Stochastic Differential Equations

模块目标

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

课程大纲

1. 01 Brownian motion 25h

   学习结果 State defining properties used in the synopsis and a non-differentiability caveat.

   形成性检查 四选一检查

2. 02 Itô integral 25h

   学习结果 Construct the integral on simple processes and name the isometry used.

   形成性检查 书面推导

3. 03 Itô’s formula 25h

   学习结果 Apply Itô’s formula to a standard function of Brownian motion.

   形成性检查 书面推导

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

   学习结果 State Lipschitz-type hypotheses and a case where they fail.

   形成性检查 反例

5. 05 Linear SDEs and examples 25h

   学习结果 Solve a linear SDE that the syllabus treats in closed form.

   形成性检查 书面推导

6. 06 Synthesis 25h

   学习结果 Write one argument: model, Itô step, and a uniqueness or interpretation caveat.

   形成性检查 书面推导

学习结果

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

学科技能

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

可迁移技能

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