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The lesbegue integral and its generalisations are useful in many places in mathematics other than its place of origin, e.g., representation theory. Is the same true for the stochastic integral? From my meager understanding, it seems to rely on a similar theoretical background, but stochastically interpreted. Or is it too new an innovation to have gained ground elsewhere?

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 They are used to study Brownian motion, which is useful outside of financial math. Did you google "stochastic differential equations"? – KConrad Dec 2 at 17:06
The Langevin equation has applications in physics, see en.wikipedia.org/wiki/Langevin_equation – Vidit Nanda Dec 2 at 17:12
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 In what sense are stochastic integrals new? Stochastic processes arose originally from physical models of heat and diffusion and are closely linked to linear elliptic and parabolic PDE's and have been quite effective in studying their properties, especially with minimal assumptions of regularity on the coefficients. They have also been used to study the spectrum of the Laplacian and the heat kernel on a Riemannian manifold. Stochastic integrals have been an extremely active area of study for many years. The application to finance is but one of many applications. – Deane Yang Dec 2 at 17:26
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 The study of continuous-time stochastic processes is where we find these. As noted: applications not only in mathematics, but also in physics. Stochastic differential equations are more general: their solutions are not always written as stochastic integrals. But it is true that the prospect of "making money" has provided new impetus to this study. Far beyond the impetus from "discover the laws of nature". – Gerald Edgar Dec 2 at 18:02
If this is not subjective and argumentative, I don't know what is. – Igor Rivin Dec 2 at 18:26
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closed as not a real question by Steve Huntsman, Igor Rivin, David Roberts, Daniel Moskovich, S. Carnahan♦ Dec 3 at 17:23

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