A is an adapted process $X$ on a prescribed filtered probability space carrying a prescribed Brownian motion $W$, satisfying
$$
X_t=X_0+\int_0^tb(X_s)ds+\int_0^t\sigma(X_s)dW_s
$$
almost surely. A <weak solution of a stochastic differential equation> consists of a filtered probability space, a Brownian motion, and an adapted process on that space satisfying the same integral equation; the space and driving Brownian motion are part of what may be chosen.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 202 / 4 / a / Solution

A strong solution is adapted to the filtration of a prescribed driving <Brownian motion> on a prescribed probability space and satisfies the stochastic integral equation almost surely.

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