The Projection-Pursuit Multivariate Transform for Improved Continuous Variable Modeling

Summary Reservoir process-performance evaluation requires the simulation of multiple continuous variables such as porosity, water saturation, and permeability. Geostatistical realizations should reproduce the univariate and multivariate statistics that are deemed representative of the reservoir. A conventional work flow that sequentially applies cosimulation and cloud transformations is frequently used for this multivariate simulation. Although it effectively reproduces univariate properties, a common issue with this work flow is its inability to reproduce all the multivariate relationships that exist between variables. To resolve this issue, the projection-pursuit multivariate transform (PPMT) is applied to reservoir modeling. The PPMT work flow requires fewer steps, no manual tuning, and fewer assumptions than the conventional work flow. Background, essential steps, and practical considerations of the conventional and PPMT work flows are outlined before comparing them in a case study. The PPMT is shown to yield multivariate reproduction that is expected to improve reservoir forecasting.

A MULTIVARIATE PURE-JUMP MODEL WITH MULTI-FACTORIAL DEPENDENCE STRUCTURE

In this work we propose a new approach to build multivariate pure jump processes. We introduce linear and nonlinear dependence, without restrictions on marginal properties, by imposing a multi-factorial structure separately on both positive and negative jumps. Such a new approach provides higher flexibility in calibrating nonlinear dependence than in other comparable Lévy models in the literature. Using the notion of multivariate subordinator, this modeling approach can be applied to the class of univariate Lévy processes which can be written as the difference of two subordinators. A common example in the financial literature is the variance gamma process, which we extend to the multivariate (multi-factorial) case. The model is tractable and a straightforward multivariate simulation procedure is available. An empirical analysis documents an accurate multivariate fit of stock index returns in terms of both linear and nonlinear dependence. An example of multi-asset option pricing emphasizes the importance of the proposed multivariate approach.