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 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.

Reflected BSDEs when the obstacle is predictable and nonlinear optimal stopping problem

2021 ◽  
pp. 2150049
Author(s):  
Siham Bouhadou ◽  
Youssef Ouknine
Keyword(s):  
Optimal Stopping ◽  
Existence And Uniqueness ◽  
Value Function ◽  
Backward Stochastic Differential Equation ◽  
Stopping Time ◽  
Stopping Times ◽  
Optimal Stopping Problem ◽  
Predictable Process ◽  
Optimal Stopping Theory ◽  
Reflected Bsdes

In the first part of this paper, we study RBSDEs in the case where the filtration is non-quasi-left-continuous and the lower obstacle is given by a predictable process. We prove the existence and uniqueness by using some results of optimal stopping theory in the predictable setting, some tools from general theory of processes as the Mertens decomposition of predictable strong supermartingale. In the second part, we introduce an optimal stopping problem indexed by predictable stopping times with the nonlinear predictable [Formula: see text] expectation induced by an appropriate backward stochastic differential equation (BSDE). We establish some useful properties of [Formula: see text]-supremartingales. Moreover, we show the existence of an optimal predictable stopping time, and we characterize the predictable value function in terms of the first component of RBSDEs studied in the first part.

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.

OPTIMAL STOPPING FOR THE LAST EXIT TIME

2018 ◽  
Vol 99 (1) ◽  
pp. 148-160
Author(s):  
DAN REN
Keyword(s):  
Diffusion Process ◽  
Unique Solution ◽  
Optimal Stopping ◽  
Exit Time ◽  
Stopping Time ◽  
Random Time ◽  
Stopping Times ◽  
One Dimensional ◽  
Last Exit Time ◽  
First Time

Given a one-dimensional downwards transient diffusion process $X$, we consider a random time $\unicode[STIX]{x1D70C}$, the last exit time when $X$ exits a certain level $\ell$, and detect the optimal stopping time for it. In particular, for this random time $\unicode[STIX]{x1D70C}$, we solve the optimisation problem $\inf _{\unicode[STIX]{x1D70F}}\mathbb{E}[\unicode[STIX]{x1D706}(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70C})_{+}+(1-\unicode[STIX]{x1D706})(\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D70F})_{+}]$ over all stopping times $\unicode[STIX]{x1D70F}$. We show that the process should stop optimally when it runs below some fixed level $\unicode[STIX]{x1D705}_{\ell }$ for the first time, where $\unicode[STIX]{x1D705}_{\ell }$ is the unique solution in the interval $(0,\unicode[STIX]{x1D706}\ell )$ of an explicitly defined equation.

On wald-type optimal stopping for Brownian motion

1997 ◽  
Vol 34 (1) ◽  
pp. 66-73 ◽  
Author(s):  
S. E. Graversen ◽  
G. Peškir
Keyword(s):  
Brownian Motion ◽  
Optimal Stopping ◽  
Stopping Time ◽  
Maximal Inequality ◽  
Stopping Times ◽  
Standard Brownian Motion ◽  
Square Root ◽  
Real Analysis ◽  
Optimal Stopping Problems ◽  
Wald's Identity

The solution is presented to all optimal stopping problems of the form supτE(G(|Β τ |) – cτ), where is standard Brownian motion and the supremum is taken over all stopping times τ for B with finite expectation, while the map G : ℝ+ → ℝ satisfies for some being given and fixed. The optimal stopping time is shown to be the hitting time by the reflecting Brownian motion of the set of all (approximate) maximum points of the map . The method of proof relies upon Wald's identity for Brownian motion and simple real analysis arguments. A simple proof of the Dubins–Jacka–Schwarz–Shepp–Shiryaev (square root of two) maximal inequality for randomly stopped Brownian motion is given as an application.

Optimal stopping and maximal inequalities for geometric Brownian motion

1998 ◽  
Vol 35 (04) ◽  
pp. 856-872 ◽  
Author(s):  
S. E. Graversen ◽  
G. Peskir
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Optimal Stopping ◽  
Moving Boundary ◽  
Stopping Time ◽  
Practical Interest ◽  
Maximal Inequality ◽  
Geometric Brownian Motion ◽  
Stopping Times ◽  
Image Position

Explicit formulas are found for the payoff and the optimal stopping strategy of the optimal stopping problem supτ E (max0≤t≤τ X t − c τ), where X = (X t ) t≥0 is geometric Brownian motion with drift μ and volatility σ > 0, and the supremum is taken over all stopping times for X. The payoff is shown to be finite, if and only if μ < 0. The optimal stopping time is given by τ* = inf {t > 0 | X t = g * (max0≤t≤s X s )} where s ↦ g *(s) is the maximal solution of the (nonlinear) differential equation under the condition 0 < g(s) < s, where Δ = 1 − 2μ / σ2 and K = Δ σ2 / 2c. The estimate is established g *(s) ∼ ((Δ − 1) / K Δ)1 / Δ s 1−1/Δ as s → ∞. Applying these results we prove the following maximal inequality: where τ may be any stopping time for X. This extends the well-known identity E (sup t>0 X t ) = 1 − (σ 2 / 2 μ) and is shown to be sharp. The method of proof relies upon a smooth pasting guess (for the Stephan problem with moving boundary) and the Itô–Tanaka formula (being applied two-dimensionally). The key point and main novelty in our approach is the maximality principle for the moving boundary (the optimal stopping boundary is the maximal solution of the differential equation obtained by a smooth pasting guess). We think that this principle is by itself of theoretical and practical interest.

scholarly journals Predicting the Supremum: Optimality of ‘Stop at Once or Not at All’

2012 ◽  
Vol 49 (3) ◽  
pp. 806-820
Author(s):  
Pieter C. Allaart
Keyword(s):  
Optimal Stopping ◽  
Continuous Time ◽  
Analogous Result ◽  
Stopping Time ◽  
Stable Processes ◽  
Finite Variation ◽  
Lévy Measure ◽  
One Dimensional ◽  
Stationary Increments ◽  
Reverse Relationship

Let (Xt)0 ≤ t ≤ T be a one-dimensional stochastic process with independent and stationary increments, either in discrete or continuous time. In this paper we consider the problem of stopping the process (Xt) ‘as close as possible’ to its eventual supremum MT := sup0 ≤ t ≤ TXt, when the reward for stopping at time τ ≤ T is a nonincreasing convex function of MT - Xτ. Under fairly general conditions on the process (Xt), it is shown that the optimal stopping time τ takes a trivial form: it is either optimal to stop at time 0 or at time T. For the case of a random walk, the rule τ ≡ T is optimal if the steps of the walk stochastically dominate their opposites, and the rule τ ≡ 0 is optimal if the reverse relationship holds. An analogous result is proved for Lévy processes with finite Lévy measure. The result is then extended to some processes with nonfinite Lévy measure, including stable processes, CGMY processes, and processes whose jump component is of finite variation.

scholarly journals A General ‘Bang-Bang’ Principle for Predicting the Maximum of a Random Walk

2010 ◽  
Vol 47 (04) ◽  
pp. 1072-1083 ◽  
Author(s):  
Pieter Allaart
Keyword(s):  
Random Walk ◽  
Optimal Stopping ◽  
Stopping Time ◽  
Stopping Times ◽  
Optimal Prediction ◽  
Prediction Problem ◽  
Brownian Motion With Drift ◽  
Natural Filtration ◽  
Mathematical Justification ◽  
Bernoulli Random Walk

Let (B t )0≤t≤T be either a Bernoulli random walk or a Brownian motion with drift, and let M t := max{B s: 0 ≤ s ≤ t}, 0 ≤ t ≤ T. In this paper we solve the general optimal prediction problem sup0≤τ≤T E[f(M T − B τ], where the supremum is over all stopping times τ adapted to the natural filtration of (B t ) and f is a nonincreasing convex function. The optimal stopping time τ* is shown to be of ‘bang-bang’ type: τ* ≡ 0 if the drift of the underlying process (B t ) is negative and τ* ≡ T if the drift is positive. This result generalizes recent findings of Toit and Peskir (2009) and Yam, Yung and Zhou (2009), and provides additional mathematical justification for the dictum in finance that one should sell bad stocks immediately, but keep good stocks as long as possible.

scholarly journals The odds algorithm based on sequential updating and its performance

2009 ◽  
Vol 41 (01) ◽  
pp. 131-153 ◽  
Author(s):  
F. Thomas Bruss ◽  
Guy Louchard
Keyword(s):  
Optimal Stopping ◽  
Stopping Time ◽  
Stopping Times ◽  
Selection Problems ◽  
Real World Applications ◽  
Sequential Updating ◽  
Indicator Functions ◽  
Sequential Information ◽  
And Performance ◽  
Independent Indicator

LetI1,I2,…,Inbe independent indicator functions on some probability spaceWe suppose that these indicators can be observed sequentially. Furthermore, letTbe the set of stopping times on (Ik),k=1,…,n, adapted to the increasing filtrationwhereThe odds algorithm solves the problem of finding a stopping time τ ∈Twhich maximises the probability of stopping on the lastIk=1, if any. To apply the algorithm, we only need the odds for the events {Ik=1}, that is,rk=pk/(1-pk), whereThe goal of this paper is to offer tractable solutions for the case where thepkare unknown and must be sequentially estimated. The motivation is that this case is important for many real-world applications of optimal stopping. We study several approaches to incorporate sequential information. Our main result is a new version of the odds algorithm based on online observation and sequential updating. Questions of speed and performance of the different approaches are studied in detail, and the conclusiveness of the comparisons allows us to propose always using this algorithm to tackle selection problems of this kind.

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