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Stochastic Analysis And Applications: From SDEs To MCMC And Generators.pdf

Notes_of_Stochastic_Analysis.pdf
Preview of Stochastic Analysis and Applications: From SDEs to MCMC and Generators
🔗 Source: zdhNarsil.github.io
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Summary

Notes of Stochastic Analysis

## Stochastic Analysis

### Ito's Formula
- Essential properties: dB²t = dt, dBtdt = 0, (dt)² = 0.
- For an Ito process Xt with dXt = b(t, Xt)dt + σ(t, Xt)dBt, and any function f(t, x):
- Perform Taylor expansion to second order:
d f(t, Xt) = ∇t f dt + ∇x f · dXt + 1/2 (dXt)ᵀ ∇²x f (dXt).
- Simplifying, Ito's formula is: d f(t, Xt) = (∇t f + ∇x f · b + 1/2 σᵀσ ∇²x f) dt + (∇x f)ᵀ σ dBt.

### Forward Equation
- Describing the evolution of probability density function p(x, t | y, s):
- For s < t, p(x, t | y, s) dx = P{Xt ∈ [x, x + dx] | Xs = y} dt.
- Using Ito's formula and expectation, we derive: ∂t p = Lp, where L is an infinite small generator defined as Lf = ∇x f · b + 1/2 σᵀσ ∇²x f.

### Backward Equation
- For s ≤ t, define u(y, s) = E[f(Xt)] given Xs = y.
- Then, du(Xτ, τ) = (∂τu + Lu) (Xτ, τ) dτ + ∇u · σ dWτ.
- Taking expectations and limits, we get: ∂s u(y, s) + Lu(y, s) = 0, which leads to the backward equation: ∂s p(x, t | y, s) + Ly*p(x, t | y, s) = 0.

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Description

Notes on Stochastic Analysis by Narsil Zhang, covering Ito's Formula, Forward and Backward Equations, SDE & MCMC methods (SGLD, Stochastic Gradient Hamiltonian MC), Wasserstein distance, and Gradient Flows with a focus on SGLD, all within the context of Peking University's CS program.

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  • File Format: PDF
  • File Size: 635 KB
  • Pages: 10
  • Language: EN
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  • Last Updated: 3 weeks ago

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