# R/ModelB11.R In mfrdixon/MLEMVD: Maximum Likelihood Estimation for Diffusion Equations

#### Documented in ModelB11

```#' Compute the maximum likelihood estimate of Model B11
#'
#'
#' @param x Observation of the state variable at time t
#' @param x0 Observation of the state variable at time t-1
#' @param del The time step between the current and previous observation
#' @param param The parameter 6-vector  (k1,k2,rho,kappa,theta,sigma)
#' @export output a list with a llk variable storing the result of the log likelihood calculation
#' @examples
#' ModelB11(0.4,0.3,0.1,c(0.1,0.2,0.3,0.4,0.5,0.6))
#'
ModelB11<-function(x,x0,del,param){

# dx1 <- (k1 + k2*x2)*dt + sqrt(x2)*(sqrt(1 - rho^2)*dW1 + rho*dW2)
# dx2 <- kappa*(theta - x2)*dt + sigma*x2*dW2

x1 <- x[1]
x2 <- x[2]
x10 <- x0[1]
x20 <- x0[2]

k1    <- param[1]
k2    <- param[2]
rho   <- param[3]
kappa <- param[4]
theta <- param[5]
sigma <- param[6]

Dv <- (1/2)*log(x2^3*(1 - rho^2)*sigma^2)
cm1 <- ((x1 - x10)^2*(x2 - x20)^3*(4 - 7*rho^2))/(32*x20^4*(-1 + rho^2)^2) -
((x1 - x10)^2*(x2 - x20))/(4*x20^2*(-1 + rho^2)) + ((x1 - x10)^2*(x2 - x20)^2*
(-8 + 11*rho^2))/(48*x20^3*(-1 + rho^2)^2) +
((x1 - x10)^2*(x2 - x20)^4*(384 - 1179*rho^2 + 815*rho^4))/(3840*x20^5*(-1 + rho^2)^3) + ((x2 - x20)^5*(20 - 21*rho^2))/(48*x20^5*(-1 + rho^2)^2*sigma^2) -
(x2 - x20)^3/(2*x20^3*(-1 + rho^2)*sigma^2) + ((x2 - x20)^4*(-44 + 45*rho^2))/(96*x20^4*(-1 + rho^2)^2*sigma^2) +
((x2 - x20)^6*(4384 - 9105*rho^2 + 4725*rho^4))/(11520*x20^6*(-1 + rho^2)^3*sigma^2) +
(7*(x1 - x10)*(x2 - x20)^4*rho*(-13 + 15*rho^2))/(192*x20^(9/2)*(-1 + rho^2)^2*sigma) +
((x1 - x10)*(x2 - x20)^3*(14*rho - 15*rho^3))/(24*x20^(7/2)*(-1 + rho^2)^2*sigma) +
((x1 - x10)*(x2 - x20)^5*(-764*rho + 1705*rho^3 - 945*rho^5))/(1920*x20^(11/2)*(-1 + rho^2)^3*sigma) +
((x1 - x10)^3*(x2 - x20)^3*rho*(153 - 173*rho^2)*sigma)/(2880*x20^(9/2)*(-1 + rho^2)^3) + (5*(x1 - x10)^3*(x2 - x20)^2*rho*sigma)/(96*x20^(7/2)*
(-1 + rho^2)^2) - ((x1 - x10)^3*(x2 - x20)*rho*sigma)/(24*x20^(5/2)*(-1 + rho^2)^2) - ((x1 - x10)^4*(x2 - x20)*
sigma^2)/(96*x20^3*(-1 + rho^2)^2) + ((x1 - x10)^4*sigma^2)/
(96*x20^2*(-1 + rho^2)^2) + ((x1 - x10)^4*(x2 - x20)^2*
(-9 + 14*rho^2)*sigma^2)/(960*x20^4*(-1 + rho^2)^3) -
((x1 - x10)^5*(x2 - x20)*rho*sigma^3)/(480*x20^(7/2)*
(-1 + rho^2)^3) + ((x1 - x10)^6*sigma^4)/(2880*x20^3*(-1 + rho^2)^3) + (3*(x1 - x10)*(x2 - x20)^2*
rho)/(x20^(5/2)*(-4*sigma + 4*rho^2*sigma)) + (1/(2*x20^2*(-1 + rho^2)*sigma^2))*(x2^2 - 2*x2*sqrt(x20)*(sqrt(x20) +
(x1 - x10)*rho*sigma) + x20*(x20 +
2*(x1 - x10)*sqrt(x20)*rho*sigma + (x1 - x10)^2*sigma^2))

c0 <- ((x1 - x10)^2*sigma^2)/
(-48*x20 + 48*x20*rho^2) + (7*(x1 - x10)^4*sigma^4)/
(11520*x20^2*(-1 + rho^2)^2) -
((x2 - x20)^2*(24*x20*kappa - 48*theta*kappa +
36*k1*sqrt(x20)*rho*sigma +
12*k2*x20^(3/2)*rho*sigma + 5*x20*sigma^2))/(48*
x20^3*(-1 + rho^2)*
sigma^2) + ((x2 - x20)*(-4*theta*kappa +
4*k1*sqrt(x20)*rho*sigma +
4*k2*x20^(3/2)*rho*sigma + x20*(4*kappa + sigma^2)))/
(4*x20^2*(-1 + rho^2)*sigma^2) -
((x1 - x10)^3*
sigma*(-12*theta*kappa*rho + 8*k1*sqrt(x20)*sigma +
x20*rho*(4*kappa + sigma^2)))/(192*
x20^(5/2)*(-1 + rho^2)^2) -
((x1 - x10)*(-4*theta*kappa*rho + 4*k1*sqrt(x20)*sigma +
4*k2*x20^(3/2)*sigma + x20*rho*(4*kappa + sigma^2)))/
(4*x20^(3/2)*(-1 + rho^2)*sigma) +
((x1 - x10)*(x2 - x20)*(-36*theta*kappa*rho +
24*k1*sqrt(x20)*sigma +
x20*rho*(12*kappa + sigma^2)))/(48*
x20^(5/2)*(-1 + rho^2)*sigma) +
((x1 - x10)^3*(x2 - x20)*sigma*(-900*theta*kappa*rho +
480*k1*sqrt(x20)*sigma +
x20*rho*(180*kappa + 17*sigma^2)))/
(11520*x20^(7/2)*(-1 + rho^2)^2) +
((x1 -
x10)*(x2 - x20)^2*(12*theta*kappa*rho*(-10 + 13*rho^2) +
8*k1*sqrt(x20)*(8 - 11*rho^2)*sigma -
3*x20*(4*kappa*rho*(-2 + 3*rho^2) + rho^3*sigma^2)))/
(192*x20^(7/2)*(-1 + rho^2)^2*sigma) +
((x2 - x20)^4*(1440*theta*kappa*(-8 + 9*rho^2) -
900*k2*x20^(3/2)*rho*
(-1 + rho^2)*sigma -
420*k1*sqrt(x20)*rho*(-13 + 15*rho^2)*sigma +
x20*(-360*kappa*(-8 + 9*rho^2) + (502 - 585*rho^2)*
sigma^2)))/
(11520*x20^5*(-1 + rho^2)^2*sigma^2) +
(1/(192*x20^3*(-1 + rho^2)^2))*((x1 - x10)^2*(x2 - x20)*
(-36*theta*kappa*rho^2 + 24*k1*sqrt(x20)*rho*sigma +
x20*(12*kappa*rho^2 + (2 + rho^2)*sigma^2))) +
((x2 - x20)^3*(12*theta*kappa*(16 - 17*rho^2) +
24*k2*x20^(3/2)*rho*
(-1 + rho^2)*sigma +
8*k1*sqrt(x20)*rho*(-14 + 15*rho^2)*sigma +
x20*(kappa*(-64 + 68*rho^2) + (-12 + 13*rho^2)*sigma^2)))/
(192*x20^4*(-1 + rho^2)^2*sigma^2) - (1/(5760*x20^4*(-1 + rho^2)^2))*(x1 - x10)^2*(x2 - x20)^2*(-1620*theta*kappa*rho^2 +
900*k1*sqrt(x20)*rho*sigma + x20*(360*kappa*rho^2 +
(38 + 31*rho^2)*sigma^2)) + ((x1 -
x10)*(x2 - x20)^3*
(-1260*theta*kappa*rho*(-5 + 8*rho^2) + 720*k1*sqrt(x20)*
(-4 + 7*rho^2)*sigma +
x20*rho*(900*kappa*(-1 + 2*rho^2) +
(47 + 150*rho^2)*sigma^2)))/(11520*
x20^(9/2)*(-1 + rho^2)^2*
sigma)

c1 <- -(((x1 - x10)*sigma*(-3*theta*kappa*rho + k1*sqrt(x20)*sigma))/(24*x20^(3/2)*(-1 + rho^2))) + (1/(48*x20^3*(-1 + rho^2)*
sigma^2))*
(x2 - x20)*(-24*theta^2*kappa^2 +
36*k1*sqrt(x20)*theta*kappa*rho*sigma +
12*k2^2*x20^3*sigma^2 +
3*k2*x20^(5/2)*rho*sigma*(4*kappa + sigma^2) -
x20^(3/2)*rho*
sigma*(-12*k2*theta*kappa + k1*(12*kappa + sigma^2)) +
12*x20*((-k1^2)*sigma^2 +
2*theta*kappa*(kappa + sigma^2 - rho^2*sigma^2))) +
(1/(96*x20^2*(-1 + rho^2)*sigma^2))*(48*theta^2*kappa^2 -
96*k1*sqrt(x20)*theta*kappa*rho*sigma +
48*k2^2*x20^3*sigma^2 +
24*k2*x20^(5/2)*rho*sigma*(4*kappa + sigma^2) +
x20^2*(48*kappa^2 + 96*k1*k2*sigma^2 -
24*kappa*(-2 + rho^2)*sigma^2 +
(13 - 10*rho^2)*sigma^4) + 24*x20^(3/2)*rho*sigma*
(-4*k2*theta*kappa + k1*(4*kappa + sigma^2)) -
24*
x20*(-2*k1^2*sigma^2 +
theta*kappa*(4*kappa + (4 - 3*rho^2)*sigma^2))) +
(1/(23040*x20^3*(-1 + rho^2)^2))*((x1 - x10)^2*
(240*theta^2*kappa^2*(4 + 9*rho^2) -
4320*k1*sqrt(x20)*theta*kappa*rho*
sigma - 480*k2^2*x20^3*sigma^2 -
120*k2*x20^(5/2)*rho*sigma*
(4*kappa + sigma^2) + x20^2*(240*kappa^2*rho^2 +
120*kappa*rho^2*sigma^2 + (4 + 11*rho^2)*sigma^4) +
120*x20^(3/2)*rho*
sigma*(-4*k2*theta*kappa + 3*k1*(4*kappa + sigma^2)) -
120*x20*(-12*k1^2*sigma^2 +
theta*kappa*(4*kappa*(2 + 3*rho^2) +
(8 - 3*rho^2)*sigma^2)))) +
(1/(11520*x20^(7/2)*(-1 + rho^2)^2*sigma))*((x1 - x10)*(x2 -
x20)*
(-240*theta^2*kappa^2*rho*(4 + 9*rho^2) +
4320*k1*sqrt(x20)*theta*kappa*
rho^2*sigma + 480*k2^2*x20^3*rho*sigma^2 +
120*k2*x20^(5/2)*rho^2*
sigma*(4*kappa + sigma^2) -
x20^2*(240*kappa^2*rho^3 +
120*kappa*rho^3*sigma^2 +
rho*(4 + 11*rho^2)*sigma^4) +
120*x20*
rho*(-12*k1^2*sigma^2 +
theta*kappa*(4*kappa*(2 + 3*rho^2) +
(17 - 12*rho^2)*sigma^2)) -
120*x20^(3/2)*sigma*
(-4*k2*theta*kappa*rho^2 +
k1*(12*kappa*rho^2 + (2 + rho^2)*
sigma^2)))) + (1/(23040*
x20^4*(-1 + rho^2)^2*sigma^2))*
((x2 - x20)^2*(240*theta^2*kappa^2*(-56 + 69*rho^2) -
1440*k1*sqrt(x20)*theta*kappa*rho*(-10 + 13*rho^2)*sigma -
480*k2^2*x20^3*(-2 + 3*rho^2)*sigma^2 -
120*k2*x20^(5/2)*rho*
(-4 + 5*rho^2)*sigma*(4*kappa + sigma^2) +
x20^2*(240*kappa^2*rho^2 + 120*kappa*rho^2*sigma^2 +
(4 + 11*rho^2)*sigma^4) +
120*x20^(3/2)*rho*sigma*
(4*k2*theta*kappa*(8 - 9*rho^2) +
3*k1*(-8*kappa + 12*kappa*rho^2 +
rho^2*sigma^2)) -
120*x20*(4*k1^2*(8 - 11*rho^2)*sigma^2 +
5*theta*
kappa*(4*
kappa*(-4 + 5*rho^2) + (-16 + 35*rho^2 - 18*rho^4)*
sigma^2))))

c2 <- (240*theta^2*kappa^2*(20 - 9*rho^2) -
2880*k1*sqrt(x20)*theta*kappa*rho*sigma -
960*k2*x20^(3/2)*theta*kappa*rho*sigma + 480*k2^2*x20^3*sigma^2 -
x20^2*(240*kappa^2*rho^2 +
120*kappa*rho^2*sigma^2 + (4 + 11*rho^2)*sigma^4) +
120*x20*(4*k1^2*sigma^2 + 3*theta*kappa*(4*kappa*(-2 + rho^2) +
(-8 + 7*rho^2)*sigma^2)))/(11520*x20^2*(-1 + rho^2))
output <- list()
output\$llk <- -log(2*pi*del) - Dv + cm1/del + c0 + c1*del + c2*(del^2/2)
return(output)
}
```
mfrdixon/MLEMVD documentation built on May 1, 2018, 11:38 p.m.