Probability Theory
Brownian Motion
The Black-Scholes formula, which earned its authors the 1997 Nobel Prize, is derived from one equation: $dS = \mu S \, dt + \sigma S \, dW_t$, where $W_t$ is Brownian motion. Without Ito's stochastic calculus, this equation has no meaning - $dW_t$ is not a differential of any differentiable function.
- Diffusion physics: the heat equation $\partial_t u = \frac{1}{2}\Delta u$ describes the distribution of Brownian motion particles. The connection between SDEs and PDEs via Ito's lemma underlies Monte Carlo methods for solving partial differential equations.
- Geometric Brownian motion (GBM) is the standard model for stock prices in options pricing. Parameters $\mu$ (drift) and $\sigma$ (volatility) are calibrated from historical data; option pricing requires $\sigma$ but not $\mu$.
- Score-based diffusion models (DDPM, Score SDE): the forward process is Brownian motion adding noise; the reverse process inverts the SDE via Anderson's formula. Stable Diffusion works exactly this way.
Цели урока
- State the axioms of the Wiener process and explain the non-differentiability of sample paths
- Apply Ito's lemma to compute the differential of a function of Brownian motion
- Derive the Black-Scholes formula from GBM via Ito's lemma and the risk-neutral argument
Предварительные знания
- Poisson process and independent increments
- Lebesgue integral and $L^2$ convergence
- Ordinary differential equations: main types
Wiener process: axioms and properties
A standard Wiener process $W_t$ is defined by: (1) $W_0 = 0$; (2) $W_t - W_s \sim \mathcal{N}(0, t-s)$ for $s < t$; (3) increments on non-overlapping intervals are independent; (4) sample paths $t \mapsto W_t$ are continuous almost surely. From these axioms: $\mathrm{Var}(W_t) = t$, $\mathrm{Cov}(W_s, W_t) = \min(s,t)$. Sample paths are nowhere differentiable with probability 1 - a consequence of axiom 2 and the Paley-Wiener theorem.
The stochastic differential equation $dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t$ under Lipschitz conditions on $b$ and $\sigma$ has a unique strong solution. Feynman-Kac formula: $u(t,x) = \mathbb{E}[g(X_T) | X_t = x]$ satisfies $\partial_t u + b \partial_x u + \frac{\sigma^2}{2} \partial_{xx} u = 0$ with $u(T,x) = g(x)$. This is the foundation of Monte Carlo methods for option pricing.
The Ito stochastic integral $\int_0^t H_s \, dW_s$ is defined only for adapted processes $H_s$ with $\mathbb{E}[\int_0^T H_s^2 \, ds] < \infty$. It is a martingale. The Stratonovich integral $\int H_s \circ dW_s$ uses a different convention and preserves the ordinary chain rule but is not a martingale. Physicists often use Stratonovich; financial mathematicians use Ito.
Kiyosi Ito developed stochastic calculus in 1942-1951 while working in Japan dur
Kiyosi Ito developed stochastic calculus in 1942-1951 while working in Japan during a period of international isolation. His work remained little known until the 1960s. The Black-Scholes formula (1973) made stochastic calculus an industrial tool: by 1987 Wall Street used it everywhere. Paul Samuelson proposed GBM as a model for stock prices in 1965.
Definition and Characterization
In 1827, botanist Robert Brown observed pollen grains in water moving in erratic, irregular paths under a microscope. Today the same mathematical object, Brownian motion, describes Tesla stock price movements with annualized volatility sigma = 0.58, the diffusion of molecules in cell membranes, and the noise driving score-based generative models in modern AI.
The covariance structure Cov(W_s, W_t) = min(s, t) is the kernel of the Wiener process Gaussian process. This kernel appears in infinite-dimensional Bayesian regression and in the theory of support vector machines on function spaces.
Given W_1 ~ N(0,1), what is the distribution of W_4 - W_1? Is it independent of W_1?
The answer follows directly from the definition and properties of the object under consideration.
Path Properties: Continuity and Quadratic Variation
Brownian paths are continuous everywhere but differentiable nowhere. This is not a curiosity; it is the source of Ito calculus. The quadratic variation of a Brownian path equals t almost surely, meaning the path accumulates squared fluctuations linearly in time. This property drives the Ito correction term that separates stochastic calculus from ordinary calculus.
Nowhere differentiability of Brownian paths is a key theoretical reason why diffusion models in generative AI (DDPM, score-matching) require stochastic differential equations rather than ordinary ones. The noise added during the forward process follows Brownian increments, and the reverse denoising score function exploits the structure of the resulting path measure.
Compute E[(W_2 - W_1)^2 + (W_1)^2]. What is the quadratic variation of W over [0, 2]?
The answer follows directly from the definition and properties of the object under consideration.
Geometric Brownian Motion and Black-Scholes
Tesla stock closed at 250 USD on a given day. With daily volatility sigma = 0.058 (annualized 0.58) and drift mu = 0.30 per year, the price tomorrow is not Gaussian. It follows a geometric Brownian motion, a process that stays positive and has log-normally distributed increments. This model is the foundation of the Black-Scholes option pricing framework used in trillions of dollars of derivatives.
Geometric Brownian motion also appears in machine learning research as a model for feature drift in non-stationary environments. The SDE framework it introduces, stochastic differential equations, generalizes to neural SDEs (Latent ODEs with diffusion), which are used for irregularly sampled time series, physical simulation, and score-based generative models such as DDPM and DDIM.
If S_0 = 100, mu = 0.05, sigma = 0.2, and T = 1, what is E[S_T] and P(S_T > 110)?
The answer follows directly from the definition and properties of the object under consideration.
Ito's lemma applied to $W_t^2$
For $f(W_t) = W_t^2$, the ordinary chain rule would give $d(W_t^2) = 2W_t \, dW_t$. Ito's lemma adds a correction: $d(W_t^2) = 2W_t \, dW_t + dt$. Integrating: $W_t^2 = 2\int_0^t W_s \, dW_s + t$. Check: $\mathbb{E}[W_t^2] = t$ (variance), and $\mathbb{E}[\int_0^t W_s \, dW_s] = 0$ (Ito integral is a martingale). The correction $dt$ is the quadratic variation: $[W]_t = t$, symbolically $(dW_t)^2 = dt$.
Итоги
- Brownian motion is a continuous process with Gaussian independent increments and nowhere differentiable paths.
- Ito's lemma: $df(W_t) = f'(W_t)dW_t + \frac{1}{2}f''(W_t)dt$.
- GBM $dS = \mu S dt + \sigma S dW_t$ solves to $S_t = S_0 \exp((\mu - \sigma^2/2)t + \sigma W_t)$ by Ito's lemma.
Connections to other topics
Diffusion models in generative AI: the forward process adds noise via an SDE like $dX = -X \, dt + \sqrt{2} \, dW_t$ (OU process), the reverse restores the image. Statistical physics (prob-28) uses the same SDEs through the Langevin equation.
- Prob 28 — related
Вопросы для размышления
- Black-Scholes assumes constant volatility $\sigma$, but the market 'volatility smile' shows $\sigma$ depends on the strike. How do SDEs with state-dependent volatility $\sigma(S_t, t)$ (local volatility models) generalize GBM, and what changes in applying Ito's lemma?