arXiv subjects

Gerard Brunick

Publications and source records attributed to Gerard Brunick.

2 recordsLinked to original sources

arXiv2013

Mimicking an Itô process by a solution of a stochastic differential equation

Given a multi-dimensional Itô process whose drift and diffusion terms are adapted processes, we construct a weak solution to a stochastic differential equation that matches the distribution of the Itô process at each fixed time. Moreover, we show how to match the distributions at each fixed time of functionals of the Itô process, including the running maximum and running average of one of the components of the process. A consequence of this result is that a wide variety of exotic derivative securities have the same prices when the underlying asset price is modeled by the original Itô process or the mimicking process that solves the stochastic differential equation.

math.PR↗

arXiv2011

Uniqueness in Law for a Class of Degenerate Diffusions with Continuous Covariance

We study the martingale problem associated with the operator $L u = \partial_s u + 1/2 \sum_{i,j=1}^{d_0} a^{ij} \partial_{ij} u + \sum_{i,j=1}^d B^{ij} x^j \partial_i u$, where $d_0 \leq d$. We show that the martingale problem is well-posed when the function $a$ is continuous and strictly positive-definite on $\bb R^{d_0}$ and the matrix $B$ takes a particular lower-diagonal, block form. We then localize this result to show that the martingale problem remains well-posed when $B$ is replaced by a sufficiently smooth vector field whose Jacobian matrix satisfies a nondegeneracy condition.

math.PR↗