1. Sample paths of a Gaussian process with the exponential covariance function are not smooth.2. :i . e ., all adapted processes with absolutely continuous sample paths . 3. Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. 4. This leads to a sample path of 5. Approximate sample paths of the Langevin diffusion can be generated by many discrete-time methods. 6. Instead of generating random paths, new sampling paths are created as slight mutations of existing ones. 7. A prevalent example of the controlled path X _ t is the sample path of a Wiener process. 8. This last characterization gives an understanding of the structure of the sample path with location and sizes of jumps. 9. An example of a continuous-time stochastic process for which sample paths are not continuous is a Poisson process. 10. The sample paths chosen can be thought of as showing discrete sampled points on an " fBm " process.
Mobile