Basic description of stochastic differential equations (SDE)
'setModel' gives a description of stochastic differential equation with or without jumps of the following form:
dXt = a(t,Xt, alpha)dt + b(t,Xt,beta)dWt + c(t,Xt,gamma)dZt, X0=x0
All functions relying on the yuima
package will get as much information as possible from the different slots of the yuima-class structure without replicating the same code twice. If there are missing pieces of information, some default values can be assumed.
setModel(drift = NULL, diffusion = NULL, hurst = 0.5, jump.coeff = NULL, measure = list(), measure.type = character(), state.variable = "x", jump.variable = "z", time.variable = "t", solve.variable, xinit)
drift: a vector of expressions (the default value is 0 when drift=NULL).diffusion: a matrix of expressions (the default value is 0 when diffusion=NULL).hurst: the Hurst parameter of the gaussian noise. If h=0.5, the default, the process is Wiener otherwise it is fractional Brownian motion with that precise value of the Hurst index. Can be set to NA for further specification.jump.coeff: a matrix of expressions for the jump component.measure: Levy measure for jump variables.measure.type: type specification for Levy measures.state.variable: a vector of names of the state variables in the drift and diffusion coefficients.jump.variable: a vector of names of the jump variables in the jump coefficient.time.variable: the name of the time variable.solve.variable: a vector of names of the variables in the left-hand-side of the equations in the model; solve.variable equals state.variable as long as we have no exogenous variable other than statistical parameters in the coefficients (drift and diffusion).xinit: a vector of numbers identifying the initial value of the solve.variable.Please refer to the vignettes and the examples or to the yuimadocs
package.
An object of yuima.model-class contains several slots:
drift:: an expression which specifies the drift coefficient (a vector).
diffusion:: an expression which specifies the diffusion coefficient (a matrix).
jump.coeff:: coefficient of the jump term.
measure:: the Levy measure of the driving Levy process.
measure.type:: specifies the type of the measure, such as CP, code or density. See below.
parameter:: a short name for ``parameters''. It is an object of model.parameter-class which is a list of vectors of names of parameters belonging to the single components of the model (drift, diffusion, jump and measure), the names of common parameters and the names of all parameters. For more details see model.parameter-class
documentation page.
solve.variable:: a vector of variable names, each element corresponds to the name of the solution variable (left-hand-side) of each equation in the model, in the corresponding order.
state.variable:: identifies the state variables in the
expression. By default, it is assumed to be `x`.
jump.variable:: the variable for the jump coefficient. By default, it is assumed to be z.
time:: the time variable. By default, it is assumed to be t.
solve.variable:: used to identify the solution variables in the expression, i.e. the variable with respect to which the stochastic differential equation has to be solved. By default, it is assumed to be x, otherwise the user can choose any other model specification.
noise.number:: denotes the number of sources of noise. Currently only for the Gaussian part.
equation.number:: denotes the dimension of the stochastic differential equation.
dimension:: the dimensions of the parameters in the parameter slot.
xinit:: denotes the initial value of the stochastic differential equation.
The yuima.model-class structure assumes that the user either uses the default names for state.variable, jump.variable, solution.variable and time.variable or specifies his/her own names. All the rest of the terms in the expressions are considered as parameters and identified accordingly in the parameter slot.
yuima.model-class.The YUIMA Project Team
There may be missing information in the model description. Please contribute with suggestions and fixings.
# Ex 1. (One-dimensional diffusion process) # To describe # dXt = -3*Xt*dt + (1/(1+Xt^2+t))dWt, # we set mod1 <- setModel(drift = "-3*x", diffusion = "1/(1+x^2+t)", solve.variable = c("x")) # We may omit the solve.variable; then the default variable x is used mod1 <- setModel(drift = "-3*x", diffusion = "1/(1+x^2+t)") # Look at the model structure by str(mod1) # Ex 2. (Two-dimensional diffusion process with three factors) # To describe # dX1t = -3*X1t*dt + dW1t +X2t*dW3t, # dX2t = -(X1t + 2*X2t)*dt + X1t*dW1t + 3*dW2t, # we set the drift coefficient a <- c("-3*x1","-x1-2*x2") # and also the diffusion coefficient b <- matrix(c("1","x1","0","3","x2","0"),2,3) # Then set mod2 <- setModel(drift = a, diffusion = b, solve.variable = c("x1","x2")) # Look at the model structure by str(mod2) # The noise.number is automatically determined by inputting the diffusion matrix expression. # If the dimensions of the drift differs from the number of the rows of the diffusion, # the error message is returned. # Ex 3. (Process with jumps (compound Poisson process)) # To describe # dXt = -theta*Xt*dt+sigma*dZt mod3 <- setModel(drift=c("-theta*x"), diffusion="sigma", jump.coeff="1", measure=list(intensity="1", df=list("dnorm(z, 0, 1)")), measure.type="CP", solve.variable="x") # Look at the model structure by str(mod3) # Ex 4. (Process with jumps (stable process)) # To describe # dXt = -theta*Xt*dt+sigma*dZt mod4 <- setModel(drift=c("-theta*x"), diffusion="sigma", jump.coeff="1", measure.type="code",measure=list(df="rstable(z,1,0,1,0)"), solve.variable="x") # Look at the model structure by str(mod4) # See rng about other candidate of Levy noises. # Ex 5. (Two-dimensional stochastic differenatial equation with Levy noise) # To describe # dX1t = (1 - X1t - X2t)*dt+dZ1t # dX2t = (0.5 - X1t - X2t)*dt+dZ2t beta<-c(.5,.5) mu<-c(0,0) Lambda<-matrix(c(1,0,0,1),2,2) mod5 <- setModel(drift=c("1 - x1-x2",".5 - x1-x2"), solve.variable=c("x1","x2"), jump.coeff=Lambda, measure.type="code", measure=list(df="rNIG(z, alpha, beta, delta0, mu, Lambda)")) # Look at the model structure by str(mod5) # Ex 6. (Process with fractional Gaussian noise) # dYt = 3*Yt*dt + dWt^h mod6 <- setModel(drift="3*y", diffusion=1, hurst=0.3, solve.variable=c("y")) # Look at the model structure by str(mod6)