# How to include a time dependent parameter while solving a system of SDEs in Julia?

Hello,

I am new to Julia, started using it in R to solve SDEs. I am trying to solve this system of SDEs, dX = (-X*a + (Y-X) b + h)dt + g dW and dY = (-Y a + (X-Y)*b)dt for time [0,200], a=0.30, b=0.2, g=1 and h is 1 for time [50,70] and 0 otherwise. I have the following code for the system for constant h=1:
(The code is from R)

``````library(diffeqr) ##for solving ODEs SDEs DDEs..
library(deSolve)
library(Julia)  ##used  for the sde
library(JuliaCall)
library(data.table)
library(minpack.lm)

tryCatch(system2(file.path(JULIA_HOME, "julia"),
"-E \"try println(JULIA_HOME) catch e println(Sys.BINDIR) end;\"", stdout = TRUE),
warning = function(war) {},
error = function(err) NULL)

julia_setup(JULIA_HOME ="/home/tim/.local/share/R/JuliaCall/julia/v1.5.0/bin")

##calling Julia
JuliaCall:::julia_locate(JULIA_HOME = "/home/tim/.local/share/R/JuliaCall/julia/v1.5.0/bin")

## setup of the packages
de <- diffeqr::diffeq_setup()
julia <- julia_setup()

#u0 is the starting value for the vector
#func1 is the ODE TO SOLVE
#f is the drift part in the sde
#g is the diffusion part in the sde
#p are the parameters associated to the sde
# noise_rate_prototype is a sparse matrix by making a dense matrix and setting some values as not zero

#starting value for the three variables
u0<-c(0,0)

## parameters
a=0.33;
b=0.2;
g=1;
h=1;

## parameters of the sde
p<-c(a, b, g, h)

## drift part in the SDE
f <- JuliaCall::julia_eval("
function f(dy,y,p,t)
dy = -y*p + (y-y)*p + p
dy = -y*p + (y-y)*p
end")

## diffusion part in the SDE
g <- JuliaCall::julia_eval("
function g(dy,y,p,t)
dy[1,1] = p
dy[2,1] = 0
end")

## matrix associated to the noise
noise_rate_prototype <- matrix(rep(1,2), nrow = 2, ncol = 1)

tspan <- c(0.,200)

JuliaCall::julia_assign("u0", u0); ## initial value
JuliaCall::julia_assign("tspan", tspan); ## grid
JuliaCall::julia_assign("p", p); ## parameters
JuliaCall::julia_assign("noise_rate_prototype", noise_rate_prototype); ## matrix associated to the noise

## call the SDE problem
prob <- JuliaCall::julia_eval("SDEProblem(f, g, u0,tspan,p, noise_rate_prototype=noise_rate_prototype)");

sol <- de\$solve(prob)

udf <- as.data.frame(t(sapply(sol\$u,identity)))

#plot the 2 figures,
plot(sol\$t,udf\$V1,type="l",col="purple", ylab="Y1") #L
lines(sol\$t,udf\$V2,col="red",ylab="Y2") #R
``````

I am not sure how to include the time dependence of `h`, I tried to do it by including a forcing variable (in the form of ‘signal’) but I got a lengthy Julia error (stating that `h` is not defined- code below) . So perhaps thats not the way to go. I’d appreciate any help regarding this.

The chunk of the code I changed to include the time dependent parameters is this (the new or changes lines have an arrow to their right):

``````## drift part in the SDE
f <- JuliaCall::julia_eval("
function f(dy,y,p,t)
MU <- signal(t)                                    # <-
dy = -y*p + (y-y)*p + MU         # <-
dy = -y*p + (y-y)*p
end")

signal <- approxfun(x = c(0, 35, 125, 200),          # <-
y = c(0, 1,  0,  0),
method = "constant", rule = 2)

## diffusion part in the SDE
g <- JuliaCall::julia_eval("
function g(dy,y,p,t)
dy[1,1] = p
dy[2,1] = 0
end")

## matrix associated to the noise
noise_rate_prototype <- matrix(rep(1,2), nrow = 2, ncol = 1)

tspan <- c(0,200)

JuliaCall::julia_assign("u0", u0); ## initial value
JuliaCall::julia_assign("tspan", tspan); ## grid
JuliaCall::julia_assign("p", p); ## parameters
JuliaCall::julia_assign("signal", signal);                   # <-
JuliaCall::julia_assign("noise_rate_prototype", noise_rate_prototype); ## matrix associated to the noise

## call the SDE problem
prob <- JuliaCall::julia_eval("SDEProblem(f, g, u0,tspan,p, noise_rate_prototype=noise_rate_prototype, signal=signal)");      # <-
``````