 Mean-square fundamental theorem of calculus

If an L2 space-valued map $X$ has a continuous mean-square derivative $Y$, then its increments equal the Bochner integral of $Y$. To see this, for each partition interval $[t_j,t_{j+1}]$, continuous linear observations and the scalar mean-value bound give $\Vert X(t_{j+1})-X(t_j)-(t_{j+1}-t_j)Y(t_j){\Vert }_2\le (t_{j+1}-t_j){\sup }_{u\in [t_j,t_{j+1}]}\Vert Y(u)-Y(t_j){\Vert }_2$. Sum these errors. Uniform continuity of $Y$ makes the total error tend to zero as the mesh tends to zero, while the Riemann sums converge to the Bochner integral. This proves the fundamental theorem of calculus in L2 space. If a Gaussian process derivative also has a continuous modification, its anchored path integral $X(s)+{\int }_s^t\tilde{Y}(u)du$ therefore produces a differentiable modification of a stochastic process of the original Gaussian process.