A harmonic function technique for the optimal stopping of diffusions

Christensen, Sören; Irle, Albrecht

doi:10.1080/17442508.2010.498915

Cited by 41 publications

(52 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section we want to study optimal stopping problems for one‐dimensional diffusions with infinite time horizon. These problems can be solved analytically using different techniques; we only refer to Mucci (1978/79), Salminen (1985), Beibel and Lerche (2000), Dayanik and Karatzas (2003), and Christensen and Irle (2011). Nonetheless in many situations it can be helpful to use numerical methods.…”

Section: One‐dimensional Diffusions With Infinite Time Horizonmentioning

confidence: 99%

“…Proof Two proofs based on different methods can be found in Helmes and Stockbridge (2010, theorem 4.2) and in Christensen and Irle (2011, corollary 2.2). This is a general fact, see Proposition 4.1 in the following section. If an optimal stopping time exists, then by the general theory the smallest is given by . Since ( X t ) t ≥ 0 has continuous sample paths the assertion holds by (b). …”

mentioning

confidence: 91%

“…Two proofs based on different methods can be found in Helmes and Stockbridge (2010, theorem 4.2) and in Christensen and Irle (2011, corollary 2.2).…”

mentioning

confidence: 99%

See 2 more Smart Citations

A Method for Pricing American Options Using Semi‐infinite Linear Programming

Christensen

2012

Mathematical Finance

Self Cite

View full text Add to dashboard Cite

We introduce a new approach for the numerical pricing of American options. The main idea is to choose a finite number of suitable excessive functions (randomly) and to find the smallest majorant of the gain function in the span of these functions. The resulting problem is a linear semi-infinite programming problem, that can be solved using standard algorithms. This leads to good upper bounds for the original problem. For our algorithms no discretization of space and time and no simulation is necessary. Furthermore it is applicable even for high-dimensional problems. The algorithm provides an approximation of the value not only for one starting point, but for the complete value function on the continuation set, so that the optimal exercise region and e.g. the Greeks can be calculated. We apply the algorithm to (one-and) multidimensional diffusions and to Lévy processes, and show it to be fast and accurate.

show abstract

Section: One‐dimensional Diffusions With Infinite Time Horizonmentioning

confidence: 99%

mentioning

confidence: 91%

See 1 more Smart Citation

A Method for Pricing American Options Using Semi‐infinite Linear Programming

Christensen

2012

Mathematical Finance

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is then shown that the optimization of the gain function over all admissible stopping points gives the value of the initial optimal stopping problem, whenever the optimal stopping time is finite almost surely with respect to the probability measure constructed by means of the positive martingale (see also Lerche and Urusov [20]). This approach is closely connected to the fundamental principle of Snell [34] of the least superharmonic characterization of the value function and provides a new characterization of the optimal stopping set (see Christensen and Irle [10]). Furthermore, it can easily be verified that the solution of the optimal stopping problem, obtained by means of the approach of Beibel and Lerche, satisfies the associated free-boundary problem.…”

Section: Introductionmentioning

confidence: 98%

On the structure of discounted optimal stopping problems for one-dimensional diffusions

Gapeev

Lerche

2011

Stochastics

View full text Add to dashboard Cite

We connect two approaches for solving discounted optimal stopping problems for one-dimensional time-homogeneous regular diffusion processes on infinite time intervals. The optimal stopping rule is assumed to be the first exit time of the underlying process from a region restricted by two constant boundaries. We provide an explicit decomposition of the reward process into a product of a gain function of the boundaries and a uniformly integrable martingale inside the continuation region. This martingale plays a key role for stating sufficient conditions for the optimality of the first exit time. We also consider several illustrating examples of rational valuation of perpetual American strangle options.

show abstract

“…Optimal stopping problems for jump-diffusions, which admit two-sided optimal stopping rules, appear also in finance and real-option theory for pricing American-type financial contracts, which, we believe, can be tackled very effectively with the same method of this paper. We refer the reader to Salminen (1985), Lerche (1997, 2001), Christensen andIrle (2011), Cissé et al (2012) for examples of and additional remarks on problems with two-sided stopping regions. For the general theory of optimal stopping for (jump) diffusions, the books by Shiryaev (1978), Peskir and Shiryaev (2006), Øksendal and Sulem (2007) and the references cited therein can be consulted.…”

Section: Introductionmentioning

confidence: 99%

Multisource Bayesian sequential binary hypothesis testing problem

Dayanık

Sezer

2012

Ann Oper Res

View full text Add to dashboard Cite

We consider the problem of testing two simple hypotheses about unknown local characteristics of several independent Brownian motions and compound Poisson processes. All of the processes may be observed simultaneously as long as desired before a final choice between hypotheses is made. The objective is to find a decision rule that identifies the correct hypothesis and strikes the optimal balance between the expected costs of sampling and choosing the wrong hypothesis. Previous work on Bayesian sequential hypothesis testing in continuous time provides a solution when the characteristics of these processes are tested separately. However, the decision of an observer can improve greatly if multiple information sources are available both in the form of continuously changing signals (Brownian motions) and marked count data (compound Poisson processes). In this paper, we combine and extend those previous efforts by considering the problem in its multisource setting. We identify a Bayes optimal rule by solving an optimal stopping problem for the likelihood-ratio process. Here, the likelihood-ratio process is a jump-diffusion, and the solution of the optimal stopping problem admits a two-sided stopping region. Therefore, instead of using the variational arguments (and smooth-fit principles) directly, we solve the problem by patching the solutions of a sequence of optimal stopping problems for the pure diffusion part of the likelihood-ratio process. We also provide a numerical algorithm and illustrate it on several examples.

show abstract

A harmonic function technique for the optimal stopping of diffusions

Cited by 41 publications

References 11 publications

A Method for Pricing American Options Using Semi‐infinite Linear Programming

A Method for Pricing American Options Using Semi‐infinite Linear Programming

On the structure of discounted optimal stopping problems for one-dimensional diffusions

Multisource Bayesian sequential binary hypothesis testing problem

Contact Info

Product

Resources

About