The integral is the area under the curve. Is there a similar notion for stochastic integrals?

As discussed in the answers to this question, the integral is defined to be the (net signed) area under the curve. The definition in terms of Riemann sums is precisely designed to accomplish this.

Now the stochastic integral in Ito calculus is more formally defined and the result of the integration is another stochastic process. Is there a similar geometric Interpretation of a stochastic integral?

Are there special cases which are simpler to understand? For example what about Brownian Motion? Are there restrictions which allow pathwise integration?

Edit: I found a related question here which asks how to compute $\int W_sdW_s$.

Solution 1:

For the geometric interpretation there is a nice result in the book by Karatzas and Shreve (chapter 3 2.29): Having a partition $\Pi$ with $0=t_0<t_1<\dots<t_n=T$ and defining $$ S_\varepsilon(\Pi) = \sum_{i=0}^{n-1} \big[(1-\varepsilon)W(t_i)+\varepsilon W(t_{i+1})\big]\big(W(t_{i+1})-W(t_i)\big) $$ then $$ \lim_{\|\Pi\|\to 0} S_\varepsilon(\Pi) = \frac 12 W^2(t)+(\varepsilon-1/2)t. $$ For $\varepsilon=0$ the right-handed side is the Itô integral $\int_0^T W(s)dW(s)$ and for $\varepsilon=1/2$ this is the Fisk-Stratonovich integral $\int_0^T W(s)\circ dW(s)$. So this is a nice interpretation for the stochastic integral as a geometric area. The only drawback is that the limit is in $L^2$-sense and we are dealing with stochastic processes (not just real functions) depending on a stochastic state $\omega$.

The general stochastic integral is first defined for simple (step-)functions exactly using the notion of area as in the Riemann case. For arbitrary functions approximations and limits are used. Thus again the notion of area is in the integral however somehow hided by the $L^2$-convergence and possible exclusions of null-sets where the integral can be defined completely arbitrary.

In classical analysis area and volume calculations are an important topic, but as Karatzas and Shreve point out in the introduction to their chapter 3, stochastic calculus was invented (and streamlined) for handling stochastic differential equations. So putting to much weight on the geometric interpretation might be misleading.

Solution 2:

Let $X$ be a semimartingale (this covers in particular the case of a Brownian motion) and let $H$ be a predictable and locally bounded process. In this case, the integral process of $H$ with respect to $X$ is well-defined. For each $n$, let $t^n_0,\ldots,t^n_{K_n}$ be a partition of $[0,t]$ and assume that the mesh (i.e. size) of the partitions tends to zero, so that the partitions become arbitrarily fine. It then holds that $$ \sum_{k=1}^{K_n} H_{t^n_{k-1}}(X_{t^n_k} - X_{t^n_{k-1}}) \to \int_0^t H_{s-}dX_s $$ where the convergence is in probability. See for example the book "Stochastic integration and differential equations" by P. Protter for more on this. The right-hand side is very similar to a Riemann sum, and represents an approximation to "the net signed area under the curve" where $H$ is the curve and $X$ is the weight assigned per time. A more precise analogy is to the Lebesgue integral of a function with respect to a measure. In any case, this yields a "geometric interpretation" of the integral very similar to the classical (Riemann / Lebesgue) case.