6 research outputs found
Spectral Density-Based and Measure-Preserving ABC for partially observed diffusion processes. An illustration on Hamiltonian SDEs
Approximate Bayesian Computation (ABC) has become one of the major tools of
likelihood-free statistical inference in complex mathematical models.
Simultaneously, stochastic differential equations (SDEs) have developed to an
established tool for modelling time dependent, real world phenomena with
underlying random effects. When applying ABC to stochastic models, two major
difficulties arise. First, the derivation of effective summary statistics and
proper distances is particularly challenging, since simulations from the
stochastic process under the same parameter configuration result in different
trajectories. Second, exact simulation schemes to generate trajectories from
the stochastic model are rarely available, requiring the derivation of suitable
numerical methods for the synthetic data generation. To obtain summaries that
are less sensitive to the intrinsic stochasticity of the model, we propose to
build up the statistical method (e.g., the choice of the summary statistics) on
the underlying structural properties of the model. Here, we focus on the
existence of an invariant measure and we map the data to their estimated
invariant density and invariant spectral density. Then, to ensure that these
model properties are kept in the synthetic data generation, we adopt
measure-preserving numerical splitting schemes. The derived property-based and
measure-preserving ABC method is illustrated on the broad class of partially
observed Hamiltonian type SDEs, both with simulated data and with real
electroencephalography (EEG) data. The proposed ingredients can be incorporated
into any type of ABC algorithm and directly applied to all SDEs that are
characterised by an invariant distribution and for which a measure-preserving
numerical method can be derived.Comment: 35 pages, 21 figure
Qualitative properties of different numerical methods for the inhomogeneous geometric Brownian motion
We provide a comparative analysis of qualitative features of different numerical methods for the inhomogeneous geometric Brownian motion (IGBM). The limit distribution of the IGBM exists, its conditional and asymptotic mean and variance are known and the process can be characterised according to Feller’s boundary classification. We compare the frequently used Euler–Maruyama and Milstein methods, two Lie–Trotter and two Strang splitting schemes and two methods based on the ordinary differential equation (ODE) approach, namely the classical Wong–Zakai approximation and the recently proposed log-ODE scheme. First, we prove that, in contrast to the Euler–Maruyama and Milstein schemes, the splitting and ODE schemes preserve the boundary properties of the process, independently of the choice of the time discretisation step. Second, we prove that the limit distribution of the splitting and ODE methods exists for all stepsize values and parameters. Third, we derive closed-form expressions for the conditional and asymptotic means and variances of all considered schemes and analyse the resulting biases. While the Euler–Maruyama and Milstein schemes are the only methods which may have an asymptotically unbiased mean, the splitting and ODE schemes perform better in terms of variance preservation. The Strang schemes outperform the Lie–Trotter splittings, and the log-ODE scheme the classical ODE method. The mean and variance biases of the log-ODE scheme are very small for many relevant parameter settings. However, in some situations the two derived Strang splittings may be a better alternative, one of them requiring considerably less computational effort than the log-ODE method. The proposed analysis may be carried out in a similar fashion on other numerical methods and stochastic differential equations with comparable features