Problem 4.1

Answer 4.1-1

Answer 4.1-2

Answer 4.1-3

Answer 4.1-4

Answer 4.1-5

Problem 4.2

Answer 4.2:1-2-3

Answer 4.2:4

Euler-Maruyama Method for a Given Linear SDE:

Integral form

Differential equation form

Euler-Maruyama(EM) method takes the form

Problem 4.3

Answer 4.3:1

Answer 4.3:2

Answer 4.3-3

Problem 4.4

Answer 4.4: 1

Answer 4.4: 2

Answer 4.4: 3

  • Euler-Maruyama method to approximate the solution of Ornstein-Uhlenbeck SDE

Conclusion

Ornstein-Uhlenbeck is a path-dependent SDE. The negative \(\mu\) in its standard stochastic calculus is representing mean-reversed force or restoring force. The disadvantages of Euler-Maruyama method have problems of weak convergence and explosion with time evolution for path-dependent SDE. The above five figures show that when negative \(\mu\) is growing larger and larger, the solution of approximation with Euler-Maruyama method will explode with time evolution. Therefore, Euler-Maruyama method is not the best method to approximate non-stationary path-dependent SDE.

Problem 4.5

Answer 4.5-1

Answer 4.5-2

Answer 4.5-3