Ambit Fields: Survey and New Challenges

Podolskij, Mark

doi:10.1007/978-3-319-13984-5_12

Cited by 20 publications

(16 citation statements)

References 55 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Hence, we obtain pointwise convergence at (28). Since the limiting process A is continuous, we now need to show that which follows by Lemma 1 and condition (26). This completes the proof of Theorem 1.…”

Section: Theorem 1 Assume That Conditions (A) (B)mentioning

confidence: 66%

“…In the past years stochastic analysis, probabilistic properties and statistical inference for Lévy semi-stationary processes have been studied in numerous papers. We refer to [2,3,6,7,11,12,15,17,20,25] for the mathematical theory as well as to [5,26] for a recent survey on theory of ambit fields and their applications. For practical applications in sciences numerical approximation of Lévy (Brownian) semi-stationary processes, or, more generally, of ambit fields, is an important issue.…”

Section: G(t S X ξ)σ S (ξ )L(ds Dξ)+ D T (X)mentioning

confidence: 99%

“…We remark that the stronger conditions (B) and (26) are not required to prove the finite dimensional version of convergence (28) and (29).…”

Section: Remarkmentioning

confidence: 99%

See 2 more Smart Citations

A Weak Limit Theorem for Numerical Approximation of Brownian Semi-stationary Processes

Podolskij

Thamrongrat

2015

Stochastics of Environmental and Financial Economics

Self Cite

View full text Add to dashboard Cite

In this paper we present a weak limit theorem for a numerical approximation of Brownian semi-stationary processes studied in [14]. In the original work of [14] the authors propose to use Fourier transformation to embed a given one dimensional (Lévy) Brownian semi-stationary process into a two-parameter stochastic field. For the latter they use a simple iteration procedure and study the strong approximation error of the resulting numerical scheme given that the volatility process is fully observed. In this work we present the corresponding weak limit theorem for the setting, where the volatility/drift process needs to be numerically simulated. In particular, weak approximation errors for smooth test functions can be obtained from our asymptotic theory.

show abstract

Section: Theorem 1 Assume That Conditions (A) (B)mentioning

confidence: 66%

Section: G(t S X ξ)σ S (ξ )L(ds Dξ)+ D T (X)mentioning

confidence: 99%

See 1 more Smart Citation

A Weak Limit Theorem for Numerical Approximation of Brownian Semi-stationary Processes

Podolskij

Thamrongrat

2015

Stochastics of Environmental and Financial Economics

Self Cite

View full text Add to dashboard Cite

show abstract

“…Finally, L denotes a Lévy basis (i.e., an independently scattered and infinitely divisible random measure). For aspects of the theory and applications of ambit processes and fields, see [8,10,12,14,15,23,34,39,52] and [55].…”

Section: Ambit Fields Volterra Fields and Lss Processesmentioning

confidence: 99%

Selfdecomposable Fields

2015

View full text Add to dashboard Cite

In the present paper, we study selfdecomposability of random fields, as defined directly rather than in terms of finite-dimensional distributions. The main tools in our analysis are the master Lévy measure and the associated Lévy-Itô representation. We give the dilation criterion for selfdecomposability analogous to the classical one. Next, we give necessary and sufficient conditions (in terms of the kernel function) for a Volterra field driven by a Lévy basis to be selfdecomposable. In this context, we also study the so-called Urbanik classes of random fields. We follow this with the study of existence and selfdecomposability of integrated Volterra fields. Finally, we introduce infinitely divisible field-valued Lévy processes, give the Lévy-Itô representation associated with them and study stochastic integration with respect to such processes. We provide examples in the form of Lévy semistationary processes with a Gamma kernel and Ornstein-Uhlenbeck processes.

show abstract

“…The methodology we use is very much inspired by [10]. Ambit fields where introduced in [6] with the aim of studying turbulence flows, see also the survey papers [5,15]. They are stochastic processes indexed by (t,…”

Section: Ambit Random Fieldsmentioning

confidence: 99%

Non-elliptic SPDEs and Ambit Fields: Existence of Densities

Sanz–Solé

Süß

2015

Stochastics of Environmental and Financial Economics

View full text Add to dashboard Cite

Relying on the method developed in [11], we prove the existence of a density for two different examples of random fields indexed by (t, x) ∈ (0, T ] × R d . The first example consists of SPDEs with Lipschitz continuous coefficients driven by a Gaussian noise white in time and with a stationary spatial covariance, in the setting of [9]. The density exists on the set where the nonlinearity σ of the noise does not vanish. This complements the results in [20] where σ is assumed to be bounded away from zero. The second example is an ambit field with a stochastic integral term having as integrator a Lévy basis of pure-jump, stable-like type.

show abstract

Ambit Fields: Survey and New Challenges

Cited by 20 publications

References 55 publications

A Weak Limit Theorem for Numerical Approximation of Brownian Semi-stationary Processes

A Weak Limit Theorem for Numerical Approximation of Brownian Semi-stationary Processes

Selfdecomposable Fields

Non-elliptic SPDEs and Ambit Fields: Existence of Densities

Contact Info

Product

Resources

About