# EULERloglik: Euler approximation of the likelihood In sde: Simulation and Inference for Stochastic Differential Equations

## Description

Euler approximation of the likelihood of a process solution of a stochastic differential equation. These functions are useful to calculate approximated maximum likelihood estimators when the transition density of the process is not known.

## Usage

 `1` ```EULERloglik(X, theta, d, s, log = TRUE) ```

## Arguments

 `X` a ts object containing a sample path of an sde. `theta` vector of parameters. `d,s` drift and diffusion coefficients; see details. `log` logical; if TRUE, the log-likelihood is returned.

## Details

The function `EULERloglik` returns the Euler approximation of the log-likelihood. The functions `s` and `d` are the drift and diffusion coefficients with arguments `(t,x,theta)`.

## Value

 `x` a number

## Author(s)

Stefano Maria Iacus

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25``` ```set.seed(123) d <- expression(-1*x) s <- expression(2) sde.sim(drift=d, sigma=s) -> X S <- function(t, x, theta) sqrt(theta[2]) B <- function(t, x, theta) -theta[1]*x true.loglik <- function(theta){ DELTA <- deltat(X) lik <- 0 for(i in 2:length(X)) lik <- lik + dnorm(X[i], mean=X[i-1]*exp(-theta[1]*DELTA), sd = sqrt((1-exp(-2*theta[1]*DELTA))* theta[2]/(2*theta[1])),TRUE) lik } xx <- seq(-3,3,length=100) sapply(xx, function(x) true.loglik(c(x,4))) -> py sapply(xx, function(x) EULERloglik(X,c(x,4),B,S)) -> pz # true likelihood plot(xx,py,type="l",xlab=expression(beta),ylab="log-likelihood") lines(xx,pz, lty=2) # Euler ```

### Example output

```Loading required package: MASS

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")
sigma.x not provided, attempting symbolic derivation.
```

sde documentation built on May 31, 2017, 3:58 a.m.