# mixedsde.sim: Simulation Of A Mixed Stochastic Differential Equation In mixedsde: Estimation Methods for Stochastic Differential Mixed Effects Models

## Description

Simulation of M independent trajectories of a mixed stochastic differential equation (SDE) with linear drift and two random effects (α_j, β_j) dX_j(t)= (α_j- β_j X_(t))dt + σ a(X_j(t)) dW_j(t), for j=1, ..., M.

## Usage

 ```1 2 3``` ```mixedsde.sim(M, T, N = 100, model, random, fixed = 0, density.phi, param, sigma, t0 = 0, X0 = 0.01, invariant = 0, delta = T/N, op.plot = 0, add.plot = FALSE) ```

## Arguments

 `M` number of trajectories `T` horizon of simulation. `N` number of simulation steps, default Tx100. `model` name of the SDE: 'OU' (Ornstein-Uhlenbeck) or 'CIR' (Cox-Ingersoll-Ross). `random` random effects in the drift: 1 if one additive random effect, 2 if one multiplicative random effect or c(1,2) if 2 random effects. `fixed` fixed effects in the drift: value of the fixed effect when there is only one random effect, 0 otherwise. If random =2, fixed can be 0 but β has to be a non negative random variable for the estimation. `density.phi` name of the density of the random effects. `param` vector of parameters of the distribution of the two random effects. `sigma` diffusion parameter `t0` time origin, default 0. `X0` initial value of the process, default X0=0. `invariant` 1 if the initial value is simulated from the invariant distribution, default 0.01 and X0 is fixed. `delta` time step of the simulation (T/N). `op.plot` 1 if a plot of the trajectories is required, default 0. `add.plot` 1 for add trajectories to an existing plot

## Details

Simulation of M independent trajectories of the SDE (the Brownian motions Wj are independent), with linear drift. Two diffusions are implemented, with one or two random effects:

#### Ornstein-Uhlenbeck model (OU)

If random = 1, β is a fixed effect: dX_j(t)= (α_j- β X_j(t))dt + σ dW_j(t)

If random = 2, α is a fixed effect: dX_j(t)= (α - β_j X_j(t))dt + σ dW_j(t)

If random = c(1,2), dX_j(t)= (α_j- β_j X_j(t))dt + σ dW_j(t)

#### Cox-Ingersoll-Ross model (CIR)

If random = 1, β is a fixed effect: dX_j(t)= (α_j- β X_j(t))dt + σ √{X_j(t)} dW_j(t)

If random = 2, α is a fixed effect: dX_j(t)= (α - β_j X_j(t))dt + σ √{X_j(t)} dW_j(t)

If random = c(1,2), dX_j(t)= (α_j- β_j X_j(t))dt + σ √{X_j(t)} dW_j(t)

The initial value of each trajectory can be simulated from the invariant distribution of the process: Normal distribution with mean α/β and variance σ^2/(2 β) for the OU, a gamma distribution Γ(2α/σ^2, σ^2/(2β)) for the C-I-R model.

#### Density of the random effects

Several densities are implemented for the random effects, depending on the number of random effects.

If two random effects, choice between

'normalnormal': Normal distributions for both α β and param=c(mean_α, sd_α, mean_β, sd_β)

'gammagamma': Gamma distributions for both α β and param=c(shape_α, scale_α, shape_β, scale_β)

'gammainvgamma': Gamma for α, Inverse Gamma for β and param=c(shape_α, scale_α, shape_β, scale_β)

'normalgamma': Normal for α, Gamma for β and param=c(mean_α, sd_α, shape_β, scale_β)

'normalinvgamma': Normal for α, Inverse Gamma for β and param=c(mean_α, sd_α, shape_β, scale_β)

'gammagamma2': Gamma +2 * σ^2 for α, Gamma + 1 for β and param=c(shape_α, scale_α, shape_β, scale_β)

'gammainvgamma2': Gamma +2 * σ^2 for α, Inverse Gamma for β and param=c(shape_α, scale_α, shape_β, scale_β)

If only α is random, choice between

'normal': Normal distribution with param=c(mean, sd)

lognormal': logNormal distribution with param=c(mean, sd)

'mixture.normal': mixture of normal distributions p N(μ1,σ1^2) + (1-p)N(μ2, σ2^2) with param=c(p, μ1, σ1, μ2, σ2)

'gamma': Gamma distribution with param=c(shape, scale)

'mixture.gamma': mixture of Gamma distribution p Γ(shape1,scale1) + (1-p)Γ(shape2,scale2) with param=c(p, shape1, scale1, shape2, scale2)

'gamma2': Gamma distribution +2 * σ^2 with param=c(shape, scale)

'mixed.gamma2': mixture of Gamma distribution p Γ(shape1,scale1) + (1-p) Γ(shape2,scale2) + +2 * σ^2 with param=c(p, shape1, scale1, shape2, scale2)

If only β is random, choice between 'normal': Normal distribution with param=c(mean, sd)

'gamma': Gamma distribution with param=c(shape, scale)

'mixture.gamma': mixture of Gamma distribution p Γ(shape1,scale1) + (1-p) Γ(shape2,scale2) with param=c(p, shape1, scale1, shape2, scale2)

## Value

 `X` matrix (M x (N+1)) of the M trajectories. `phi` vector (or matrix) of the M simulated random effects.

## References

This function mixedsde.sim is based on the package sde, function sde.sim. See Simulation and Inference for stochastic differential equation, S.Iacus, Springer Series in Statistics 2008 Chapter 2

## Examples

 ```1 2 3 4 5 6 7``` ```#Simulation of 5 trajectories of the OU SDE with random =1, and a Gamma distribution. simuOU <- mixedsde.sim(M=5, T=10,N=1000,model='OU', random=1,fixed=0.5, density.phi='gamma', param=c(1.8, 0.8) , sigma=0.1,op.plot=1) X <- simuOU\$X ; phi <- simuOU\$phi hist(phi) ```

### Example output

```Loading required package: sde

Attaching package: 'fda'

The following object is masked from 'package:graphics':

matplot

Attaching package: 'zoo'

The following objects are masked from 'package:base':

as.Date, as.Date.numeric

sde 2.0.15
Companion package to the book
'Simulation and Inference for Stochastic Differential Equations With R Examples'
Iacus, Springer NY, (2008)
To check the errata corrige of the book, type vignette("sde.errata")
be careful, X0 and invariant are missing thus the initial value X0=0.01 is used

T set to = 10.000000

T set to = 10.000000

T set to = 10.000000

T set to = 10.000000

T set to = 10.000000
```

mixedsde documentation built on May 1, 2019, 7:33 p.m.