Tools from Stochastic Calculus 2
summary
Constructs a continuous bit-sampling martingale from Brownian stopping times whose coordinates are iid copies and whose one-step jump probability is asymptotically jump rate dt/(2t), so near 0 it jumps infinitely often but only finitely many times on [ε,1]. Defines Mt = f(xt) and uses Ito isometry to get quadratic variation equal in expectation to Var(f) and pointwise to quadratic variation = ∫_0^1 t||∇f(xt)||^2 dt. Proves a martingale lemma: P([M]_1 > c) ≥ c'·Var(f), and uses that to derive Talagrand’s improvement replacing squared influences by linear influences.
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chapters (11)
- 0:00Continuous sampling methods
- 1:02Martingale process setup
- 4:03Brownian motion connection
- 7:05Heuristic jump calculations
- 14:26Multidimensional martingales
- 18:51Martingale properties
- 23:39Quadratic variation
- 26:55Variance representation
- 40:14Talagrand's theorem
- 46:04Main lemma proof
- 59:50Q&A and discussion
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