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Konstantinos Dareiotis

Publications and source records attributed to Konstantinos Dareiotis.

27 records · Page 2Linked to original sources
arXiv2018

Density symmetries for a class of 2-D diffusions with applications to finance

We study densities of two-dimensional diffusion processes with one non-negative component. For such diffusions, the density may explode at the boundary, thus making a precise specification of the boundary condition in the corresponding forward Kolmogorov equation problematic. We overcome this by extending a classical symmetry result for densities of one-dimensional diffusions to our case, thereby reducing the study of forward equations with exploding boundary data to the study of a related backward equation with non-exploding boundary data. We also discuss important applications of this symmetry for option pricing in stochastic volatility models and in stochastic short rate models.

math.PR↗
arXiv2016

Local $L_\infty$-estimates, weak Harnack inequality, and stochastic continuity of solutions of SPDEs

We consider stochastic partial differential equations under minimal assumptions: the coefficients are merely bounded and measurable and satisfy the stochastic parabolicity condition. In particular, the diffusion term is allowed to be scaling-critical. We derive local supremum estimates with a stochastic adaptation of De Giorgi's iteration and establish a weak Harnack inequality for the solutions. The latter is then used to obtain pointwise almost sure continuity.

math.PR↗
arXiv2016

On Finite difference schemes for partial integro-differential equations of Lévy type

In this article we introduce a finite difference approximation for integro-differential operators of Lévy type. We approximate solutions of integro-differential equations, where the second order operator is allowed to degenerate. In the existing literature, the Lévy operator is treated as a zero/first order operator outside of a centered ball of radius $δ$, leading to error estimates of order $ξ(δ)+N(δ)(h+\sqrtτ)$, where $h$ and $τ$ are the spatial and temporal discretization parameters respectively. In these estimates $ξ(δ) \downarrow 0$, but $N(δ)\uparrow \infty$ as $δ\downarrow 0$. In contrast, we treat the integro-differential operator as a second order operator on the whole unit ball. By this method we obtain error estimates of order $(h+τ^k)$ for $k\in \{1/2,1\}$, eliminating the additional errors and the blowing up constants. Moreover, we do not pose any conditions on the Lévy measure.

math.NA↗
arXiv2016

Symmetrization of exterior parabolic problems and probabilistic interpretation

We prove a comparison theorem for the averages of the solutions of two exterior parabolic problems, the second being the "symmetrization" of the first one, by using approximation of the Schwarz symmetrization by polarizations, as it was introduced in [4]. This comparison provides an alternative proof, based on PDEs, of the isoperimetric inequality for the Wiener sausage, which was proved in [14].

math.AP↗
arXiv2015

On Tamed Euler Approximations of SDEs Driven by Lévy Noise with Applications to Delay Equations

We extend the taming techniques for explicit Euler approximations of stochastic differential equations (SDEs) driven by Lévy noise with super-linearly growing drift coefficients. Strong convergence results are presented for the case of locally Lipschitz coefficients. Moreover, rate of convergence results are obtained in agreement with classical literature when the local Lipschitz continuity assumptions are replaced by global and, in addition, the drift coefficients satisfy polynomial Lipschitz continuity. Finally, we further extend these techniques to the case of delay equations.

math.PR↗
arXiv2015

A comparison principle for stochastic integro-differential equations

A comparison principle for stochastic integro-differential equations driven by Levy processes is proved. This result is obtained via an extension of an Ito formula from [11] for the square of the norm of the positive part of $L_2-$valued, continuous semimartingales, to the case of discontinuous semimartingales.

math.PR↗
arXiv2015

On the boundedness of solutions of SPDEs

In this paper estimates for the uniform norm of solutions of parabolic SPDEs are derived. The result is obtained through iteration techniques, motivated by the work of Moser in deterministic settings. As an application of the main result, solvability of a class of semilinear SPDEs is derived.

math.PR↗