a strong solution
A strong solution of the SDE dX_t = b(X_t) dt + sigma(X_t) dB_t is a solution built on top of a Brownian motion that is given to you in advance. You fix a probability space, a filtration satisfying the usual conditions, and a Brownian motion B_t adapted to it; a strong solution is then an adapted continuous process X_t, defined on that same space and driven by that same B, which satisfies the integral equation for all t almost surely. The point is that the randomness comes only from the prescribed noise B — the solution is, in principle, a deterministic functional of the Brownian path and the initial condition.
More precisely, X_t must be adapted to the (augmented) filtration generated by B and the initial value X_0; this is what 'strong' means — no extra randomness beyond the input noise, and the solution is measurable with respect to the past of B. Strong uniqueness (pathwise uniqueness) says any two strong solutions with the same B and same X_0 are indistinguishable, i.e. P(X_t = X'_t for all t) = 1. The existence-and-uniqueness theorem under Lipschitz and linear-growth conditions delivers a strong solution, constructed by Picard iteration: define X^(0)_t = X_0 and X^(n+1)_t = X_0 + integral_0^t b(X^(n)_s) ds + integral_0^t sigma(X^(n)_s) dB_s, then show this contracts in an L^2 norm (using the Ito isometry to control the stochastic integral) and converges to a fixed point.
The strong notion is what you need for simulation and for any statement of the form 'the output is a function of the input noise' — for filtering, for control where you observe the driving noise, and for numerical schemes like Euler-Maruyama, which approximate the strong solution path. The honest subtlety is that strong solutions can fail to exist even when weak solutions do: the Tanaka SDE dX = sign(X) dB has a weak solution (which is just a Brownian motion) but NO strong solution and no pathwise uniqueness, because the driving noise cannot be recovered from the path. So 'strong' is a genuinely stronger demand than 'weak'.
For the linear SDE dX_t = -theta X_t dt + sigma dB_t (the OU process), Picard iteration converges, and the strong solution can be written explicitly as X_t = e^(-theta t) X_0 + sigma integral_0^t e^(-theta (t-s)) dB_s. Every value of X_t is a measurable functional of the past of the one Brownian motion B.
A strong solution as an explicit functional of the prescribed Brownian path.
Strong existence is strictly stronger than weak existence: Tanaka's equation dX = sign(X) dB has a weak solution but no strong solution. Picard iteration delivers strong solutions only under Lipschitz/growth; without them you may have to settle for weak.