R_t^alpha = Gamma(alpha + 1/2)^(-1) int_0^t (t - s)^(alpha - 1/2) dB_s
is Gaussian and self-similar of index alpha, so var(R_t) is
proportional to t^(2 alpha). Fractional integration of white noise is
how the chapter builds a process of any prescribed smoothness, and the
variance-growth exponent is the observable signature of that
smoothness; it is recovered here from a discretised stochastic
integral.
Usage
Ghosalrlprocess(alpha = 0.75, n_grid = 200, n_sim = 300, seed = 42)
Arguments
- alpha
Self-similarity index, positive.
- n_grid
Discretisation of [0, 1].
- n_sim
Number of simulated paths.
- seed
Seed for the deterministic draws.
Value
List with estimate (empirical growth exponent),
expected, gap, method.
Details
Formula: growth = log(var(R_1) / var(R_{1/4})) / log 4, which equals
2 alpha exactly for the continuum process.
References
Ghosal & van der Vaart (2017), Fundamentals of
Nonparametric Bayesian Inference, CUP, Example 11.6, eq. (11.2).
Examples
Ghosalrlprocess()
#> $estimate
#> [1] 1.608322
#>
#> $expected
#> [1] 1.5
#>
#> $gap
#> [1] 0.1083215
#>
#> $method
#> [1] "Riemann-Liouville process (GvdV 2017 eq. 11.2)"
#>
#> attr(,"class")
#> [1] "morie_rich_result" "list"
