stopping rule allows us to prove strong duality between these problems and, therefore, allows us to either optimize over these simpler stopping rules or to solve the restricted dual program. Each optimization problem is parameterized by the initial value of the diffusion and, thus, we are able to construct the value function by solving the family of optimization problems. This methodology requires little regularity of the terminal reward function. When the reward function is smooth, the optimal stopping locations are shown to satisfy the smooth pasting principle. The procedure is illustrated using two examples.

scholarly journals Optimal Stopping for Processes with Independent Increments, and Applications

2009 ◽  
Vol 46 (04) ◽  
pp. 1130-1145 ◽  
Author(s):  
G. Deligiannidis ◽  
H. Le ◽  
S. Utev
Keyword(s):  
Optimal Stopping ◽  
Optimization Problems ◽  
Infinite Horizon ◽  
Unified Approach ◽  
Reward Function ◽  
Finance Industry ◽  
Independent Increments ◽  
Reward Functions ◽  
First Time ◽  
The Value Function

In this paper we present an explicit solution to the infinite-horizon optimal stopping problem for processes with stationary independent increments, where reward functions admit a certain representation in terms of the process at a random time. It is shown that it is optimal to stop at the first time the process crosses a level defined as the root of an equation obtained from the representation of the reward function. We obtain an explicit formula for the value function in terms of the infimum and supremum of the process, by making use of the Wiener–Hopf factorization. The main results are applied to several problems considered in the literature, to give a unified approach, and to new optimization problems from the finance industry.

scholarly journals Discretionary stopping of one-dimensional Itô diffusions with a staircase reward function

2006 ◽  
Vol 43 (4) ◽  
pp. 984-996 ◽  
Author(s):  
Anne Laure Bronstein ◽  
Lane P. Hughston ◽  
Martijn R. Pistorius ◽  
Mihail Zervos
Keyword(s):  
Asset Pricing ◽  
Optimal Stopping ◽  
Value Function ◽  
Stopping Time ◽  
Performance Criterion ◽  
Stopping Times ◽  
Financial Asset ◽  
One Dimensional ◽  
Reward Function ◽  
Discretionary Stopping

We consider the problem of optimally stopping a general one-dimensional Itô diffusion X. In particular, we solve the problem that aims at maximising the performance criterion Ex[exp(-∫0τr(Xs)ds)f(Xτ)] over all stopping times τ, where the reward function f can take only a finite number of values and has a ‘staircase’ form. This problem is partly motivated by applications to financial asset pricing. Our results are of an explicit analytic nature and completely characterise the optimal stopping time. Also, it turns out that the problem's value function is not C1, which is due to the fact that the reward function f is not continuous.

scholarly journals Discretionary stopping of one-dimensional Itô diffusions with a staircase reward function

2006 ◽  
Vol 43 (04) ◽  
pp. 984-996 ◽  
Author(s):  
Anne Laure Bronstein ◽  
Lane P. Hughston ◽  
Martijn R. Pistorius ◽  
Mihail Zervos
Keyword(s):  
Asset Pricing ◽  
Optimal Stopping ◽  
Value Function ◽  
Stopping Time ◽  
Performance Criterion ◽  
Stopping Times ◽  
Financial Asset ◽  
One Dimensional ◽  
Reward Function ◽  
Discretionary Stopping

We consider the problem of optimally stopping a general one-dimensional Itô diffusion X. In particular, we solve the problem that aims at maximising the performance criterion E x [exp(-∫0 τ r(X s )ds)f(X τ)] over all stopping times τ, where the reward function f can take only a finite number of values and has a ‘staircase’ form. This problem is partly motivated by applications to financial asset pricing. Our results are of an explicit analytic nature and completely characterise the optimal stopping time. Also, it turns out that the problem's value function is not C 1, which is due to the fact that the reward function f is not continuous.

scholarly journals Optimal Stopping for Processes with Independent Increments, and Applications

2009 ◽  
Vol 46 (4) ◽  
pp. 1130-1145 ◽  
Author(s):  
G. Deligiannidis ◽  
H. Le ◽  
S. Utev
Keyword(s):  
Optimal Stopping ◽  
Optimization Problems ◽  
Infinite Horizon ◽  
Unified Approach ◽  
Reward Function ◽  
Finance Industry ◽  
Independent Increments ◽  
Reward Functions ◽  
First Time ◽  
The Value Function

In this paper we present an explicit solution to the infinite-horizon optimal stopping problem for processes with stationary independent increments, where reward functions admit a certain representation in terms of the process at a random time. It is shown that it is optimal to stop at the first time the process crosses a level defined as the root of an equation obtained from the representation of the reward function. We obtain an explicit formula for the value function in terms of the infimum and supremum of the process, by making use of the Wiener–Hopf factorization. The main results are applied to several problems considered in the literature, to give a unified approach, and to new optimization problems from the finance industry.

scholarly journals The First Passage Time and the Dividend Value Function for One-Dimensional Diffusion Processes between Two Reflecting Barriers

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Chuancun Yin ◽  
Huiqing Wang
Keyword(s):  
Boundary Conditions ◽  
Value Function ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
One Dimensional ◽  
The Laplace Transform ◽  
The Value Function ◽  
Reflecting Barriers

We consider the general one-dimensional time-homogeneous regular diffusion process between two reflecting barriers. An approach based on the Itô formula with corresponding boundary conditions allows us to derive the differential equations with boundary conditions for the Laplace transform of the first passage time and the value function. As examples, the explicit solutions of them for several popular diffusions are obtained. In addition, some applications to risk theory are considered.

scholarly journals Parameter Dependent Optimal Thresholds, Indifference Levels and Inverse Optimal Stopping Problems

2014 ◽  
Vol 51 (02) ◽  
pp. 492-511
Author(s):  
Martin Klimmek
Keyword(s):  
Optimal Stopping ◽  
Infinite Horizon ◽  
Gittins Index ◽  
Dynamic Allocation ◽  
One Dimensional ◽  
Optimal Thresholds ◽  
The Family ◽  
Natural Generalisation ◽  
Optimal Stopping Problems ◽  
Parameter Dependent

Consider the classic infinite-horizon problem of stopping a one-dimensional diffusion to optimise between running and terminal rewards, and suppose that we are given a parametrised family of such problems. We provide a general theory of parameter dependence in infinite-horizon stopping problems for which threshold strategies are optimal. The crux of the approach is a supermodularity condition which guarantees that the family of problems is indexable by a set-valued map which we call the indifference map. This map is a natural generalisation of the allocation (Gittins) index, a classical quantity in the theory of dynamic allocation. Importantly, the notion of indexability leads to a framework for inverse optimal stopping problems.

scholarly journals American options in an imperfect complete market with default

2018 ◽  
Vol 64 ◽  
pp. 93-110 ◽  
Author(s):  
Roxana Dumitrescu ◽  
Marie-Claire Quenez ◽  
Agnès Sulem
Keyword(s):  
Optimal Stopping ◽  
Value Function ◽  
American Options ◽  
American Option ◽  
Stopping Times ◽  
Optimal Stopping Problem ◽  
Nonlinear Expectation ◽  
Portfolio Strategy ◽  
Reflected Bsde ◽  
The Value Function

We study pricing and hedging for American options in an imperfect market model with default, where the imperfections are taken into account via the nonlinearity of the wealth dynamics. The payoff is given by an RCLL adapted process (ξt). We define the seller's price of the American option as the minimum of the initial capitals which allow the seller to build up a superhedging portfolio. We prove that this price coincides with the value function of an optimal stopping problem with a nonlinear expectation 𝓔g (induced by a BSDE), which corresponds to the solution of a nonlinear reflected BSDE with obstacle (ξt). Moreover, we show the existence of a superhedging portfolio strategy. We then consider the buyer's price of the American option, which is defined as the supremum of the initial prices which allow the buyer to select an exercise time τ and a portfolio strategy φ so that he/she is superhedged. We show that the buyer's price is equal to the value function of an optimal stopping problem with a nonlinear expectation, and that it can be characterized via the solution of a reflected BSDE with obstacle (ξt). Under the additional assumption of left upper semicontinuity along stopping times of (ξt), we show the existence of a super-hedge (τ, φ) for the buyer.

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