Penetration times and skorohod stopping

Fitzsimmons, P. J.

doi:10.1007/bfb0084134

Cited by 6 publications

(4 citation statements)

References 12 publications

(14 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…An important continuation of Rost's work is found in two papers of Fitzsimmons. In [38] (with some minor correction recorded in [39]) an existence theorem for embedding in a general Markov process is obtained as a by-product of investigation of a natural extention of Hunt's balayage L B m (see [38] for appropriate definitions). The second paper [40], is devoted to the Skorokhod problem for general right Markov processes.…”

Section: Rost (1971)mentioning

confidence: 99%

The Skorokhod embedding problem and its offspring

Obłój¹

2004

Probab. Surveys

236

120

View full text Add to dashboard Cite

This is a survey about the Skorokhod embedding problem. It presents all known solutions together with their properties and some applications. Some of the solutions are just described, while others are studied in detail and their proofs are presented. A certain unification of proofs, based on one-dimensional potential theory, is made. Some new facts which appeared in a natural way when different solutions were cross-examined, are reported. Az\'ema and Yor's and Root's solutions are studied extensively. A possible use of the latter is suggested together with a conjecture.Comment: Published at http://dx.doi.org/10.1214/154957804100000060 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

show abstract

Section: Rost (1971)mentioning

confidence: 99%

The Skorokhod embedding problem and its offspring

Obłój¹

2004

Probab. Surveys

236

120

View full text Add to dashboard Cite

show abstract

“…□ We shall use Corollary (7.7) to obtain an analogue of Hunt's balayage theorem. A "crude" result of this type can be found in [18]: (7.9) LArn = inf{£ G Exc: £ > m on ,4} iff Qm(TA / nA) = 0.…”

Section: Jdmentioning

confidence: 99%

Homogeneous Random Measures and a Weak Order for the Excessive Measures of a Markov Process

Fitzsimmons¹

1987

Transactions of the American Mathematical Society

Self Cite

View full text Add to dashboard Cite

ABSTRACT. Let X = (Xt,Px) be a right Markov process and let m be an excessive measure for X. Associated with the pair (X, ra) is a stationary strong Markov process (Yt, Qm) with random times of birth and death, with the same transition function as X, and with m as one dimensional distribution.We use (Yt,Qm) to study the cone of excessive measures for X. A "weak order" is defined on this cone: an excessive measure £ is weakly dominated by m if and only if there is a suitable homogeneous random measure re such that (Yt,Qç) is obtained by "birthing" (Yt,Qm), birth in [t,t + dt] occurring at rate re(di). Random measures such as re are studied through the use of Palm measures. We also develop aspects of the "general theory of processes" over (Yt,Qm), including the moderate Markov property of {Yt,Qm) when the arrow of time is reversed. Applications to balayage and capacity are suggested. Introduction.Let (Ps : s > 0) be a Borel right semigroup on a Lusin state space (E, £) and let (Xt, Px) denote the associated right continuous strong Markov process. Recall that a measure m on (E, £) is excessive for (Ps) if m is tr-finite and if mPs < m for all s > 0. Since (Ps) is a right semigroup, one then has mPs ] m as s J. 0, as is well known. Let Exc denote the convex cone of excessive measures for (Pa)-Given m E Exc, according to a theorem of Kuznetsov [31], one can construct a stationary process (Yt : t E R) having random birth and death times and a tr-finite governing measure Qm, such that for ii < t2 < • ■ ■ < tn (i» E R)

show abstract

“…The réduite of it, Rir is then defined as the greatest lower bound (in the lattice Exc with the simple order) of the set {7 G Exc: 7 > 7r}. In particular, if £ G Exc and A E £ we write RaC for R(Ia • 0-^ is shown in [18] that if £ G Exc then í?a£ = £n(A)£, where n(^4) is the Lebesgue penetration time of A.…”

Section: ]Oi]mentioning

confidence: 99%

“…The following is an adaptation of a result that Heath [42] ascribes to Mokobodski. For a proof (modeled on [42]) see [18]. (6.35) PROPOSITION.…”

Section: ]Oi]mentioning

confidence: 99%

Untitled

1987

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Let X = (Xt,Px) be a right Markov process and let m be an excessive measure for X. Associated with the pair (X, ra) is a stationary strong Markov process (Yt, Qm) with random times of birth and death, with the same transition function as X, and with m as one dimensional distribution.We use (Yt,Qm) to study the cone of excessive measures for X. A "weak order" is defined on this cone: an excessive measure £ is weakly dominated by m if and only if there is a suitable homogeneous random measure re such that (Yt,Qç) is obtained by "birthing" (Yt,Qm), birth in [t,t + dt] occurring at rate re(di). Random measures such as re are studied through the use of Palm measures. We also develop aspects of the "general theory of processes" over (Yt,Qm), including the moderate Markov property of {Yt,Qm) when the arrow of time is reversed. Applications to balayage and capacity are suggested.

show abstract

Penetration times and skorohod stopping

Cited by 6 publications

References 12 publications

The Skorokhod embedding problem and its offspring

The Skorokhod embedding problem and its offspring

Homogeneous Random Measures and a Weak Order for the Excessive Measures of a Markov Process

Untitled

Contact Info

Product

Resources

About