Stochastic Differential Equations in Infinite Dimensions

Gawarecki, Leszek; Mandrekar, V.

doi:10.1007/978-3-642-16194-0

Cited by 154 publications

(118 citation statements)

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“…OU processes with values in Hilbert space provide a natural infinite dimensional formulation for many linear (parabolic) stochastic partial differential equations (see, e.g., Da Prato and Zabczyk [14], Gawarecki and Mandrekar [17] and Peszat and Zabczyk [22]). Our main motivation for studying Hilbert space-valued OU processes comes from the modelling of futures prices in commodity markets, where the dynamics follow a class of hyperbolic stochastic partial differential equations (see Benth and Krühner [9,10]).…”

Section: Dx(t) = Ax(t) Dt + σ(T) Db(t)mentioning

confidence: 99%

Ornstein–Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility

Benth

Rüdiger

Süß

2018

Stochastic Processes and their Applications

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ABSTRACT. We propose a non-Gaussian operator-valued extension of the Barndorff-Nielsen and Shephard stochastic volatility dynamics, defined as the square-root of an operator-valued Ornstein-Uhlenbeck process with Lévy noise and bounded drift. We derive conditions for the positive definiteness of the Ornstein-Uhlenbeck process, where in particular we must restrict to operator-valued Lévy processes with "non-decreasing paths". It turns out that the volatility model allows for an explicit calculation of its characteristic function, showing an affine structure. We introduce another Hilbert space-valued Ornstein-Uhlenbeck process with Wiener noise perturbed by this class of stochastic volatility dynamics. Under a strong commutativity condition between the covariance operator of the Wiener process and the stochastic volatility, we can derive an analytical expression for the characteristic functional of the Ornstein-Uhlenbeck process perturbed by stochastic volatility if the noises are independent. The case of operator-valued compound Poisson processes as driving noise in the volatility is discussed as a particular example of interest. We apply our results to futures prices in commodity markets, where we discuss our proposed stochastic volatility model in light of ambit fields.

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Section: Dx(t) = Ax(t) Dt + σ(T) Db(t)mentioning

confidence: 99%

Ornstein–Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility

Benth

Rüdiger

Süß

2018

Stochastic Processes and their Applications

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“…1) have widely been studied in the literature, see e.g. [4,23,27,10]. In equation (1.1), A denotes the generator of a strongly continuous semigroup, and W is a trace class Wiener process.…”

Section: Introductionmentioning

confidence: 99%

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

Tappe

2012

International Journal of Stochastic Analysis

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We provide existence and uniqueness of global (and local) mild solutions for a general class of semilinear stochastic partial differential equations driven by Wiener processes and Poisson random measures under local Lipschitz and linear growth (or local boundedness, resp.) conditions. The socalled "method of the moving frame" allows us to reduce the SPDE problems to SDE problems.2010 Mathematics Subject Classification. 60H15, 60G57. Key words and phrases. Semilinear SPDE with jumps, Poisson random measure, mild solution, method of the moving frame.The author is grateful to an anonymous referee for valuable comments and suggestions.

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“…The necessary condition of existence of the integrals in (3.1) is B 0 (u), B 1 (u) ∈ L 2 (H; H) (the space of Hilbert-Schmidt operators acting in H) for any u ∈ H. It is not the case here, therefore it is impossible to obtain theorems on existence and uniqueness of solution (weak, or mild) for the problem (3.3) (see, for example, Theorem 6.7, p. 164 in [5], Theorem 3.3, p. 97 in [6]). The way out can be found in setting the problem in the space of generalized Hilbert-spacevalued random variables (S) −ρ (H) ⊃ L 2 (Ω, F, P ; H), ρ ∈ [0; 1] (see the definition and properties of this space in [9]).…”

Section: Differential Equationmentioning

confidence: 99%

“…In section 3 we discuss difficulties that arise when we attempt to convert the difference equation into a stochastic differential equation in the Hilbert space H. We show that the use of the theory of Itô-type stochastic differential equations in infinite dimensional Hilbert spaces (see the review of the theory in [5,6]) is limited due to the properties of the operator coefficients in the difference equation obtained on the previous step. The necessary requirement for the operator-valued integrand of a well defined Itô integral with respect to a cylindrical Wiener process is the condition of being a Hilbert-Schmidt operator, which is not the case here.…”

Section: Introductionmentioning

confidence: 99%

A Model of Age–structured Population Under Stochastic Perturbation of Death and Birth Rates

Alshanskiy

2018

UMJ

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Under consideration is construction of a model of age-structured population reflecting random oscillations of the death and birth rate functions. We arrive at an Itô-type difference equation in a Hilbert space of functions which can not be transformed into a proper Itô equation via passing to the limit procedure due to the properties of the operator coefficients. We suggest overcoming the obstacle by building the model in a space of Hilbert space valued generalized random variables where it has the form of an operator-differential equation with multiplicative noise. The result on existence and uniqueness of the solution to the obtained equation is stated.

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Stochastic Differential Equations in Infinite Dimensions

Cited by 154 publications

References 0 publications

Ornstein–Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility

Ornstein–Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility

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

A Model of Age–structured Population Under Stochastic Perturbation of Death and Birth Rates

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