rcss (version 1.8)

AcceleratedBellman: Bellman recursion accelerated with k nearest neighbours

Description

Approximate the value functions using k nearest neighbours.

Usage

AcceleratedBellman(grid, reward, scrap, control, disturb, weight, k = 1)

Arguments

grid

Matrix representing the grid. The i-th row corresponds to i-th point of the grid. The j-th column captures the dimensions. The first column must equal to 1.

reward

5-D array representing the tangent approximation of the reward. Entry [i,,a,p,t] captures the tangent at grid point i for action a taken in position p at time t. The intercept is given by [i,1,a,p,t] and slope by [i,-1,a,p,t].

scrap

3-D array representing the tangent approximation of the scrap. Entry [i,,p] captures the tangent at grid point i for position p. The intercept is given by [i,1,p] and slope by [i,-1,p].

control

Array representing the transition probabilities of the controlled Markov chain. Two possible inputs:

• Matrix of dimension n_pos \(\times\) n_action, where entry [i,j] describes the next position after selecting action j at position i.

• 3-D array with dimensions n_pos \(\times\) n_action \(\times\) n_pos, where entry [i,j,k] is the probability of moving to position k after applying action j to position i.

disturb

3-D array containing the disturbance matrices. Matrix [,,i] specifies the i-th disturbance matrix.

weight

Array containing the probability weights of the disturbance matrices.

k

Number of nearest neighbours used for each grid point.

Value

value

4-D array tangent approximation of the value function, where the intercept [i,1,p,t] and slope [i,-1,p,t] describes a tangent of the value function at grid point i for position p at time t.

expected

4-D array representing the expected value functions.

Examples

# NOT RUN {
## Bermuda put option
grid <- as.matrix(cbind(rep(1, 81), c(seq(20, 60, length = 81))))
disturb <- array(0, dim = c(2, 2, 100))
disturb[1, 1,] <- 1
quantile <- qnorm(seq(0, 1, length = (100 + 2))[c(-1, -(100 + 2))])
disturb[2, 2,] <- exp((0.06 -0.5 * 0.2^2) * 0.02 + 0.2 * sqrt(0.02) * quantile)
weight <- rep(1 / 100, 100)
control <- matrix(c(c(1, 2),c(1, 1)), nrow = 2)
reward <- array(data = 0, dim = c(81, 2, 2, 2, 50))
in_money <- grid[, 2] <= 40
reward[in_money, 1, 2, 2,] <- 40
reward[in_money, 2, 2, 2,] <- -1
for (tt in 1:50){
  reward[,,2,2,tt] <- exp(-0.06 * 0.02 * (tt - 1)) * reward[,,2,2,tt] 
}
scrap <- array(data = 0, dim = c(81, 2, 2))
scrap[in_money, 1, 2] <- 40
scrap[in_money, 2, 2] <- -1
scrap[,,2] <- exp(-0.06 * 0.02 * 50) * scrap[,,2]
bellman <- AcceleratedBellman(grid, reward, scrap, control, disturb, weight)
# }