Some Refinements of Existence Results for SPDEs Driven by Wiener Processes and Poisson Random Measures

The evolution of finitely many particles obeying Langevin dynamics is described by Dean-Kawasaki equations, a class of stochastic equations featuring a non-Lipschitz multiplicative noise in divergence form. We derive a regularised Dean-Kawasaki model based on second order Langevin dynamics by analysing a system of particles interacting via a pairwise potential. Key tools of our analysis are the propagation of chaos and Simon's compactness criterion. The model we obtain is a small-noise stochastic perturbation of the undamped McKean-Vlasov equation. We also provide a high-probability result for existence and uniqueness for our model.

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“…where the notation is in analogy to (27). The convergence T 2 → 0 is settled as in the first part of the proof, and we omit the details.…”

Section: A Useful Application Of the Propagation Of Chaosmentioning

confidence: 99%

From weakly interacting particles to a regularised Dean–Kawasaki model

2020

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“…In the literature for stochastic partial differential equations several criteria are known for t → X(t), viewed as a process with values in a Hilbert space, to have continuous or càdlàg sample paths (see for instance Theorem 11.8 of [25] or Theorem 4.5 and Remark 4.6 of [29]). By contrast, the random field approach to the VOU equation allows for a detailed analysis of the temporospatial path properties of (t, x) → X(t, x) as in Section 5.…”

Section: Example 37 (Vou Process With Finite Speed Propagation Of Nomentioning

confidence: 99%

Volterra-type Ornstein–Uhlenbeck processes in space and time

Pham

Chong

2018

Stochastic Processes and their Applications

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We propose a novel class of temporo-spatial Ornstein-Uhlenbeck processes as solutions to Lévy-driven Volterra equations with additive noise and multiplicative drift. After formulating conditions for the existence and uniqueness of solutions, we derive an explicit solution formula and discuss distributional properties such as stationarity, second-order structure and short versus long memory. Furthermore, we analyze in detail the path properties of the solution process. In particular, we introduce different notions of càdlàg paths in space and time and establish conditions for the existence of versions with these regularity properties. The theoretical results are accompanied by illustrative examples.

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“…Hence, referring to Thm. 4.5 in Tappe [35], there exists a unique mild solution of (3.8) for s ≥ t, that is, a càdlàg process g ∈ H α satisfying…”

Section: European Options On Energy Forwards and Futuresmentioning

confidence: 99%

“…(f, g) , we can relate to the theory of existence and uniqueness of mild solutions of SPDEs given by Tappe [35]: there exists a unique mild solution satisfying the integral equations…”

Section: Cross-commodity Modellingmentioning

confidence: 99%

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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Some Refinements of Existence Results for SPDEs Driven by Wiener Processes and Poisson Random Measures

Cited by 15 publications

References 20 publications

From weakly interacting particles to a regularised Dean–Kawasaki model

From weakly interacting particles to a regularised Dean–Kawasaki model

Volterra-type Ornstein–Uhlenbeck processes in space and time

Derivatives Pricing in Energy Markets: An Infinite-Dimensional Approach

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