Visualizing Brownian Stochastic Integrals and Their Approximation

Summary

The document asks how to picture an integral of a function against Brownian motion. It explains that an ordinary integral can be drawn as areas in two dimensions, while an integral against a varying integrator is more revealing in three dimensions, with a two-dimensional projection offering another view. Brownian motion's jagged paths make this visualization especially irregular.

For a deterministic differentiable function, the response gives an integration-by-parts identity that rewrites the integral against Brownian motion using a terminal product term and an ordinary time integral involving the Brownian path. This offers a conceptual route to visualization, but the document does not provide a worked example or a convergence proof. It also notes that stochastic integration commonly defines the integral through simple-function approximations rather than rectangle-area intuition, because the path irregularity complicates convergence arguments. The identity's stated differentiability condition matters; the discussion does not cover more general stochastic integrands.

Key ideas

• A Riemann integral can be visualized as area in two dimensions, while an integral against a varying function can benefit from a three-dimensional view.
• Brownian motion's jagged sample paths make direct geometric intuition and convergence reasoning more difficult.
• For deterministic differentiable integrands, integration by parts relates an integral against Brownian motion to a terminal product and a time integral.
• Stochastic integrals are often constructed through simple-function approximations rather than ordinary rectangle sums.

Tags

# Stochastic Integral Graph


# 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)

## Answer by Magic is in the chain (score 1, accepted)

https://quant.stackexchange.com/a/49607

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.

## Answer by user34971 (score 4)

https://quant.stackexchange.com/a/49602

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.