On the one-sided Tanaka equation with drift

Karatzas, Ioannis; Shiryaev, Albert N.; Shkolnikov, Mykhaylo

doi:10.1214/ecp.v16-1665

Cited by 21 publications

(13 citation statements)

References 23 publications

(35 reference statements)

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“…We now explain in more detail what is included in Definition (20). For all y ∈ I • , we define a I (y) = min{y −l, r −y} (∈ (0, ∞]) and notice that, for a ≥ 0, it holds y ±a ∈ I • if and only if a < a I (y).…”

Section: Resultsmentioning

confidence: 99%

Approximating exit times of continuous Markov processes

Kruse¹,

Urusov²

2020

Discrete &Amp; Continuous Dynamical Systems - B

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The time at which a one-dimensional continuous strong Markov process attains a boundary point of its state space is a discontinuous path functional and it is, therefore, unclear whether the exit time can be approximated by hitting times of approximations of the process. We prove a functional limit theorem for approximating weakly both the paths of the Markov process and its exit times. In contrast to the functional limit theorem in [3] for approximating the paths, we impose a stronger assumption here. This is essential, as we present an example showing that the theorem extended with the convergence of the exit times does not hold under the assumption in [3]. However, the EMCEL scheme introduced in [3] satisfies the assumption of our theorem, and hence we have a scheme capable of approximating both the process and its exit times for every one-dimensional continuous strong Markov process, even with irregular behavior (e.g., a solution of an SDE with irregular coefficients or a Markov process with sticky features). Moreover, our main result can be used to check for some other schemes whether the exit times converge. As an application we verify that the weak Euler scheme is capable of approximating the absorption time of the CEV diffusion and that the scale-transformed weak Euler scheme for a squared Bessel process is capable of approximating the time when the squared Bessel process hits zero.

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Section: Resultsmentioning

confidence: 99%

Approximating exit times of continuous Markov processes

Kruse¹,

Urusov²

2020

Discrete &Amp; Continuous Dynamical Systems - B

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“…Nonexistence of strong solutions to equations (1.1) and (1.2) was first shown in [24] and [82] (see also [32] for a more canonical arguments which would more easily generalize to other sticky processes). Several other works have been published on the existence of solutions to similar SDEs with indicator functions as the coefficient of dB t or dt including [48,10]. A more complete history of these SDEs can be found in [32].…”

Section: Remark 12 (Time Change)mentioning

confidence: 99%

Large deviations for sticky Brownian motions

Barraquand¹,

Rychnovsky²

2020

Electron. J. Probab.

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We consider n-point sticky Brownian motions: a family of n diffusions that evolve as independent Brownian motions when they are apart, and interact locally so that the set of coincidence times has positive Lebesgue measure with positive probability.These diffusions can also be seen as n random motions in a random environment whose distribution is given by so-called stochastic flows of kernels. For a specific type of sticky interaction, we prove exact formulas characterizing the stochastic flow and show that in the large deviations regime, the random fluctuations of these stochastic flows are Tracy-Widom GUE distributed. An equivalent formulation of this result states that the extremal particle among n sticky Brownian motions has Tracy-Widom distributed fluctuations in the large n and large time limit. These results are proved by viewing sticky Brownian motions as a (previously known) limit of the exactly solvable beta random walk in random environment.

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“…At this point we remark that (3.5) or (3.7) has unique weak solutions. One does not expect to have strong solutions or pathwise uniqueness, see the recent works of [2,3], as well as the older works of [10] and of [20][21][22] on sticky Brownian motion.…”

Section: S) Is a Martingale With Quadratic Variationmentioning

confidence: 99%

Markov processes with spatial delay: Path space characterization, occupation time and properties

Salins

Spiliopoulos

2017

Stoch. Dyn.

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In this paper, we study one-dimensional Markov processes with spatial delay. Since the seminal work of Feller, we know that virtually any one-dimensional, strong, homogeneous, continuous Markov process can be uniquely characterized via its infinitesimal generator and the generator's domain of definition. Unlike standard diffusions like Brownian motion, processes with spatial delay spend positive time at a single point of space. Interestingly, the set of times that a delay process spends at its delay point is nowhere dense and forms a positive measure Cantor set. The domain of definition of the generator has restrictions involving second derivatives. In this paper we provide a pathwise characterization for processes with delay in terms of an SDE and an occupation time formula involving the symmetric local time. This characterization provides an explicit Doob-Meyer decomposition, demonstrating that such processes are semi-martingales and that all of stochastic calculus including Itô formula and Girsanov formula applies. We also establish an occupation time formula linking the time that the process spends at a delay point with its symmetric local time there. A physical example of a stochastic dynamical system with delay is lastly presented and analyzed.

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On the one-sided Tanaka equation with drift

Cited by 21 publications

References 23 publications

Approximating exit times of continuous Markov processes

Approximating exit times of continuous Markov processes

Large deviations for sticky Brownian motions

Markov processes with spatial delay: Path space characterization, occupation time and properties

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