Description Details References

Parametric estimation in mixed-effects stochastic differential equations

This package is dedicated to parametric estimation in the following mixed-effects stochastic differential equations:

*dX_j(t)= (α_j- β_j X_j(t))dt + σ_j \ a(X_j(t)) dW_j(t),*

*j=1,…,M*, where the *(W_j(t))* are independant Wiener processes and the *(X_j(t))* are observed without noise. The volatility function *a(x)* is known and can be either *a(x)=1* (Ornstein-Uhlenbeck process) or *a(x)=√{x}* (Cox-Ingersoll-Ross process).

Different estimation methods are implemented depending on whether there are random effects in the drift and/or in the diffusion coefficient:

The diffusion coefficient is fixed

*σ_j \equiv σ*and the parameters in the drift are Gaussian random variables:either

*α_j \equiv α*and*β_j \sim N(μ,Ω), j=1,…,M*,or

*β_j \equiv β*and*α_j \sim N(μ,Ω), j=1,…,M*,or

*(α_j,β_j) \sim N(μ,Ω), j=1,…,M*.

*μ*,*Ω*and potentially the fixed effects*σ*,*α*,*β*are estimated as proposed in [1] and [4]. The extension to mixtures of Gaussian distributions is also implemented by following [3].The coefficients in the drift are fixed

*α_j \equiv α*and*β_j \equiv β*and the diffusion coefficient*1/σ_j^2*follows a Gamma distribution*1/σ_j^2 \sim Γ(a,λ), j=1,…,M*.*a*,*λ*, and potentially the fixed effects*α*and*β*are estimated by the method published in [2].There are random effects in the drift and in the diffusion, such that

*1/σ_j^2 \sim Γ(a,λ)*andeither

*α_j \equiv α*and*β_j|σ_j \sim N(μ,σ_j^2 Ω)*,or

*β_j \equiv β*and*α_j|σ_j \sim N(μ,σ_j^2 Ω)*,or

*(α_j,β_j)|σ_j \sim N(μ,σ_j^2 Ω)*.

*a*,*λ*,*μ*,*Ω*and potentially the fixed effects*α*and*β*are estimated by following [5].

**[1]** Maximum Likelihood Estimation for Stochastic Differential Equations with Random Effects, Delattre, M., Genon-Catalot, V. and Samson, A. *Scandinavian Journal of Statistics 40(2) 2012* **322-343**.

**[2]** Estimation of population parameters in stochastic differential equations with random effects in the diffusion coefficient, Delattre, M., Genon-Catalot, V. and Samson, A. *ESAIM:PS 19 2015* **671-688**.

**[3]** Mixtures of stochastic differential equations with random effects: application to data clustering, Delattre, M., Genon-Catalot, V. and Samson, A. *Journal of Statistical Planning and Inference 173 2016* **109-124**.

**[4]** Parametric inference for discrete observations of diffusion processes with mixed effects, Delattre, M., Genon-Catalot, V. and Laredo, C. *Stochastic Processes and their Applications*. To appear. (Available pre-publication hal-0133263, 2016).

**[5]** Estimation of the joint distribution of random effects for a discretely observed diffusion with random effects, Delattre, M., Genon-Catalot, V. and Laredo, C. *hal-01446063 2017*.

Embedding an R snippet on your website

Add the following code to your website.

For more information on customizing the embed code, read Embedding Snippets.