Master the art of simulating stochastic processes with comprehensive recipes for processes including the Brownian bridge and fractional Brownian motion.
Essential methods for simulating Gaussian processes
The classical Wiener process, foundation of stochastic calculus and Gaussian process simulation.
Conditioned Brownian motion that starts and ends at fixed values, useful for path-dependent simulations.
Long-memory Gaussian process with tunable Hurst parameter for modeling persistence and roughness.
Causal representation of fractional Brownian motion with explicit filtration structure and iterative construction.
Selected research papers and preprints by Chef Roman Paolucci
This note discusses the necessary steps to simulate a stochastic process with a desired covariance structure. In this context, I outline the Karhunen–Loéve theorem and provide intuition and general recipes to decompose a stochastic process by establishing the integral eigenvalue problem (continuous-time) or, equivalently, the covariance matrix diagonalization problem (discrete-time). I then apply these general recipes to a Brownian motion and Brownian bridge to numerically simulate paths.
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