Vector Autoregressive Process as Discretized Vector OrnsteinUhlenbeck Process 

Improved support for computation with process slices, as well as support for arbitrary mean time series processes and time processes with initial values, allows the matching of a uniformly discretized Gaussian Ito process to a vector-valued autoregressive process.

Define a 2D Ito process with linear drift coefficients and constant diffusion coefficients.

In[1]:=
Click for copyable input
X

Define a bivariate autoregressive process with initial values.

In[2]:=
Click for copyable input
X

Since both processes are Gaussian, they are completely specified by their mean and covariance functions.

In[3]:=
Click for copyable input
X
In[4]:=
Click for copyable input
X
In[5]:=
Click for copyable input
X
In[6]:=
Click for copyable input
X

Construct equations of moments by equating the Ito process moment functions at regularly spaced times and the VAR moment functions at consecutive integers.

In[7]:=
Click for copyable input
X
In[8]:=
Click for copyable input
X

Solve the equations.

In[9]:=
Click for copyable input
X
Out[9]=

Simulation of the VAR process gives exact simulation of the Ito process of the regular grid.

In[10]:=
Click for copyable input
X
Out[10]=

Visualize the path.

show complete Wolfram Language input
Out[11]=