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Torstein Nilssen

Publications and source records attributed to Torstein Nilssen.

26 records · Page 2Linked to original sources
arXiv2018

Rough linear PDE's with discontinuous coefficients - existence of solutions via regularization by fractional Brownian motion

We consider two related linear PDE's perturbed by a fractional Brownian motion. We allow the drift to be discontinuous, in which case the corresponding deterministic equation is ill-posed. However, the noise will be shown to have a regularizing effect on the equations in the sense that we can prove existence of solutions for almost all paths of the fractional Brownian motion.

math.PR↗
arXiv2018

Strong existence and higher order Fréchet differentiability of stochastic flows of fractional Brownian motion driven SDE's with singular drift

In this paper we present a new method for the construction of strong solutions of SDE's with merely integrable drift coefficients driven by a multidimensional fractional Brownian motion with Hurst parameter H < 1/2. Furthermore, we prove the rather surprising result of the higher order Frechet differentiability of stochastic flows of such SDE's in the case of a small Hurst parameter. In establishing these results we use techniques from Malliavin calculus combined with new ideas based on a "local time variational calculus". We expect that our general approach can be also applied to the study of certain types of stochastic partial differential equations as e.g. stochastic conservation laws driven by rough paths.

math.PR↗
arXiv2018

Weak solutions of rough path SDE's via Girsanov

We consider a differential equation driven by a Brownian motion as well as a rough path. We prove a Girsanov-type result for this equation to construct a weak solution in the probabilistic sense.

math.PR↗
arXiv2018

Generalized Burgers equation with rough transport noise

We introduce a new technique for studying well posedness and energy estimates for evolution equations with a rough transport term. The technique is based on finding suitable space-time weight functions for the equations at hand. As an example we study the well posedness of the generalized viscous Burgers equation perturbed by a rough path transport noise.

math.PR↗
arXiv2015

Malliavin and flow regularity of SDEs. Application to the study of densities and the stochastic transport equation

In this work we present a condition for the regularity, in both space and Malliavin sense, of strong solutions to SDEs driven by Brownian motion. We conjecture that this condition is optimal. As a consequence, we are able to improve the regularity of densities of such solutions. We also apply these results to construct a classical solution to the stochastic transport equation when the drift is Lipschitz.

math.PR↗
arXiv2014

Quasi-linear Stochastic Partial Differential Equations with irregular Coefficients - Malliavin regularity of the Solutions

We study quasi-linear stochastic partial differential equations with discontinuous drift coefficients. Existence and uniqueness of a solution is already known under weaker conditions on the drift, but we are interested in the regularity of the solution in terms of Malliavin calculus. We prove that when the drift is bounded and measurable the solution is directional Malliavin differentiable.

math.PR↗
arXiv2014

A Note on the Malliavin Differentiability of One-Dimensional Reflected Stochastic Differential Equations with Discontinuous Drift

We consider a one-dimensional Stochastic Differential Equation with reflection where we allow the drift to be merely bounded and measurable. It is already known that such equations have a unique strong solution. Recently, it has been shown that non-reflected SDE's with discontinuous drift possess more regularity than one could expect, namely they are Malliavin differentiable and weakly differentiable w.r.t. the initial value. In this paper we show that similar results hold for one-dimensional SDE's with reflection. We then apply the results to get a Bismut-Elworthy-Li formula for the corresponding Kolmogorov equation.

math.PR↗