# Stochastic Integral Graph

As we can represent the integration of $$f(x)$$ on $$[a,b]$$ with the graph below,

I was wondering how to represent the following integral with $$X(t)$$ a Brownian motion, $$f(t)$$ any function and $$t_j = \frac{jt}{n}$$ (source : Willmot)

Best visualised in 3D. 2D works for Riemann (when the integrator is x, as in dx) but for Riemann-Stieltjes (when the integrator is a function of x, e.g., $$\int{f(x)dg(x)}$$), visualisation in 3D is more revealing. You can also then interpret the 3D chart in terms of its projection in 2D.
When the integrator is Brownian, as @ilovevolatility pointed out it, the $$dX(t)$$ will be very zigzaggy- you can visualise the integral, but this zigzags makes the proof of convergence hard. Hence that is why the interpretation of the stochastic integral in terms of the simple functions, as opposed to the sum of rectangles, is used in stochastic integration.
Is $$f$$ a deterministic and differentiable function function of time? If so, write $$fdX = d(fX) - X (df/dt) dt$$ The integral of the first term on the right is just the terminal value, the second term looks like your graph but it will be jagged because $$X$$ has jagged paths.