Martingales Locales sur un Ouvert Droit Optionnel<sup>†</sup>

Jing, Yan

doi:10.1080/17442508208833236

Cited by 2 publications

(3 citation statements)

References 2 publications

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“…At this juncture we refer to [10], [6, Sec. V.1], [17], [19], [15], and [16], where several classes of processes on stochastic intervals (or even on optional random sets) are considered. In particular, in [10] (also see [14, Ch.…”

Section: 2mentioning

confidence: 99%

“…IV, Ex. 1.48]) the notion of a continuous local martingale on a stochastic interval [0, τ ) is introduced, where τ is a stopping time (not necessarily predictable), and in [17] a way of extending this notion to càdlàg processes is suggested. An important and delicate point in these works is precisely the definition of the notion of a local martingale on the stochastic interval [0, τ ), when τ is a non-predictable stopping time.…”

Section: 2mentioning

confidence: 99%

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On the Loss of the Semimartingale Property at the Hitting Time of a Level

Mijatović

Urusov

2013

J Theor Probab

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Abstract. This paper studies the loss of the semimartingale property of the process g(Y )at the time a one-dimensional diffusion Y hits a level, where g is a difference of two convex functions. We show that the process g(Y ) can fail to be a semimartingale in two ways only, which leads to a natural definition of non-semimartingales of the first and second kind. We give a deterministic if and only if condition (in terms of g and the coefficients of Y ) for g(Y ) to fall into one of the two classes of processes, which yields a characterisation for the loss of the semimartingale property. A number of applications of the results in the theory of stochastic processes and real analysis are given: e.g. we construct an adapted diffusion Y on [0, ∞) and a predictable finite stopping time ζ, such that Y is a local semimartingale on the stochastic interval [0, ζ), continuous at ζ and constant after ζ, but is not a semimartingale on [0, ∞).

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Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

On the Loss of the Semimartingale Property at the Hitting Time of a Level

Mijatović

Urusov

2013

J Theor Probab

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“…The case of an optional rightopen interval of the form 0, ζ was discussed first by Maisonneuve [Ma77] for continuous martingales, and by Sharpe [Sh92] in the general case. See also [Ya82], [Zh82].…”

Section: (01) Problem Find Necessary and Sufficient Conditions On Xmentioning

confidence: 99%

Martingales on Random Sets and the Strong Martingale Property

Sharpe

2000

Electron. J. Probab.

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Abstract. Let X be a process defined on an optional random set. The paper develops two different conditions on X guaranteeing that it is the restriction of a uniformly integrable martingale. In each case, it is supposed that X is the restriction to Λ of some special semimartingale Z with canonical decomposition Z = M + A. The first condition, which is both necessary and sufficient, is an absolute continuity condition on A. Under additional hypotheses, the existence of a martingale extension can be characterized by a strong martingale property of X on Λ. Uniqueness of the extension is also considered. IntroductionLet Λ be an optional random set and let X t (ω) be defined for (t, ω) ∈ Λ. We consider the following and some of its extensions. (0.1) Problem. Find necessary and sufficient conditions on X guaranteeing that it is the restriction to Λ of a globally defined, right continuous uniformly integrable martingale.For an example where this formulation may be natural, consider a process (Y t ) t≥0 with values in a manifold. Given a coordinate patch V , let Λ := {(t, ω) : Y t (ω) ∈ V } and let X t (ω) denote a real component of Y t (ω) for (t, ω) ∈ Λ. A second natural example is provided by X = f •W , where W is a Markov process in a state space E and f is a function defined on a subset S of E, Λ denoting in this case {(t, ω) :The solution is obvious if there is an increasing sequence of stopping times T n which are complete sections of Λ (i.e., T n ⊂ Λ and P{T n < ∞} = 1) such that Λ ⊂ ∪ n 0, T n . A number of other cases may now be found in the literature. In sections 2 and 3, we give a complete solution for the discrete parameter problem under mild conditions on X. The continuous parameter case, treated in sections 4 and 5, involves considerable additional complication. Roughly speaking, in the discrete parameter case the condition is that X have a "strong martingale property" on Λ (defined in (3.4)), but in the continuous parameter context, an example is given in section 5 to show that this condition is not sufficient. Theorem (4.1), one of the main results of the paper, assumes X is the restriction of a special semimartingale Z = M + A (with A predictable and of integrable 1991 Mathematics Subject Classification. 60J30.

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Martingales Locales sur un Ouvert Droit Optionnel^†

Cited by 2 publications

References 2 publications

On the Loss of the Semimartingale Property at the Hitting Time of a Level

On the Loss of the Semimartingale Property at the Hitting Time of a Level

Martingales on Random Sets and the Strong Martingale Property

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Martingales Locales sur un Ouvert Droit Optionnel†

Cited by 2 publications

References 2 publications

On the Loss of the Semimartingale Property at the Hitting Time of a Level

On the Loss of the Semimartingale Property at the Hitting Time of a Level

Martingales on Random Sets and the Strong Martingale Property

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Product

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Martingales Locales sur un Ouvert Droit Optionnel^†