scholarly journals An optimal stopping problem for a geometric Brownian motion with poissonian jumps

2003 ◽  
Vol 38 (11-13) ◽  
pp. 1381-1390 ◽  
Author(s):  
Masamitsu Ohnishi
Keyword(s):  
Brownian Motion ◽  
Optimal Stopping ◽  
Geometric Brownian Motion ◽  
Optimal Stopping Problem

scholarly journals Stopping at the maximum of geometric Brownian motion when signals are received

2005 ◽  
Vol 42 (03) ◽  
pp. 826-838 ◽  
Author(s):  
X. Guo ◽  
J. Liu
Keyword(s):  
Brownian Motion ◽  
Optimal Stopping ◽  
Stopping Time ◽  
Geometric Brownian Motion ◽  
Stopping Rules ◽  
Optimal Stopping Problem ◽  
Signal Process ◽  
Optimal Stopping Time ◽  
A Free Boundary Problem ◽  
Random Times

Consider a geometric Brownian motion X t (ω) with drift. Suppose that there is an independent source that sends signals at random times τ 1 < τ 2 < ⋯. Upon receiving each signal, a decision has to be made as to whether to stop or to continue. Stopping at time τ will bring a reward S τ , where S t = max(max0≤u≤t X u , s) for some constant s ≥ X 0. The objective is to choose an optimal stopping time to maximize the discounted expected reward E[e−r τ i S τ i | X 0 = x, S 0 = s], where r is a discount factor. This problem can be viewed as a randomized version of the Bermudan look-back option pricing problem. In this paper, we derive explicit solutions to this optimal stopping problem, assuming that signal arrival is a Poisson process with parameter λ. Optimal stopping rules are differentiated by the frequency of the signal process. Specifically, there exists a threshold λ* such that if λ>λ*, the optimal stopping problem is solved via the standard formulation of a ‘free boundary’ problem and the optimal stopping time τ * is governed by a threshold a * such that τ * = inf{τ n : X τ n ≤a * S τ n }. If λ≤λ* then it is optimal to stop immediately a signal is received, i.e. at τ * = τ 1. Mathematically, it is intriguing that a smooth fit is critical in the former case while irrelevant in the latter.

scholarly journals Stopping at the maximum of geometric Brownian motion when signals are received

2005 ◽  
Vol 42 (3) ◽  
pp. 826-838 ◽  
Author(s):  
X. Guo ◽  
J. Liu
Keyword(s):  
Brownian Motion ◽  
Optimal Stopping ◽  
Stopping Time ◽  
Geometric Brownian Motion ◽  
Stopping Rules ◽  
Optimal Stopping Problem ◽  
Signal Process ◽  
Optimal Stopping Time ◽  
A Free Boundary Problem ◽  
Random Times

Consider a geometric Brownian motion Xt(ω) with drift. Suppose that there is an independent source that sends signals at random times τ1 < τ2 < ⋯. Upon receiving each signal, a decision has to be made as to whether to stop or to continue. Stopping at time τ will bring a reward Sτ, where St = max(max0≤u≤tXu, s) for some constant s ≥ X0. The objective is to choose an optimal stopping time to maximize the discounted expected reward E[e−rτiSτi | X0 = x, S0 = s], where r is a discount factor. This problem can be viewed as a randomized version of the Bermudan look-back option pricing problem. In this paper, we derive explicit solutions to this optimal stopping problem, assuming that signal arrival is a Poisson process with parameter λ. Optimal stopping rules are differentiated by the frequency of the signal process. Specifically, there exists a threshold λ* such that if λ>λ*, the optimal stopping problem is solved via the standard formulation of a ‘free boundary’ problem and the optimal stopping time τ* is governed by a threshold a* such that τ* = inf{τn: Xτn≤a*Sτn}. If λ≤λ* then it is optimal to stop immediately a signal is received, i.e. at τ* = τ1. Mathematically, it is intriguing that a smooth fit is critical in the former case while irrelevant in the latter.

Bounds for the American perpetual put on a stock index

2001 ◽  
Vol 38 (01) ◽  
pp. 55-66 ◽  
Author(s):  
V. Paulsen
Keyword(s):  
Brownian Motion ◽  
Optimal Stopping ◽  
Geometric Brownian Motion ◽  
Stock Index ◽  
Decomposition Technique ◽  
Early Exercise

Let us consider n stocks with dependent price processes each following a geometric Brownian motion. We want to investigate the American perpetual put on an index of those stocks. We will provide inner and outer boundaries for its early exercise region by using a decomposition technique for optimal stopping.

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 &gt; 0, and the supremum is taken over all stopping times for X. The payoff is shown to be finite, if and only if μ &lt; 0. The optimal stopping time is given by τ* = inf {t &gt; 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 &lt; g(s) &lt; 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&gt;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 Investing and Stopping

2014 ◽  
Vol 51 (4) ◽  
pp. 898-909
Author(s):  
Moritz Duembgen ◽  
L. C. G. Rogers
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Optimal Stopping ◽  
Numerical Scheme ◽  
Optimization Problem ◽  
Hedge Fund ◽  
Optimal Stopping Problem ◽  
Over Time

In this paper we solve the hedge fund manager's optimization problem in a model that allows for investors to enter and leave the fund over time depending on its performance. The manager's payoff at the end of the year will then depend not just on the terminal value of the fund level, but also on the lowest and the highest value reached over that time. We establish equivalence to an optimal stopping problem for Brownian motion; by approximating this problem with the corresponding optimal stopping problem for a random walk we are led to a simple and efficient numerical scheme to find the solution, which we then illustrate with some examples.

Optimal stopping and maximal inequalities for geometric Brownian motion

1998 ◽  
Vol 35 (4) ◽  
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 supτE (max0≤t≤τXt − c τ), where X = (Xt)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 | Xt = g* (max0≤t≤sXs)} 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 / Δs1−1/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 (supt>0Xt) = 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.

A class of singular stochastic control problems

1983 ◽  
Vol 15 (02) ◽  
pp. 225-254 ◽  
Author(s):  
Ioannis Karatzas
Keyword(s):  
Brownian Motion ◽  
Optimal Stopping ◽  
Moving Boundary ◽  
Upper And Lower Bounds ◽  
Control Problems ◽  
Explicit Solutions ◽  
Value Functions ◽  
Singular Stochastic Control ◽  
Optimal Stopping Problem ◽  
The Third

We consider the problem of tracking a Brownian motion by a process of bounded variation, in such a way as to minimize total expected cost of both ‘action' and ‘deviation from a target state 0'. The former is proportional to the amount of control exerted to date, while the latter is being measured by a function which can be viewed, for simplicity, as quadratic. We discuss the discounted, stationary and finite-horizon variants of the problem. The answer to all three questions takes the form of exerting control in asingularmanner, in order not to exit from a certain region. Explicit solutions are found for the first and second questions, while the third is reduced to an appropriate optimal stopping problem. This reduction yields properties, as well as global upper and lower bounds, for the associated moving boundary. The pertinent Abelian and ergodic relationships for the corresponding value functions are also derived.

A class of singular stochastic control problems

1983 ◽  
Vol 15 (2) ◽  
pp. 225-254 ◽  
Author(s):  
Ioannis Karatzas
Keyword(s):  
Brownian Motion ◽  
Optimal Stopping ◽  
Moving Boundary ◽  
Upper And Lower Bounds ◽  
Control Problems ◽  
Explicit Solutions ◽  
Value Functions ◽  
Singular Stochastic Control ◽  
Optimal Stopping Problem ◽  
The Third

We consider the problem of tracking a Brownian motion by a process of bounded variation, in such a way as to minimize total expected cost of both ‘action' and ‘deviation from a target state 0'. The former is proportional to the amount of control exerted to date, while the latter is being measured by a function which can be viewed, for simplicity, as quadratic. We discuss the discounted, stationary and finite-horizon variants of the problem. The answer to all three questions takes the form of exerting control in a singular manner, in order not to exit from a certain region. Explicit solutions are found for the first and second questions, while the third is reduced to an appropriate optimal stopping problem. This reduction yields properties, as well as global upper and lower bounds, for the associated moving boundary. The pertinent Abelian and ergodic relationships for the corresponding value functions are also derived.

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