Ambit Processes and Stochastic Partial Differential Equations

Barndorff–Nielsen, Ole E.; Benth, Fred Espen; Veraart, Almut E. D.

doi:10.2139/ssrn.1597697

Cited by 28 publications

(71 citation statements)

References 34 publications

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“…In a recent paper, Barndorff-Nielsen, Benth and Veraart [7] has extended the ambit field idea to cross-commodity market modelling and the pricing of spread options. We remark that there is a close relationship between ambit fields and stochastic partial differential equations (see Barndorff-Nielsen, Benth and Veraart [5]). …”

Section: Introductionmentioning

confidence: 81%

Derivatives Pricing in Energy Markets: An Infinite-Dimensional Approach

Benth

Krühner

2015

SIAM J. Finan. Math.

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ABSTRACT. Based on forward curves modelled as Hilbert-space valued processes, we analyse the pricing of various options relevant in energy markets. In particular, we connect empirical evidence about energy forward prices known from the literature to propose stochastic models. Forward prices can be represented as linear functions on a Hilbert space, and options can thus be viewed as derivatives on the whole curve. The value of these options are computed under various specifications, in addition to their deltas. In a second part, crosscommodity models are investigated, leading to a study of square integrable random variables with values in a "two-dimensional" Hilbert space. We analyse the covariance operator and representations of such variables, as well as presenting applications to pricing of spread and energy quanto options.

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Section: Introductionmentioning

confidence: 81%

Derivatives Pricing in Energy Markets: An Infinite-Dimensional Approach

Benth

Krühner

2015

SIAM J. Finan. Math.

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“…In particular, the canonical decomposition of Λ determines its Lévy-Itô decomposition as derived in [44]. Consequently, the integrability criteria obtained in Theorem 4.1 extend the corresponding result of [46], Theorem 2.7, for deterministic functions (or, as used in [5], for integrands which are independent of Λ) to allow for predictable integrands.…”

mentioning

confidence: 77%

“…If the characteristics of M in the sense of Theorem 3.2 are known, (5.1) exists if and only if the conditions of Theorem 4.1 are satisfied for each pair (t, x) ∈ R× R d . We call processes of the form (5.1) ambit processes although the original definition in [5] requires the random measure to be a volatility modulated Lévy basis, that is, M = σ.Λ where Λ is a Lévy basis and σ ∈P. As already explained in the Introduction, this class of models is relevant in many different areas of applications.…”

Section: Ambit Processesmentioning

confidence: 99%

“…The connection between stochastic PDEs and ambit processes is exemplified in [5] relying on the integration theory of [46] or [51]. Let U be an open subset of R × R d with boundary ∂U , P a polynomial in 1 + d variables with constant coefficients and M a random measure with different times of discontinuity.…”

Section: Ambit Processesmentioning

confidence: 99%

“…Moreover, solutions to stochastic partial differential equations driven by random noise are often of the form (1.3), cf. [5,51] and Section 5.2. Furthermore, stochastic processes like forward contracts in bond and electricity markets based on a Heath-Jarrow-Morten approach also rely on a spatial structure, cf.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Integrability conditions for space–time stochastic integrals: Theory and applications

Chong¹,

Klüppelberg²

2015

Bernoulli

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We derive explicit integrability conditions for stochastic integrals taken over time and space driven by a random measure. Our main tool is a canonical decomposition of a random measure which extends the results from the purely temporal case. We show that the characteristics of this decomposition can be chosen as predictable strict random measures, and we compute the characteristics of the stochastic integral process. We apply our conditions to a variety of examples, in particular to ambit processes, which represent a rich model class.

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Simulation of Stochastic Volterra Equations Driven by Space–Time Lévy Noise

Chen

Chong

Klüppelberg

2015

The Fascination of Probability, Statistics and Their Applications

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In this paper we investigate two numerical schemes for the simulation of stochastic Volterra equations driven by space-time Lévy noise of pure-jump type. The first one is based on truncating the small jumps of the noise, while the second one relies on series representation techniques for infinitely divisible random variables. Under reasonable assumptions, we prove for both methods L p -and almost sure convergence of the approximations to the true solution of the Volterra equation. We give explicit convergence rates in terms of the Volterra kernel and the characteristics of the noise. A simulation study visualizes the most important path properties of the investigated processes.

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Ambit Processes and Stochastic Partial Differential Equations

Cited by 28 publications

References 34 publications

Derivatives Pricing in Energy Markets: An Infinite-Dimensional Approach

Derivatives Pricing in Energy Markets: An Infinite-Dimensional Approach

Integrability conditions for space–time stochastic integrals: Theory and applications

Simulation of Stochastic Volterra Equations Driven by Space–Time Lévy Noise

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