Functional limit theorems for processes pieced together from excursions

Yano, Kouji

doi:10.2969/jmsj/06741859

Cited by 7 publications

(6 citation statements)

References 23 publications

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“…As far as we know there are no other functional limit theorems that would make an explicit link between random walks with a random-jump reflection at 0 and W (x) α . Also relevant to the present work are the papers [13,23,24] and references therein, in which functional limit theorems are obtained for processes merged together from certain excursions.…”

Section: The Limit Process W (X)mentioning

confidence: 99%

Functional limit theorems for random walks perturbed by positive alpha-stable jumps

Iksanov¹,

Povar²

2021

Preprint

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Let ξ 1 , ξ 2 , . . . be i.i.d. random variables of zero mean and finite variance and η 1 , η 2 , . . . positive i.i.d. random variables whose distribution belongs to the domain of attraction of an α-stable distribution, α ∈ (0, 1). The two collections are assumed independent. We consider a Markov chain with jumps of two types. If the present position of the Markov chain is positive, then the jump ξ k occurs; if the present position of the Markov chain is nonpositive, then its next position is η j . We prove a functional limit theorem for this Markov chain under Donsker's scaling. The weak limit is a nonnegative process (X(t)) t≥0 satisfying a stochastic equation dX(t) = dW (t) + dUα(L (0)where W is a Brownian motion, Uα is an α-stable subordinator which is independent of W , and L (0) X is a local time of X at 0. Also, we explain that X is a Feller Brownian motion with a 'jump-type' exit from 0.

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Section: The Limit Process W (X)mentioning

confidence: 99%

Functional limit theorems for random walks perturbed by positive alpha-stable jumps

Iksanov¹,

Povar²

2021

Preprint

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“…Unlike Ito's original synthesis theorem, our construction leads to Markov processes on an enlarged state space. Compared with [26,44], we retain a discrete construction and are able to incorporate dependence in the sequence, as shown in the examples below, without losing analytical tractability.…”

Section: Modeling Stochastic Processes Via δ-Excursionsmentioning

confidence: 99%

“…x one can solve boundary value problem (44) (e λy − 1) dy = e λ x + p(λ)t Φ x + λσ 2 t σ √ t − e −λ x + p(λ)t Φ −x + λσ 2 t σ √ t + 2Φ −x σ √ t .…”

Section: A Technical Proofs A1 Proof Of Proposition 48mentioning

confidence: 99%

Excursion Risk

Ananova¹,

Cont²,

Xu³

2020

SSRN Journal

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The risk and return profiles of a broad class of dynamic trading strategies, including pairs trading and other statistical arbitrage strategies, may be characterized in terms of excursions of the market price of a portfolio away from a reference level. We propose a mathematical framework for the risk analysis of such strategies, based on a description in terms of price excursions, first in a pathwise setting, without probabilistic assumptions, then in a Markovian setting. We introduce the notion of δ-excursion, defined as a path which deviates by δ from a reference level before returning to this level. We show that every continuous path has a unique decomposition into δ-excursions, which is useful for the scenario analysis of dynamic trading strategies, leading to simple expressions for the number of trades, realized profit, maximum loss and drawdown. As δ is decreased to zero, properties of this decomposition relate to the local time of the path. When the underlying asset follows a Markov process, we combine these results with Ito's excursion theory to obtain a tractable decomposition of the process as a concatenation of independent δ-excursions, whose distribution is described in terms of Ito's excursion measure. We provide analytical results for linear diffusions and give new examples of stochastic processes for flexible and tractable modeling of excursions. Finally, we describe a non-parametric scenario simulation method for generating paths whose excursion properties match those observed in empirical data.

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“…Our approach is based on the idea that approximating the excursions of a process ensures approximating the process, and, in particular, various functionals of the process [27], [34].…”

Section: Two Point Padé Approximations For the Downward Ladder Time Omentioning

confidence: 99%

On the central management of risk networks

Avram

Minca

2017

Adv. Appl. Probab.

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In this paper we identify three questions concerning the management of risk networks with a central branch, which may be solved using the extensive machinery available for one-dimensional risk models. First, we propose a criterion for judging whether a subsidiary is viable by its readiness to pay dividends to the central branch, as reflected by the optimality of the zero-level dividend barrier. Next, for a deterministic central branch which must bailout a single subsidiary each time its surplus becomes negative, we determine the optimal bailout policy, as well as the ruin probability and other risk measures, in closed form. Moreover, we extend these results to the case of hierarchical networks. Finally, for nondeterministic central branches with one subsidiary, we compute approximate risk measures by applying rational approximations, and by using the recently developed matrix scale methodology.

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Functional limit theorems for processes pieced together from excursions

Cited by 7 publications

References 23 publications

Functional limit theorems for random walks perturbed by positive alpha-stable jumps

Functional limit theorems for random walks perturbed by positive alpha-stable jumps

Excursion Risk

On the central management of risk networks

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