| 1. |
- Belomestny, Denis, et al.
(författare)
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Nonparametric Bayesian inference for Gamma-type Lévy subordinators
- 2019
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Ingår i: Communications in Mathematical Sciences. - 1539-6746 .- 1945-0796. ; 17:3, s. 781-816
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Tidskriftsartikel (refereegranskat)abstract
- Given discrete time observations over a growing time interval, we consider a nonparametric Bayesian approach to estimation of the Levy density of a Levy process belonging to a flexible class of infinite activity subordinators. Posterior inference is performed via MCMC, and we circumvent the problem of the intractable likelihood via the data augmentation device, that in our case relies on bridge process sampling via Gamma process bridges. Our approach also requires the use of a new infinite-dimensional form of a reversible jump MCMC algorithm. We show that our method leads to good practical results in challenging simulation examples. On the theoretical side, we establish that our nonparametric Bayesian procedure is consistent: in the low frequency data setting, with equispaced in time observations and intervals between successive observations remaining fixed, the posterior asymptotically, as the sample size n ->infinity, concentrates around the Levy density under which the data have been generated. Finally, we test our method on a classical insurance dataset.
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| 2. |
- Belomestny, Denis, et al.
(författare)
-
Nonparametric Bayesian volatility estimation for gamma-driven stochastic differential equations
- 2022
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Ingår i: Bernoulli. - 1350-7265. ; 28:4, s. 2151-2180
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Tidskriftsartikel (refereegranskat)abstract
- We study a nonparametric Bayesian approach to estimation of the volatility function of a stochastic differential equation driven by a gamma process. The volatility function is modelled a priori as piecewise constant, and we specify a gamma prior on its values. This leads to a straightforward procedure for posterior inference via an MCMC procedure. We give theoretical performance guarantees (minimax optimal contraction rates for the posterior) for the Bayesian estimate in terms of the regularity of the unknown volatility function. We illustrate the method on synthetic and real data examples.
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| 3. |
- Belomestny, Denis, et al.
(författare)
-
Weak solutions to gamma-driven stochastic differential equations
- 2023
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Ingår i: Indagationes Mathematicae. - 0019-3577. ; 34:4, s. 820-829
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Tidskriftsartikel (refereegranskat)abstract
- We study a stochastic differential equation driven by a gamma process, for which we give results on the existence of weak solutions under conditions on the volatility function. To that end we provide results on the density process between the laws of solutions with different volatility functions.
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| 4. |
- Gugushvili, Shota, et al.
(författare)
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Nonparametric Bayesian estimation of a Hölder continuous diffusion coefficient
- 2020
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Ingår i: Brazilian Journal of Probability and Statistics. - 0103-0752. ; 34:3, s. 537-579
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Tidskriftsartikel (refereegranskat)abstract
- We consider a nonparametric Bayesian approach to estimate the diffusion coefficient of a stochastic differential equation given discrete time observations over a fixed time interval. As a prior on the diffusion coefficient, we employ a histogram-type prior with piecewise constant realisations on bins forming a partition of the time interval. Specifically, these constants are realizations of independent inverse Gamma distributed randoma variables. We justify our approach by deriving the rate at which the corresponding posterior distribution asymptotically concentrates around the data-generating diffusion coefficient. This posterior contraction rate turns out to be optimal for estimation of a Hölder-continuous diffusion coefficient with smoothness parameter 0<λ≤1. Our approach is straightforward to implement, as the posterior distributions turn out to be inverse Gamma again, and leads to good practical results in a wide range of simulation examples. Finally, we apply our method on exchange rate data sets.
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| 5. |
- Gugushvili, Shota, et al.
(författare)
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Nonparametric Bayesian volatility estimation
- 2019
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Ingår i: 2017 MATRIX Annals. - Cham : Springer International Publishing. - 9783030041601 ; , s. 279-302
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Bokkapitel (övrigt vetenskapligt/konstnärligt)
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