I have question regarding simulating under mentioned 1D Stochastic Differential Equation in R using Sim.DiffProc package:
dx1 = (b1*x1 − d1*x1) dt + Sqrt(b1*x1 + d1*x1) dW1(t)
I have taken this equation from book: Modeling with Ito Stochastic Differential Equations by E. Allen. In the deterministic and diffusion part of equation, b1 and d1 are model parameters representing birth and death rates (for single population approximation of two interacting populations compartment model). Relevant lines of my code are as under (note that i,ve used theta's to represent parameters in my code):
Code (1):
> fx gx fitmod fx gx fitmod
This problem can be solved using Euler Maruyama method.
you may check the following link for further details and examples
https://cran.r-project.org/web/packages/Sim.DiffProc/Sim.DiffProc.pdf
https://cran.r-project.org/web/packages/smfsb/smfsb.pdf
https://hal.archives-ouvertes.fr/hal-00629841/document
It should not matter whether you use 4 parameter or only 2.
As it does not seem to be necessary, I would only use 2 parameters.
Only if you want to reuse the code later with different parameters, I would go for 4 thetas.
I would use code [2} above with two parameters theta[1] and theta[2].
Also, it is very easy to code this directly without using any packages by applying the Euler Maruyama approximation method (which is described in E. Allen's book).
Also, see the book by Linda J.S. Allen which has ALL THE CODES for such examples problems given in the book . So may copy into R directly and implement:
Linda J.S. Allen, An Introduction to Stochastic Processes with Applications to Biology, Second Edition
Also, the following papers contain more examples of somewhat more complicated stochastic differential equations which have been solve in MATLAB (similar to R) using Euler Maruyama approximation:
A.S. Ackleh and S. Hu, Comparison between Stochastic and Deterministic Selection-Mutation Models. Mathematical Biosciences and Engineering, 4(2007), 133-157.
A.S. Ackleh, K. Deng and Q. Huang, Stochastic Juvenile-Adult Models with Application to a Green Tree Frog Population. Journal of Biological Dynamics, 5(2011), 64-83.