 Solution

ID: past-exam-of-the-stochastic-differential-equations-course-of-the-university-of-oxford/2025/1/a/i/solution

A convenient sufficient condition is that

f \in C^{2} (R^{d})

has a bounded gradient. The Itô formula for standard

d

-dimensional Brownian motion gives

f (B_{t}) - \frac{1}{2} \int_{0}^{t} Δ f (B_{s}) d s = f (0) + \sum_{j = 1}^{d} \int_{0}^{t} \partial_{j} f (B_{s}) d B_{s}^{j} .

∣\nabla f ∣ \leq C

E \sum_{j} \int_{0}^{t} \partial_{j} f (B_{s}) d B_{s}^{j}^{2} = E \int_{0}^{t} ∣\nabla f (B_{s}) ∣^{2} d s \leq C^{2} t .

Thus the right side is a square-integrable martingale, rather than merely a local martingale. In particular,

f \in C^{2} (R^{d}), x sup ∣\nabla f (x) ∣ < \infty ⟹ M^{f} is a martingale .

There are many nonconstant examples, such as

f (x) = sin x_{1}

f (x) = x_{1}

. More generally it is enough that

E \int_{0}^{t} ∣\nabla f (B_{s}) ∣^{2} d s < \infty

for each finite

t

. Smooth test functions with compact support satisfy the bounded-gradient condition and will provide the domain in the next part.

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