The scale function s(x) of a diffusion dX = μ(x)dt + σ(x)dW transforms the process into a local martingale: s(X(t)) is a local martingale. What does this mean for the original process?
AThe process X(t) is itself a martingale in disguise
BThe drift has been 'factored out' — s maps X to a process with no directional tendency, isolating the effect of volatility
CThe process X(t) has zero quadratic variation after the transformation
DThe scale function s is constant, meaning X is already a local martingale
By Itô's formula, s(X) satisfies ds(X) = s'(X)σ(X)dW (the drift term vanishes by construction of s). This means s(X) is a local martingale — a process with no drift, only diffusion. The scale function 'straightens out' the drift: in the s-coordinate, the process is directionless and moves like a time-changed Brownian motion. The original drift μ(x) is encoded in the nonlinearity of s — regions where s changes rapidly correspond to regions where the drift is strong.
Question 2 Multiple Choice
For the Ornstein-Uhlenbeck process dX = -θX dt + σ dW, the scale function is s(x) = ∫₀ˣ exp(θy²/σ²) dy. Since s(x) → ±∞ as x → ±∞, both boundaries ±∞ are:
AEntrance boundaries — the process can start there but never reach them
BNatural boundaries — the process can never reach ±∞, and the boundaries are inaccessible
CExit boundaries — the process reaches ±∞ in finite time
DRegular boundaries — the process can reach and return from ±∞
Feller's boundary classification uses the scale function and speed measure. For the OU process, s(x) → ±∞ and the speed measure integral ∫m(dx) over any neighborhood of ±∞ diverges. This means ±∞ are natural boundaries: the process cannot reach them in finite time (from either direction), and they are completely inaccessible. This is consistent with the OU process being positive recurrent — it always returns to the center and has a stationary Gaussian distribution.
Question 3 Short Answer
Explain why one-dimensional diffusions are much more tractable than higher-dimensional ones, and what specific tools are available in one dimension that fail in higher dimensions.
Think about your answer, then reveal below.
Model answer: In one dimension, the scale function s(x) and speed measure m(dx) completely characterize the diffusion: boundary behavior (Feller's classification), hitting probabilities (P(hit a before b | start at x) = (s(x)-s(b))/(s(a)-s(b))), stationary distribution (proportional to m(dx) when boundaries are natural), and Green's functions all have explicit formulas. These rely on the fact that a one-dimensional process must pass through every intermediate point to go from a to b — there's no way to 'go around.' In two or more dimensions, the process can bypass points, the scale function doesn't exist as a scalar function, and classification requires PDE methods (Dirichlet problems) rather than ODE methods.
The one-dimensional theory is a complete theory: every question about a diffusion on an interval can be answered in terms of s and m. The passage from ODE (one dimension) to PDE (higher dimensions) is the fundamental reason the theory becomes harder. Higher-dimensional diffusions are studied primarily through their generators (elliptic operators) and Kolmogorov equations.