A Class of Lévy Driven SDEs and their Explicit Invariant Measures

Albeverio, Sergio; Persio, Luca Di; Mastrogiacomo, Elisa; Smii, Boubaker

doi:10.1007/s11118-016-9544-3

Cited by 24 publications

(13 citation statements)

References 130 publications

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“…Also the connections between the Hunt processes constructed by the non-local Dirichlet forms on infinite dimensional topological vector spaces and solutions of SPDEs seem to provide an area of possible extensions of our work (cf. e.g., [7] and references therein).…”

Section: Future Developmentsmentioning

confidence: 99%

Non-local Markovian Symmetric Forms on Infinite Dimensional Spaces I. The closability and quasi-regularity

Albeverio

Kagawa

Yahagi

et al. 2021

Commun. Math. Phys.

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General theorems on the closability and quasi-regularity of non-local Markovian symmetric forms on probability spaces (S, B(S), μ), with S Fréchet spaces such that S ⊂ R N , B(S) is the Borel σ -field of S, and μ is a Borel probability measure on S, are introduced. Firstly, a family of non-local Markovian symmetric forms E (α) , 0 < α < 2, acting in each given L 2 (S; μ) is defined, the index α characterizing the order of the non-locality. Then, it is shown that all the forms E (α) defined on n∈N C ∞ 0 (R n ) are closable in L 2 (S; μ). Moreover, sufficient conditions under which the closure of the closable forms, that are Dirichlet forms, become strictly quasi-regular, are given. Finally, an existence theorem for Hunt processes properly associated to the Dirichlet forms is given. The application of the above theorems to the problem of stochastic quantizations of Euclidean Φ 4 d fields, for d = 2, 3, by means of these Hunt processes is indicated.

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Section: Future Developmentsmentioning

confidence: 99%

Non-local Markovian Symmetric Forms on Infinite Dimensional Spaces I. The closability and quasi-regularity

Albeverio

Kagawa

Yahagi

et al. 2021

Commun. Math. Phys.

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“…Moreover it remains rather not analyzed the case of financial models aiming at capture the dynamics of structured derivatives characterized by jumps and abrupt changes in their sample paths. Concerning the latters, insights can be gained looking at the invariant measures associated to their theoretical counterpart, see, e.g., [1] and references therein, or [9] for a more financially oriented discussion.…”

Section: Resultsmentioning

confidence: 99%

“…In The results in terms of accuracy of the Delta estimation are displayed in Fig. (1) and in Table ( 1). The hardware used is a CPU Intel Core i5, 2.20 GHz x 2 and 8 GB of RAM.…”

Section: Numerical Simulationmentioning

confidence: 99%

Polynomial Chaos Expansion Approach to Malliavin Calculus Analysis of Bond Options Sensitivity

Marconi¹

2016

Int. J. of Pure and Appl. Math.

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Abstract:The present paper provides a sensitivity analysis of bond options exploiting the probabilistic properties of Malliavin Calculus and the computational benefits of the Polynomial Chaos Expansion. The purpose is to use the integration by parts formula of Malliavin Calculus in order to obtain the Greeks, of a given financial derivative, in a form which is suitable for numerical simulation. In particular, such computations will be performed usign the so called Polynomial Chaos Expansion technique, then comparing obtained results versus those retrieved using both the usual Monte-Carlo approach and the analytical formula.

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“…In the present paper we shall provide small noise asymptotic expansions for some local volatility models (LVMs) arising in finance. Our approach is based on the rigorous results on asymptotic expansions for solutions of finite dimensional SDE's obtained in [Albeverio et al(2013)] (following the approach proposed in [Gardiner (2004), Sec.6.2]); some extensions to a class of SPDE's and infinite dimensional SDE's have been presented in [Albeverio et al(2011), Albeverio et al(2016a), Albeverio et al(2016b)]. In particular we consider underlyings whose behavior is characterized by a stochastic volatility term of small amplitude ǫ with respect to which we perform a formal, based on [Gardiner (2004), Sec.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic expansion for some local volatility models arising in finance

Albeverio

Cordoni

Persio

et al. 2019

Decisions Econ Finan

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In this paper we study the small noise asymptotic expansions for certain classes of local volatility models arising in finance. We provide explicit expressions for the involved coefficients as well as accurate estimates on the remainders. Moreover, we perform a detailed numerical analysis, with accuracy comparisons, of the obtained results by mean of the standard Monte Carlo technique as well as exploiting the polynomial Chaos Expansion approach.

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A Class of Lévy Driven SDEs and their Explicit Invariant Measures

Cited by 24 publications

References 130 publications

Non-local Markovian Symmetric Forms on Infinite Dimensional Spaces I. The closability and quasi-regularity

Non-local Markovian Symmetric Forms on Infinite Dimensional Spaces I. The closability and quasi-regularity

Polynomial Chaos Expansion Approach to Malliavin Calculus Analysis of Bond Options Sensitivity

Asymptotic expansion for some local volatility models arising in finance

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