1. Random WALK, BROWNIAN MOTION and SDEs
Continuation…

3. SDE Modeling
Example 1: Population dynamics (single species) with demographic variability 𝑑𝑥=(𝑏𝑥−𝑑𝑥)𝑑𝑡+ 𝑏𝑥+𝑑𝑥 𝑑 𝐵 𝑡 𝑥: population size 𝑏: birth rate 𝑑: death rate

4. SDE Modeling
Example 2: Lotka-Volterra Predator-Prey 𝑑 𝑥 1 = 𝑏 1 − 𝑑 1 𝑥 2 𝑥 1 𝑑𝑡+ 𝑏 1 + 𝑑 1 𝑥 2 𝑥 1 𝑑 𝐵 1𝑡 𝑑 𝑥 2 =( 𝑏 2 𝑥 1 − 𝑑 2 ) 𝑥 2 𝑑𝑡+ ( 𝑏 2 𝑥 1 + 𝑑 2 ) 𝑥 2 𝑑 𝐵 2𝑡 𝑥 1 : prey population size; 𝑥 2 : predator population size 𝑏 1 : prey birth rate; 𝑏 2 : predator birth rate due to prey utilization 𝑑 1 : prey death rate due to predation; 𝑑 1 : predator death rate

5. SDE Modeling
Example 3: Population dynamics (single species) with parameter (environmental) variability 𝑑𝑥= 𝑏𝑥−𝑑𝑥 𝑑𝑡 𝑑𝑏= 𝛽 1 ( 𝜇 1 −𝑏)𝑑𝑡+ 𝜎 1 𝑑 𝐵 1𝑡 𝑑𝑑= 𝛽 2 ( 𝜇 2 −𝑑)𝑑𝑡+ 𝜎 2 𝑑 𝐵 2𝑡 𝛽 1 ( 𝜇 1 −𝑏) represents the probability associated with drift toward the mean value 𝜇 1 (average per capita birth rate in the environment). 𝛽 2 ( 𝜇 2 −𝑑) represents the probability associated with drift toward the mean value 𝜇 2 (average per capita death rate in the environment). The dynamics of the parameters follow the Ornstein-Uhlenbeck process.

6. SDE Modeling
Example 4: Vasicek model for interest rates 𝑑𝑟=𝛽(𝜇−𝑟)𝑑𝑡+𝜎𝑑 𝐵 𝑡 Example 5: Cox-Ingersoll-Ross (CIR) model for interest rates 𝑑𝑟=𝛽(𝜇−𝑟)𝑑𝑡+𝜎 𝑟 𝑑 𝐵 𝑡