asymptotic_term: asymptotic expansion of the expected value of the functional

In yuima: The YUIMA Project Package for SDEs

asymptotic expansion of the expected value of the functional

Description

calculate the fisrt and second term of asymptotic expansion of the functional mean.

Usage

asymptotic_term(yuima, block=100, rho, g, expand.var="e")

Arguments

yuima	an yuima object containing model and functional.
block	the number of trapezoids for integrals.
rho	specify discounting factor in mean integral.
g	arbitrary measurable function for mean integral.
expand.var	default expand.var="e".

Details

Calculate the first and second term of asymptotic expansion of the expected value of the functional associated with a sde. The returned value d0 + epsilon * d1 is approximation of the expected value.

Value

terms

list of 1st and 2nd asymptotic terms, terms$d0 and terms$d1.

Note

we need to fix this routine.

Author(s)

YUIMA Project Team

Examples

## Not run: 
# to the Black-Scholes economy:
# dXt^e = Xt^e * dt + e * Xt^e * dWt
diff.matrix <- "x*e"
model <- setModel(drift = "x", diffusion = diff.matrix)
# call option is evaluated by averating
# max{ (1/T)*int_0^T Xt^e dt, 0}, the first argument is the functional of interest:
Terminal <- 1
xinit <- c(1)
f <- list( c(expression(x/Terminal)), c(expression(0)))
F <- 0
division <- 1000
e <- .3
yuima <- setYuima(model = model, sampling = setSampling(Terminal=Terminal, n=division))
yuima <- setFunctional( yuima, f=f,F=F, xinit=xinit,e=e)

# asymptotic expansion
rho <- expression(0)
F0 <- F0(yuima)
get_ge <- function(x,epsilon,K,F0){
  tmp <- (F0 - K) + (epsilon * x) 
  tmp[(epsilon * x) < (K-F0)] <- 0
  return( tmp )
}
g <- function(x) get_ge(x,epsilon=e,K=1,F0=F0)
set.seed(123)
asymp <- asymptotic_term(yuima, block=10, rho,g)
asymp
sum(asymp$d0 + e * asymp$d1)


### An example of multivariate case: Heston model
## a <- 1;C <- 1;d <- 10;R<-.1
## diff.matrix <- matrix( c("x1*sqrt(x2)*e", "e*R*sqrt(x2)",0,"sqrt(x2*(1-R^2))*e"), 2,2)
## model <- setModel(drift = c("a*x1","C*(10-x2)"), 
## diffusion = diff.matrix,solve.variable=c("x1","x2"),state.variable=c("x1","x2"))
## call option is evaluated by averating
## max{ (1/T)*int_0^T Xt^e dt, 0}, the first argument is the functional of interest:
##
## Terminal <- 1
## xinit <- c(1,1)
## 
## f <- list( c(expression(0), expression(0)),   
## c(expression(0), expression(0)) , c(expression(0), expression(0))  )
## F <- expression(x1,x2)
## 
## division <- 1000
## e <- .3
## 
## yuima <- setYuima(model = model, sampling = setSampling(Terminal=Terminal, n=division))
## yuima <- setFunctional( yuima, f=f,F=F, xinit=xinit,e=e)
## 
## rho <- expression(x1)
## F0 <- F0(yuima)
## get_ge <- function(x){
##  return( max(x[1],0))
## }
## g <- function(x) get_ge(x)
## set.seed(123)
## asymp <- asymptotic_term(yuima, block=10, rho,g)
## sum(asymp$d0 + e * asymp$d1)


## End(Not run)

yuima documentation built on Nov. 14, 2022, 3:02 p.m.