### A Brownian sheet martingale with the same marginals as the arithmetic average of geometric Brownian motion.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

A generic control variate method is proposed to price options under stochastic volatility models by Monte Carlo simulations. This method provides a constructive way to select control variates which are martingales in order to reduce the variance of unbiased option price estimators. We apply a singular and regular perturbation analysis to characterize the variance reduced by martingale control variates. This variance analysis is done in the regime where time scales of associated driving volatility...

Let (Ω, $\mathcal{F}$, (${\mathcal{F}}_{t}$)t≥0, $\mathbb{P}$) be a filtered probability space satisfying the usual assumptions: it is usually not possible to extend to ${\mathcal{F}}_{\infty}$ (theσ-algebra generated by (${\mathcal{F}}_{t}$)t≥0) a coherent family of probability measures (${\mathbb{Q}}_{t}$) indexed byt≥0, each of them being defined on ${\mathcal{F}}_{t}$. It is known that for instance, on the Wiener space, this extension problem has a positive answer if one takes the filtration generated by the coordinate process, made right-continuous, but can have a negative answer if one takes its usual augmentation....

Let (Ω, $\mathcal{F}$, (${\mathcal{F}}_{t}$)t≥0, $\mathbb{P}$) be a filtered probability space satisfying the usual assumptions: it is usually not possible to extend to ${\mathcal{F}}_{\infty}$ (the σ-algebra generated by (${\mathcal{F}}_{t}$)t≥0) a coherent family of probability measures (${\mathbb{Q}}_{t}$) indexed by t≥0, each of them being defined on ${\mathcal{F}}_{t}$. It is known that for instance, on the Wiener space, this extension problem has a positive answer if one takes the filtration generated by the coordinate process, made right-continuous, but can have a negative answer if one takes its usual...

In this paper, we present a new proof of the celebrated theorem of Kellerer, stating that every integrable process, which increases in the convex order, has the same one-dimensional marginals as a martingale. Our proof proceeds by approximations, and calls upon martingales constructed as solutions of stochastic differential equations. It relies on a uniqueness result, due to Pierre, for a Fokker-Planck equation.