Semimartingales and Markov processes

Çınlar, Erhan; Jacod, Jean; Protter, Philip; Sharpe, M. J.

doi:10.1007/bf00531446

“…Several of these results (or variants thereof) can be found in Dynkin [14] and Mitro [38]. See also [11] and [12] as general references.…”

Section: Preliminariesmentioning

confidence: 88%

Homogeneous random measures and a weak order for the excessive measures of a Markov process

Fitzsimmons¹

1987

Trans. Amer. Math. Soc.

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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)

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“…is a right-continuous semimartingale, by Çinlar, Jacod, Protter, and Sharpe (1980) the function π L is locally the difference of two 1-excessive functions. For a Borel right process, its excessive functions are generally only universally measurable, but not necessarily Borel measurable.…”

Section: Qedmentioning

confidence: 99%

Long-Term Risk: A Martingale Approach

Qin¹,

Linetsky²

2017

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A. PROOFS FOR SECTION 3 WE FIRST RECALL SOME RESULTS about semimartingale topology originally introduced by Émery (1979) (see Czichowsky and Schweizer (2006), Kardaras (2013), andCuchiero andTeichmann (2015) for recent applications in mathematical finance). The semimartingale topology is stronger than the topology of uniform convergence in probability on compacts (ucp). In the latter case, the supremum in Eq. (2.1) is only taken over integrands in the form η t = 1 [0 s] (t) for every s > 0:The following inequality due to Burkholder is useful for proving convergence in the semimartingale topology in Theorem 3.1 (see Meyer (1972, Theorem 47, p. 50) for discrete martingales and Cuchiero and Teichmann (2015) for continuous martingales, where a proof is provided inside the proof of their Lemma 4.7).LEMMA A.1: For every martingale X and every predictable process η bounded by 1, |η t | ≤ 1, it holds that aP supfor all a ≥ 0 and t > 0.We will also use the following result (see Kardaras (2013, Proposition 2.10)).We will also make use of the following lemma.LEMMA A.3: Let (X n t ) t≥0 be a sequence of martingales such that X n t

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“…Under hypotheses close to ours, Dynkin and Getoor [17] have considered diffuse random measures. For related matters see [2,5,14,11,21,22,35].…”

Section: Random Measures; Palm Measuresmentioning

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

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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)

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Semimartingales and Markov processes

Cited by 176 publications

References 27 publications

Homogeneous random measures and a weak order for the excessive measures of a Markov process

Homogeneous random measures and a weak order for the excessive measures of a Markov process

Long-Term Risk: A Martingale Approach

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

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