Abstract
In recent years subordinated processes have been widely considered in the literature. These processes not only have wide applications but also have interesting theoretical properties. In this paper we consider fractional Brownian motion (FBM) time-changed by two processes, tempered stable and inverse tempered stable. We present main properties of the subordinated FBM such as long range dependence and associated fractional partial differential equations for the probability density functions. Moreover, we present how to simulate both subordinated processes.
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This work is supported by National Center of Science Opus Grant No. 2016/21/B/ST1/ 00929 “Anomalous diffusion processes and their applications in real data modelling”.
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Appendix A
Appendix A
1.1 Laplace-Erdelyi Theorem
Consider the integral
where (a, b) is a real (finite or infinite) interval. Further, t is large positive and the functions f and g are continuous. Suppose f have a single minima at x = a and functions f and g are such that
and
Then, the asymptotic expansion of the integral I(t) is given by
where cn is given by
where \(\hat {B}_{j,i}\) are the partial (or incomplete) ordinary Bell polynomials (Andrews 1998).
1.2 Partial Exponential Bell Polynomials
The partial (or incomplete) exponential Bell polynomials are a triangular array of polynomials given by
where the sum is taken over the sequences satisfying
where j1, j2, j3,..., jn−k+ 1 are non-negative integers. Further, the sum
is called the n-th complete exponential Bell polynomial.
1.3 Partial Ordinary Bell Polynomial
The partial ordinary Bell polynomial is given by
where the sum runs over all sequences j1, j2, j3,⋯ , jn−k+ 1 of non-negative integers such that
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Kumar, A., Gajda, J., Wyłomańska, A. et al. Fractional Brownian Motion Delayed by Tempered and Inverse Tempered Stable Subordinators. Methodol Comput Appl Probab 21, 185–202 (2019). https://doi.org/10.1007/s11009-018-9648-x
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DOI: https://doi.org/10.1007/s11009-018-9648-x
Keywords
- Subordination
- Tempered stable process
- Inverse tempered stable process
- Fractional Brownian motion
- Simulation