The Symbol Associated with the Solution of a Stochastic Differential Equation

Schilling, René L.; Schnurr, Alexander

doi:10.1214/ejp.v15-807

Cited by 73 publications

(77 citation statements)

References 27 publications

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“…We refer to [29,30] where many examples of this kind are studied. Moreover, we refer to [27] and, in particular, to [39,40] where q(x, ξ) was calculated as q(x, ξ) = − lim t→0 E x e iξ(Xt−x) − 1 t .…”

Section: Towards a Geometric Understanding Of Transition Functions Ofmentioning

confidence: 99%

A geometric interpretation of the transition density of a symmetric Lévy process

et al. 2012

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Abstract. We study for a class of symmetric Lévy processes with state space R n the transition density p t (x) in terms of two one-parameter families of metrics, (d t ) t>0 and (δ t ) t>0 . The first family of metrics describes the diagonal term p t (0); it is induced by the characteristic exponent ψ of the Lévy process by d t (x, y) = tψ(x − y). The second and new family of metrics δ t relates to √ tψ through the formulawhere F denotes the Fourier transform. Thus we obtain the following "Gaussian" representation of the transition density:t (x,0) where p t (0) corresponds to a volume term related to √ tψ and where an "exponential" decay is governed by δ 2 t . This gives a complete and new geometric, intrinsic interpretation of p t (x). MSC 2010: Primary: 60J35. Secondary: 60E07; 60E10; 60G51; 60J45; 47D07; 31E05.

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Section: Towards a Geometric Understanding Of Transition Functions Ofmentioning

confidence: 99%

A geometric interpretation of the transition density of a symmetric Lévy process

et al. 2012

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“…Note that λ t (q, p) = e −tH(q,p) in general. 3 Some possibilities of constructing and approximating the symbol of the semigroup starting from the symbol of the generator can be found in [6,13,37]. Many sufficient conditions for a function −H(q, p) to be the 1-symbol of a Feller generator are known.…”

Section: Hamiltonian Feynman Formula For Feller Semigroupsmentioning

confidence: 99%

Feynman formulas and path integrals for some evolution semigroups related to τ-quantization

Böttcher

Butko

Schilling

et al. 2011

Russ. J. Math. Phys.

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This note is devoted to Feynman formulas (i.e., representations of semigroups by limits of n-fold iterated integrals as n → ∞) and their connections with phase space Feynman path integrals. Some pseudodifferential operators corresponding to different types of quantization of a quadratic Hamiltonian function are considered. Lagrangian and Hamiltonian Feynman formulas for semigroups generated by these operators are obtained. Further, a construction of Hamiltonian (phase space) Feynman path integrals is introduced. Due to this construction, the Hamiltonian Feynman formulas obtained here and in our previous papers do coincide with Hamiltonian Feynman path integrals. This connects phase space Feynman path integrals with some integrals with respect to probability measures. These connections enable us to make a contribution to the theory of phase space Feynman path integrals, to prove the existence of some of these integrals, and to study their properties by means of stochastic analysis. The Feynman path integrals thus obtained are different for different types of quantization. This makes it possible to distinguish the process of quantization in the language of Feynman path integrals.

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“…[16], chapter 8) it is sometimes hard to transform it into the Skorokhod-type. The symbol on the other hand can occasionally be written down directly and in a neat way: In [21], it is shown that the symbol of the solution of the Lévy driven SDE X t = X 0 + t 0 f X s− dZ s is f x where is the characteristic exponent of the driving Lévy process Z s . (iv) While the coefficients depend on the choice of the SDE-type and the truncation function (in (2) we have chosen 1 u ≤1 ; someone interested in limit theorems would probably choose a continuous function), this is not the case for the symbol.…”

Section: Böttcher and Schnurrmentioning

confidence: 99%

“…In [22], theorem 5.7 (see also [21]), it is shown that, given (6), the Itô process X t t≥0 has the symbol p d × d → given by…”

Section: Proof Of the Theoremmentioning

confidence: 99%

The Euler Scheme for Feller Processes

Böttcher

Schnurr

2011

Stochastic Analysis and Applications

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We consider the Euler scheme for stochastic differential equations with jumps, whose intensity might be infinite and the jump structure may depend on the position. This general type of SDE is explicitly given for Feller processes and a general convergence condition is presented.In particular, the characteristic functions of the increments of the Euler scheme are calculated in terms of the symbol of the Feller process in a closed form. These increments are increments of Lévy processes and, thus, the Euler scheme can be used for simulation by applying standard techniques from Lévy processes.

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The Symbol Associated with the Solution of a Stochastic Differential Equation

Cited by 73 publications

References 27 publications

A geometric interpretation of the transition density of a symmetric Lévy process

A geometric interpretation of the transition density of a symmetric Lévy process

Feynman formulas and path integrals for some evolution semigroups related to τ-quantization

The Euler Scheme for Feller Processes

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