Brownian Motion: Construction and Properties

Brownian motion (Wiener process) is a mathematical model of chaotic motion, described by Brown in 1827 while observing pollen on water. It is the limit of random walks and the foundation of stochastic analysis.

Definition and Construction

Standard Brownian motion: The process {W_t, t ≥ 0} such that:

1. W₀ = 0
2. Independence of increments: W_{t₄}-W_{t₃} ⊥ W_{t₂}-W_{t₁} for 0≤t₁<t₂≤t₃<t₄
3. Normality: W_t - W_s ~ N(0, t-s)
4. Continuity of trajectories: t → W_t is continuous

Construction (Levy–Centsov): Via Gaussian series over Haar functions: W_t = Σ_{n,k} Z_{nk}·H_{nk}(t). Or via the CLT: W_t = lim_{n→∞} S_{⌊nt⌋}/√n (scaling limit of the random walk).

Key Properties

Nowhere differentiability: W_t is everywhere continuous, but nowhere differentiable (almost surely). Variation: total variation is infinite, quadratic variation [W]_t = t (finite!). [W,W]_t = t — “key fact” underpinning Ito’s lemma.

Self-similarity: {W_{ct}} =_d {√c W_t} (in the sense of finite-dimensional distributions). Scaling.

Reflection principle: P(max_{s≤t} W_s ≥ a) = 2P(W_t ≥ a) = 2(1-Φ(a/√t)).

Assignment: (a) Simulate Brownian motion as the limit of the random walk: n=1000 steps over T=1. Draw 10 trajectories. (b) Compute E[W_t²], E[W_t⁴], Var[W_t²]. (c) P(W_1 > 1, W_2 > 2)? Use bivariate normal.

Brownian Motion: Construction and Properties

Levy–Khintchine construction: W_t = Σₙ Zₙ hₙ(t), where hₙ are Haar functions, Zₙ ~ N(0,1) i.i.d. Convergence in L² and almost surely. In the limit — continuous, but nowhere differentiable trajectories.

Variation of Brownian motion: Total variation |W_t| = ∞ almost surely. Quadratic variation: [W]_t = t (deterministic!). This is fundamental for stochastic integration.

Reflection principle: For τ = min{t: W_t = a}: P(max_{s≤t} W_s ≥ a) = 2P(W_t ≥ a). Time of first reaching level a: τ ~ Levy inverse Gaussian. Used in the theory of insurance ruin.

Path Space and Wiener Measure

Wiener measure W on C[0,∞) — probability measure of Brownian paths. Finite-dimensional distributions: (W_{t₁},...,W_{tₙ}) ~ N(0, Σ), Σᵢⱼ = min(tᵢ, tⱼ). Completely characterizes W (Kolmogorov’s theorem).

Conditional Brownian motion (Brownian bridge): W_t|(W_T=0): Brownian bridge. Used in statistics: Kolmogorov–Smirnov test — distribution of the maximum of the Brownian bridge. E[W_t|W_T=0] = 0; Cov(t,s|W_T=0) = min(t,s) − ts/T.

Multidimensional Brownian Motion and Correlation

For d-dimensional GBM with drift vector μ and correlation matrix Σ: dS = diag(S)(μdt + Σ^{1/2}dW). Asset portfolio modeling. Cholesky decomposition: Σ = LLᵀ. Simulation: dS = diag(S)(μdt + L·dZ), where Z ~ N(0,Idt).

Stochastic Differential Equations: Existence and Uniqueness

For SDE dX = b(X,t)dt + σ(X,t)dW: under Lipschitz and linear growth conditions: |b(x,t)−b(y,t)| + |σ(x,t)−σ(y,t)| ≤ K|x−y|, |b(x,t)| + |σ(x,t)| ≤ K(1+|x|) — there exists a unique strong solution. Weak solutions exist under weaker conditions (Stroock–Varadhan theorem). Example of violation: dX = sign(X)|X|^{1/2}dW — no uniqueness (Tanaka SDE).

Processes with Jumps: Lévy Models

Generalization of Brownian motion by inclusion of jumps: X_t = μt + σW_t + Σᵢ Yᵢ·I(Nₜ ≥ i), where N is a Poisson process, Yᵢ ~ F. Lévy process: independent stationary increments, stochastically continuous. Characteristic triple (μ, σ², ν): ν — jump measure (Lévy measure). Lévy–Khintchine formula: E[e^{iuX_t}] = exp{t·(iμu − σ²u²/2 + ∫(e^{iuy}−1−iuy·I(|y|<1))ν(dy))}.

Risk Measurement: VaR and CVaR

Value at Risk (VaR_α): VaR_α(X) = inf{m: P(X+m < 0) ≤ 1−α}. Loss quantile at level α. Issue: not subadditive — diversification may increase VaR. Expected Shortfall (CVaR): CVaR_α = (1/(1−α))∫_{α}^1 VaR_u du = E[X|X > VaR_α]. Coherent risk measure: subadditivity, monotonicity, homogeneity, translation invariance. Basel III requires CVaR₉₇.₅% instead of VaR₉₉%.

Numerical Methods in Stochastic Problems

Euler–Maruyama scheme for SDE: Xₜ₊Δ ≈ Xₜ + b(Xₜ)Δt + σ(Xₜ)ΔWₜ. Order of convergence 1/2 (weak) and 1 (for functionals). Milstein scheme: adds correction term σ·σ'·(ΔW²−Δt)/2 — order 1. Runge–Kutta method for SDE — higher order, but more complex. Quasi-Monte-Carlo for SDE: replace stochastic drivers with low-discrepancy sequences → better convergence for smooth functionals.

Feynman–Kac Formula

Links PDEs and stochastic processes. If V satisfies ∂V/∂t + LV + f = 0 (L — generator), then V(x,t) = E[∫ₜᵀ f(X_s,s)ds + g(X_T) | X_t=x]. This is a powerful tool: derivative pricing problem (B-S equation) is solved via expectation over trajectories. Conversely: expectation over SDE trajectories is the solution to the corresponding PDE. Used in quantum physics (Feynman functional integral) and stochastic models.

Filtering and State Estimation

Nonlinear filtering: particle filter (Sequential Monte Carlo). At each step: weight wᵢ ∝ P(y_t|xᵢ_t)·P(xᵢ_t|xᵢ_{t-1})/q(xᵢ_t|y_t). Resampling when ESS < n/2. Del Moral’s theorem: as n→∞, the empirical distribution of the particles → posterior filter. Used in navigation (GPS + IMU), financial volatility filtering, robotics.

Numerical Example: Geometric Brownian Motion

Problem: Stock: S₀=100, μ=0.10, σ=0.20. Find E[S₁] and P(S₁>120) in one year.

Step 1: Ito SDE: dS=μS dt+σS dW. Exact solution: S_t=S₀·exp((μ−σ²/2)t+σW_t).

Step 2: E[S₁]=S₀·e^{μ}=100·e^{0.10}=100·1.1052≈110.52. (Expected value grows at rate μ.)

Step 3: ln(S₁/100)~N((0.10−0.02)·1, 0.04·1)=N(0.08, 0.04). σ_{log}=0.20.

Step 4: P(S₁>120)=P(ln(S₁/100)>ln(1.2))=P(Z>(0.1823−0.08)/0.20)=P(Z>0.512)≈1−0.696=0.304. For σ=0 (deterministic growth) P=P(100·e^{0.10}>120)=P(110.5>120)=0. Volatility increases the chance of high gain.