Stochastic Differential Equations

Module aims

Manipulate Itô integrals and apply Itô’s formula in examples the syllabus licenses, without treating every formal SDE as a pathwise ODE.

Course outline

1. 01 Brownian motion 25h

   Learning result State defining properties used in the synopsis and a non-differentiability caveat.

   Formative check Four-option check

2. 02 Itô integral 25h

   Learning result Construct the integral on simple processes and name the isometry used.

   Formative check Written derivation

3. 03 Itô’s formula 25h

   Learning result Apply Itô’s formula to a standard function of Brownian motion.

   Formative check Written derivation

4. 04 SDEs: existence and uniqueness (taught grain) 25h

   Learning result State Lipschitz-type hypotheses and a case where they fail.

   Formative check Counterexample

5. 05 Linear SDEs and examples 25h

   Learning result Solve a linear SDE that the syllabus treats in closed form.

   Formative check Written derivation

6. 06 Synthesis 25h

   Learning result Write one argument: model, Itô step, and a uniqueness or interpretation caveat.

   Formative check Written derivation

Learning results

Write an SDE, name the noise, apply Itô’s formula in a standard example, and state an existence/uniqueness hypothesis.

Subject skills

Brownian motion, filtrations at taught grain, Itô integral, Itô formula, linear SDEs.

Transferable skills

Distinguish formal calculus from justified stochastic calculus; keep filtration and quadratic variation in view.