Stochastic ODEs

Introduction

See my video.
See Itô integral#Next step SDEs.

2. Stochastic Differential Equations (SDEs)

• SDEs generalize ordinary differential equations by incorporating a random noise term, often modeled by Brownian motion.
• General form:

Think of $\frac{dW}{dt}$ as if it were a random number from a normal $N(0,\mathrm{\infty })$ distribution. Then,

is a random number from a $N(0,dt)$ distribution.

• The solution to an SDE is not a single function but a stochastic process — i.e., a curve-valued random variable.
• The simplest example:$$dx(t)=adt+bdW(t)\Rightarrow x(t)=x_0+at+bW(t)$$

Related: Itô integral.

3. Real Example: Stock Prices

• Stock prices are modeled using an SDE called Geometric Brownian Motion:

• This models:

  • $\mu$: expected growth rate (drift)
  • $\sigma$: market volatility (randomness)
  • $W(t)$: Brownian motion (random shocks)

• The solution is:

This is a random curve over time — each realization gives one possible stock price trajectory.

• Check:
  If you take the usual derivative of

you might expect the chain rule to apply like in standard calculus, and wonder why this form solves the SDE:

But you're forgetting that Itô lemma (the stochastic analog of the chain rule) introduces extra terms due to the randomness of $W(t)$.

Let:

Then:

• $\frac{\partial f}{\partial t}=Af$
• $\frac{\partial f}{\partial W}=\sigma f$
• $\frac{{\partial }^{2}f}{\partial W^2}={\sigma }^{2}f$

Now plug into Ito’s formula:

Recall $A=\mu -\frac{{\sigma }^2}{2}$, so:

Therefore:

Stratonovich form

From Stratonovich integral#Conversion rule we can show the following. Suppose you have an Itô SDE

with $X_t\in \mathbb{R}$, $W_t$ one-dimensional Brownian motion.
The Stratonovich version is

That is:

Another example: bottle cap race

See bottle cap race SDE.

Another example: growth of a tree

See growth of a tree SDE.

Exercises

See [[SDEs exercises.pdf]] for the solutions

Block 1 — Basic manipulations with Itô’s formula

Goal: Learn to differentiate stochastic processes correctly.

1. Let $X_t=W_t^2$. Compute $d{X}_t$ using Itô’s formula.
2. Let $X_t=e^{W_t}$. Compute $d{X}_t$.
3. Let $X_t=W_t^3-3t{W}_t$. Show that $X_t$ is a martingale.
4. Let $X_t=\sin (W_t)$. Compute $d{X}_t$ and find $\mathbb{E}[\sin (W_t)]$.
5. For a constant $a\in \mathbb{R}$, let $X_t=e^{a{W}_t-\frac{1}{2}{a}^{2}t}$. Prove that $X_t$ is a martingale. (Hint: apply Itô’s formula.)

Block 2 — Solving linear SDEs

6. Solve $d{X}_t=a{X}_{t}dt+b{X}_{t}d{W}_t,X_0=x_0$.

7. Solve $d{X}_t=adt+bd{W}_t,X_0=0$.

8. Show that the solution in (6) satisfies

   $$X_t=x_0\exp ((a-\frac{1}{2}{b}^2)t+b{W}_t)$$

9. Compute $\mathbb{E}[X_t]$ and $\text{Var}(X_t)$ for (6).

10. Compute $\mathbb{E}[X_t]$ and $\mathbb{E}[X_t^2]$ for (7).

Block 3 — Ornstein–Uhlenbeck and mean reversion

Goal: Learn integration factor method.

11. Solve $d{X}_t=-\theta X_{t}dt+\sigma d{W}_t,X_0=x_0$.

12. Compute $\mathbb{E}[X_t]$ and $\text{Var}(X_t)$.

13. Show that $X_t$ has a stationary Gaussian distribution as $t\to \mathrm{\infty }$, and find its mean and variance.

14. Compute the covariance $\mathbb{E}[X_t{X}_s]$ for $s<t$.

Block 4 — Itô vs Stratonovich

Goal: Be comfortable switching between the two forms.

15. Convert the Itô SDE

    $$d{X}_t=a(X_t)dt+b(X_t)d{W}_t$$

    into the Stratonovich form.

16. Apply your formula to $d{X}_t=X_{t}d{W}_t$ and write both Itô and Stratonovich versions explicitly.

17. For the Stratonovich SDE

    $$d{X}_t=X_t\circ d{W}_t$$

    compute $d(X_t^2)$ using ordinary calculus and check that it matches the Itô version.

18. Check that Stratonovich calculus satisfies the usual chain rule for $f(X_t)$.

Block 5 — Multidimensional SDEs

Goal: Manipulate SDEs with several Brownian components.

Let $W_t=(W_t^1,W_t^2)$ be a 2D Brownian motion.

19. Compute $d(W_t^1{W}_t^2)$.

20. Given $d{X}_t=W_t^{1}d{W}_t^2$, compute $d(X_t^2)$.

21. Solve the linear vector SDE

    $$d{X}_t=A{X}_{t}dt+B{X}_{t}d{W}_t$$

    for constant $2\times 2$ matrices $A,B$ that commute.

22. Compute $d|W_t{|}^2$ for $W_t\in {\mathbb{R}}^n$.

23. Show that $R_t=|W_t|$ satisfies

    $$d{R}_t=\frac{n-1}{2R_t}dt+d{\beta }_t$$

    where ${\beta }_t$ is a 1D Brownian motion.

Block 6 — Long-term behaviour and generators

Goal: Connect SDEs to their infinitesimal generators.

24. For $d{X}_t=a(X_t)dt+b(X_t)d{W}_t$, write the generator $L$ acting on smooth $f$:

    $$Lf=a{f}^{\prime }+\frac{1}{2}{b}^2{f}^{″}$$

25. Compute $L$ for the Ornstein–Uhlenbeck process and find the invariant measure by solving $L^{\ast }p=0$.

26. For the geometric Brownian motion $d{X}_t=a{X}_{t}dt+b{X}_{t}d{W}_t$, compute $L$ and verify that $\mathbb{E}[X_t]=e^{at}{X}_0$.

27. Check the martingale property: find a function $f$ such that $f(X_t)$ is a martingale for the OU process.

28. (Optional) Verify that for any smooth $f$,

    $$M_t=f(X_t)-f(X_0)-{\int }_0^{t}Lf(X_s)ds$$

    is a martingale.