Essays on strong and weak approximations of stochastic differential equations

The thesis is composed of two projects on approximations of stochastic differential equations. In the first project, we present a method to construct positivity-preserving strong approximation schemes for jump-extended CEV and CIR processes where the jumps are governed by a compensated spectrally positive alpha-stable process with alpha in (1, 2). To the best of our knowledge, the proposed scheme is the first of its kind, i.e. a positivity preserving scheme for alpha-stable-extended CEV processes, and it has the advantage that at each discretisation step, an explicit form of the scheme is available and it's given by the positive solution of a quadratic equation. We show that the proposed scheme converges and theoretically achieves a strong convergence rate that we believe is very close to the optimal rate. The second project is on the weak approximation and density estimates for a skew diffusion with coefficients depending on its local time at zero. In the existing literature, the parametrix method has been applied to obtain density estimates for skew diffusion processes, and more recently, for Ito diffusion processes with coefficients depending on local time. The goal of the second project is to extend the class of processes for which one can apply the parametrix method. As our main contribution, We obtain an explicit representation and Gaussian estimates of the joint transition density, which can lead to an exact simulation method for the skew diffusion process and it's local time at zero

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282