scholarly journals Entry and Exit Decision Problem with Implementation Delay

2008 ◽  
Vol 45 (4) ◽  
pp. 1039-1059 ◽  
Author(s):  
Marius Costeniuc ◽  
Michaela Schnetzer ◽  
Luca Taschini
Keyword(s):  
Decision Problem ◽  
Analytic Solution ◽  
Time Lag ◽  
Probabilistic Approach ◽  
Stopping Time ◽  
Stopping Times ◽  
Maximization Problem ◽  
Entry And Exit ◽  
Brownian Motion With Drift ◽  
Implementation Delay

We study investment and disinvestment decisions in situations where there is a time lagd> 0 from the timetwhen the decision is taken to the timet+dwhen the decision is implemented. In this paper we apply the probabilistic approach to the combined entry and exit decisions under the Parisian implementation delay. In particular, we prove the independence between Parisian stopping times and a general Brownian motion with drift stopped at the stopping time. Relying on this result, we solve the constrained maximization problem, obtaining an analytic solution to the optimal ‘starting’ and ‘stopping’ levels. We compare our results with the instantaneous entry and exit situation, and show that an increase in the uncertainty of the underlying process hastens the decision to invest or disinvest, extending a result of Bar-Ilan and Strange (1996).

scholarly journals Entry and Exit Decision Problem with Implementation Delay

2008 ◽  
Vol 45 (04) ◽  
pp. 1039-1059 ◽  
Author(s):  
Marius Costeniuc ◽  
Michaela Schnetzer ◽  
Luca Taschini
Keyword(s):  
Decision Problem ◽  
Analytic Solution ◽  
Time Lag ◽  
Probabilistic Approach ◽  
Stopping Time ◽  
Stopping Times ◽  
Maximization Problem ◽  
Entry And Exit ◽  
Brownian Motion With Drift ◽  
Implementation Delay

We study investment and disinvestment decisions in situations where there is a time lagd> 0 from the timetwhen the decision is taken to the timet+dwhen the decision is implemented. In this paper we apply the probabilistic approach to the combined entry and exit decisions under the Parisian implementation delay. In particular, we prove the independence between Parisian stopping times and a general Brownian motion with drift stopped at the stopping time. Relying on this result, we solve the constrained maximization problem, obtaining an analytic solution to the optimal ‘starting’ and ‘stopping’ levels. We compare our results with the instantaneous entry and exit situation, and show that an increase in the uncertainty of the underlying process hastens the decision to invest or disinvest, extending a result of Bar-Ilan and Strange (1996).

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 Optimal Investment, Consumption and Leisure with an Option to File for Bankruptcy

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 827
Author(s):  
Byung Hwa Lim ◽  
Ho-Seok Lee
Keyword(s):  
Decision Problem ◽  
Optimal Strategies ◽  
Optimal Investment ◽  
Mathematical Finance ◽  
Stopping Times ◽  
Dynamic Programming Method ◽  
Maximization Problem ◽  
Portfolio Decisions ◽  
Discounted Utility ◽  
Utility Maximization Problem

This paper investigates the optimal personal bankruptcy decision of a debtor who participates in the labor market. This paper is based on a mathematical finance model that assumes a Black-Scholes financial market and describes a decision problem as an expected discounted utility maximization problem. Our optimization problem can be cast into a mixed optimal stopping and control problem, and has a symmetry feature with a voluntary retirement decision problem in characterizing the stopping times. To obtain value function and optimal strategies, we use dynamic programming method and transform the relevant nonlinear Bellman equation into a linear equation. Numerical illustrations from our explicit expressions for the optimal strategies reveal how an opportunity to file for bankruptcy affects debtor’s consumption, leisure, and portfolio decisions.

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

2010 ◽  
Vol 47 (4) ◽  
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 (Bt)0≤t≤T be either a Bernoulli random walk or a Brownian motion with drift, and let Mt := max{Bs: 0 ≤ s ≤ t}, 0 ≤ t ≤ T. In this paper we solve the general optimal prediction problem sup0≤τ≤TE[f(MT − Bτ], where the supremum is over all stopping times τ adapted to the natural filtration of (Bt) 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 (Bt) 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.

SKOROKHOD EMBEDDINGS IN BOUNDED TIME

2011 ◽  
Vol 11 (02n03) ◽  
pp. 215-226 ◽  
Author(s):  
STEFAN ANKIRCHNER ◽  
PHILIPP STRACK
Keyword(s):  
Brownian Motion ◽  
Time Horizon ◽  
Sufficient Conditions ◽  
Stopping Time ◽  
Fixed Time ◽  
Stopping Times ◽  
Embedding Problem ◽  
Brownian Motion With Drift ◽  
Necessary And Sufficient ◽  
Skorokhod Embedding Problem

This article deals with the Skorokhod embedding problem in bounded time for the Brownian motion with drift Xt = κt + Wt: Given a probability measure μ we aim at finding a stopping time τ such that the law of Xτ is μ, and τ is almost surely smaller than some given fixed time horizon T > 0. We provide necessary and sufficient conditions on the distribution μ for the existence of such bounded stopping times.

Entry and Exit Decision Problem with Implementation Delay

2008 ◽  
Author(s):  
Marius Costeniuc ◽  
Michaela Schnetzer ◽  
Luca Taschini
Keyword(s):  
Decision Problem ◽  
Entry And Exit ◽  
Implementation Delay

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.

An investment model with entry and exit decisions

2000 ◽  
Vol 37 (2) ◽  
pp. 547-559 ◽  
Author(s):  
J. Kate Duckworth ◽  
Mihail Zervos
Keyword(s):  
Dynamic Programming ◽  
Closed Form ◽  
Production Scheduling ◽  
Analytic Solution ◽  
Investment Model ◽  
Programming Approach ◽  
Entry And Exit ◽  
Dynamic Programming Approach ◽  
Exit Decisions ◽  
Parameter Values

We consider an investment model which generalizes a number of models that have been studied in the literature. The model involves entry and exit decisions as well as decisions relating to production scheduling. We then address the problem of its valuation from the standpoint of the dynamic programming approach. Our analysis results in a closed form analytic solution that can take qualitatively different forms depending on parameter values.

Optimal dispatching of a poisson process

1969 ◽  
Vol 6 (03) ◽  
pp. 692-699 ◽  
Author(s):  
Sheldon M. Ross
Keyword(s):  
Poisson Process ◽  
Time Lag ◽  
Stopping Time ◽  
Processing Plant ◽  
Homogeneous Poisson Process ◽  
Batch Arrivals ◽  
Optimal Dispatch ◽  
Non Homogeneous Poisson Process

Items arrive at a processing plant at a Poisson rate λ. At time T, all items are dispatched from the system. An intermediate dispatch time is to be chosen to minimize the total wait of all items. It is shown that if the dispatch time must be chosen at time 0 then T/2 not only minimizes the expected total wait but it also maximizes the probability that the total wait is less than a for every a > 0. If the intermediate dispatch time is allowed to be a (random) stopping time, then it is shown that the policy which dispatches at time t iff N(t) > λ(T – t) is optimal, where N(t) denotes the number of items present at time t. The distribution of the optimal dispatch time and the optimal expected total wait are determined. A generalization to the case of a non-homogeneous Poisson process, a time lag, and batch arrivals is given. Finally, the case where the process goes on indefinitely and any number of dispatches are allowed (at a cost K per dispatch) is considered.

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