Stochastic Calculus Simplified Part 5: Linear Stochastic Differential Equations

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Timestamps:
0:00 Intro
0:58 About the Books
1:34 ODE Review
5:15 Linear SDE
6:34 Example 1
10:16 Example 2
16:33 Exercise
17:48 Two Important SDE

[MathematicalToolbox]

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Sorry for the audio glitches in the intro.

[yannledu]

Great fun again! Some comments:

1. Intro about integration factor is the same as what concludes the preceding video.

2. For the variance in example 18, you should have divided by -2 instead of multiplying when integrating e^(-2s), meaning that Var[Xt] = (e^2t - 1)/2

3. In problem 8, I also used the isometry property together with E[e^(\alpha Wt)]

4. I find the same E and Var as you for the mean reverting OU process but then the variance goes to a/2 as t goes to inf, and not 0.

5. As for the Brownian bridge:
Xt = b + (1-t) (a-b) + (1-t) \int_0^t Ws/(1-s)
E[Xt] = b + (1-t)(a-b)
Var[Xt] = t(1-t)

[MathematicalToolbox]

ERRATA:

In example 18 ( 16:05 ) the Variance should be:
Var[Xt] = (e^2t - 1)/2


For the mean reverting Orstein-Uhlenbeck Process ( 18:55 ), the limit is taken in the mean-square sense.

My apologies.