153,558 research outputs found
Discounting and Patience in Optimal Stopping and Control Problems
This paper establishes that the optimal stopping time of virtually any optimal stopping problem is increasing in "patience," understood as a particular partial order on discount rate functions. With Markov dynamics, the result holds in a continuation- domain sense even if stopping is combined with an optimal control problem. Under intuitive additional assumptions, we obtain comparative statics on both the optimal control and optimal stopping time for one-dimensional diusions. We provide a simple example where, without these assumptions, increased patience can precipitate stopping. We also show that, with optimal stopping and control, a project's expected value is decreasing in the interest rate, generalizing analogous results in a deterministic context. All our results are robust to the presence of a salvage value. As an application we show that the internal rate of return of any endogenously-interrupted project is essentially unique, even if the project also involves a management problem until its interruption. We also apply our results to the theory of optimal growth and capital deepening and to optimal bankruptcy decisions.capital growth, comparative statics, discounting, internal rate of return, optimal control, optimal stopping, patience, present value, project valuation
A nonparametric algorithm for optimal stopping based on robust optimization
Optimal stopping is a fundamental class of stochastic dynamic optimization
problems with numerous applications in finance and operations management. We
introduce a new approach for solving computationally-demanding stochastic
optimal stopping problems with known probability distributions. The approach
uses simulation to construct a robust optimization problem that approximates
the stochastic optimal stopping problem to any arbitrary accuracy; we then
solve the robust optimization problem to obtain near-optimal Markovian stopping
rules for the stochastic optimal stopping problem. In this paper, we focus on
designing algorithms for solving the robust optimization problems that
approximate the stochastic optimal stopping problems. These robust optimization
problems are challenging to solve because they require optimizing over the
infinite-dimensional space of all Markovian stopping rules. We overcome this
challenge by characterizing the structure of optimal Markovian stopping rules
for the robust optimization problems. In particular, we show that optimal
Markovian stopping rules for the robust optimization problems have a structure
that is surprisingly simple and finite-dimensional. We leverage this structure
to develop an exact reformulation of the robust optimization problem as a
zero-one bilinear program over totally unimodular constraints. We show that the
bilinear program can be solved in polynomial time in special cases, establish
computational complexity results for general cases, and develop polynomial-time
heuristics by relating the bilinear program to the maximal closure problem from
graph theory. Numerical experiments demonstrate that our algorithms for solving
the robust optimization problems are practical and can outperform
state-of-the-art simulation-based algorithms in the context of widely-studied
stochastic optimal stopping problems from high-dimensional option pricing
Sequential testing problems for some diffusion processes
We study the Bayesian problem of sequential testing of two simple hypotheses about the local drift of an observed diffusion process. The optimal stopping time is found as the first time when the a posteriori probability process leaves the region defined by two stochastic boundaries depending on the observation process. It is shown that under some nontrivial relationships on the coefficients of the observed diffusion the problem admits a closed form solution. The method of proof is based on embedding the initial problem into a two-dimensional optimal stopping problem and solving the equivalent free-boundary problem by means of the smooth-fit conditions
A Linear Programming Approach to Sequential Hypothesis Testing
Under some mild Markov assumptions it is shown that the problem of designing
optimal sequential tests for two simple hypotheses can be formulated as a
linear program. The result is derived by investigating the Lagrangian dual of
the sequential testing problem, which is an unconstrained optimal stopping
problem, depending on two unknown Lagrangian multipliers. It is shown that the
derivative of the optimal cost function with respect to these multipliers
coincides with the error probabilities of the corresponding sequential test.
This property is used to formulate an optimization problem that is jointly
linear in the cost function and the Lagrangian multipliers and an be solved for
both with off-the-shelf algorithms. To illustrate the procedure, optimal
sequential tests for Gaussian random sequences with different dependency
structures are derived, including the Gaussian AR(1) process.Comment: 25 pages, 4 figures, accepted for publication in Sequential Analysi
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