Ito and Stratonovich Solutions of the Linear Growth Model
Define Ito and Stratonovich processes with the same SDE
![](Files/ItoAndStratonovichSolutionsOfTheLinearGrowthModel.en/1.png)
:
Find mean and variance functions for the Ito process:
The mean and variance functions for the Stratonovich process are different:
When
![](Files/ItoAndStratonovichSolutionsOfTheLinearGrowthModel.en/2.png)
, the Ito solution almost surely converges to zero, i.e. the large
![](Files/ItoAndStratonovichSolutionsOfTheLinearGrowthModel.en/3.png)
limit of probability that process value is
![](Files/ItoAndStratonovichSolutionsOfTheLinearGrowthModel.en/4.png)
does not exceed
![](Files/ItoAndStratonovichSolutionsOfTheLinearGrowthModel.en/5.png)
equals
![](Files/ItoAndStratonovichSolutionsOfTheLinearGrowthModel.en/6.png)
:
Confirm it using simulations:
When
![](Files/ItoAndStratonovichSolutionsOfTheLinearGrowthModel.en/7.png)
, the Stratonovich solution, however, almost surely diverges, i.e. the large
![](Files/ItoAndStratonovichSolutionsOfTheLinearGrowthModel.en/8.png)
limit of the probability that process value
![](Files/ItoAndStratonovichSolutionsOfTheLinearGrowthModel.en/9.png)
exceeds
![](Files/ItoAndStratonovichSolutionsOfTheLinearGrowthModel.en/10.png)
equals
![](Files/ItoAndStratonovichSolutionsOfTheLinearGrowthModel.en/11.png)
:
Confirm it using simulations: