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Approximation of stochastic differential equations driven by α-stable Lévy motion

Aleksander Janicki, Zbigniew Michna, Aleksander Weron (1997)

Applicationes Mathematicae

In this paper we present a result on convergence of approximate solutions of stochastic differential equations involving integrals with respect to α-stable Lévy motion. We prove an appropriate weak limit theorem, which does not follow from known results on stability properties of stochastic differential equations driven by semimartingales. It assures convergence in law in the Skorokhod topology of sequences of approximate solutions and justifies discrete time schemes applied in computer simulations....

Approximation of the Snell envelope and american options prices in dimension one

Vlad Bally, Bruno Saussereau (2002)

ESAIM: Probability and Statistics

We establish some error estimates for the approximation of an optimal stopping problem along the paths of the Black–Scholes model. This approximation is based on a tree method. Moreover, we give a global approximation result for the related obstacle problem.

Approximation of the Snell Envelope and American Options Prices in dimension one

Vlad Bally, Bruno Saussereau (2010)

ESAIM: Probability and Statistics

We establish some error estimates for the approximation of an optimal stopping problem along the paths of the Black–Scholes model. This approximation is based on a tree method. Moreover, we give a global approximation result for the related obstacle problem.

Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes

Bernard Roynette, Pierre Vallois, Agnès Volpi (2008)

ESAIM: Probability and Statistics

Let ( X t , t 0 ) be a Lévy process started at 0 , with Lévy measure ν . We consider the first passage time T x of ( X t , t 0 ) to level x > 0 , and K x : = X T x - 𝑥 the overshoot and L x : = x - X T 𝑥 - the undershoot. We first prove that the Laplace transform of the random triple ( T x , K x , L x ) satisfies some kind of integral equation. Second, assuming that ν admits exponential moments, we show that ( T x ˜ , K x , L x ) converges in distribution as x , where T x ˜ denotes a suitable renormalization of T x .

Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes

Bernard Roynette, Pierre Vallois, Agnès Volpi (2007)

ESAIM: Probability and Statistics

Let (Xt, t ≥ 0) be a Lévy process started at 0, with Lévy measure ν. We consider the first passage time Tx of (Xt, t ≥ 0) to level x > 0, and Kx := XTx - x the overshoot and Lx := x- XTx- the undershoot. We first prove that the Laplace transform of the random triple (Tx,Kx,Lx) satisfies some kind of integral equation. Second, assuming that ν admits exponential moments, we show that ( T x ˜ , K x , L x ) converges in distribution as x → ∞, where T x ˜ denotes a suitable renormalization of Tx.


Asymptotic Feynman–Kac formulae for large symmetrised systems of random walks

Stefan Adams, Tony Dorlas (2008)

Annales de l'I.H.P. Probabilités et statistiques

We study large deviations principles for N random processes on the lattice ℤd with finite time horizon [0, β] under a symmetrised measure where all initial and terminal points are uniformly averaged over random permutations. That is, given a permutation σ of N elements and a vector (x1, …, xN) of N initial points we let the random processes terminate in the points (xσ(1), …, xσ(N)) and then sum over all possible permutations and initial points, weighted with an initial distribution. We prove level-two...

Asymptotic windings over the trefoil knot.

Jacques Franchi (2005)

Revista Matemática Iberoamericana

Consider the group G:=PSL2(R) and its subgroups Γ:= PSL2(Z) and Γ':=DSL2(Z). G/Γ is a canonical realization (up to an homeomorphism) of the complement S3T of the trefoil knot T, and G/Γ' is a canonical realization of the 6-fold branched cyclic cover of S3T, which has a 3-dimensional cohomology of 1-forms.Putting natural left-invariant Riemannian metrics on G, it makes sense to ask which is the asymptotic homology performed by the Brownian motion in G/Γ', describing thereby in an intrinsic way part...

Asymptotics for the L p -deviation of the variance estimator under diffusion

Paul Doukhan, José R. León (2004)

ESAIM: Probability and Statistics

We consider a diffusion process X t smoothed with (small) sampling parameter ε . As in Berzin, León and Ortega (2001), we consider a kernel estimate α ^ ε with window h ( ε ) of a function α of its variance. In order to exhibit global tests of hypothesis, we derive here central limit theorems for the L p deviations such as 1 h h ε p 2 α ^ ε - α p p - 𝔼 α ^ ε - α p p .

Asymptotics for the Lp-deviation of the variance estimator under diffusion

Paul Doukhan, José R. León (2010)

ESAIM: Probability and Statistics

We consider a diffusion process Xt smoothed with (small) sampling parameter ε. As in Berzin, León and Ortega (2001), we consider a kernel estimate α ^ ε with window h(ε) of a function α of its variance. In order to exhibit global tests of hypothesis, we derive here central limit theorems for the Lp deviations such as 1 h h ε p 2 α ^ ε - α p p - I E α ^ ε - α p p .

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