# Convergence and Divergencenon-convergence of left-point and mid-point Riemann sums

In standard calculus it is a well known fact that left-point and mid-point Riemann sums do become equal in the limit. When it comes to stochastic integration this is no longer the case.
Taking the left-hand sums renders the Ito integral with an extra term, taking the midpoints renders the Stratonovich integral (see for example: Higham, p 531).
While the Ito integral is the usual choice in applied math, the Stratonovich integral is frequently used in physics. Unlike the Itō calculus, Stratonovich integrals are defined such that e.g. the chain rule of ordinary calculus holds.

My question
1.) What is/are the deeper reason(s) that we have a convergence in ordinary calculus but have a divergence non-convergence here?
2.) Are there examples in non-stochastic (or ordinary) calculus where we also have a divergence between non-convergence of these limiting cases? If yes, how do they look like (and why)? Could you give a toy example of such a function?

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In standard calculus it is a well known fact that left-point and mid-point Riemann sums do become equal in the limit. When it comes to stochastic integration this is no longer the case.
Taking the left-hand sums renders the Ito integral with an extra term, taking the midpoints renders the Stratonovich integral (see for example: Higham, p 531).
While the Ito integral is the usual choice in applied math, the Stratonovich integral is frequently used in physics. Unlike the Itō calculus, Stratonovich integrals are defined such that the chain rule of ordinary calculus holds.

My question
1.) What is/are the deeper reason(s) that we have a convergence in ordinary calculus but have a divergence here?
2.) Are there examples in non-stochastic (or ordinary) calculus where we also have a divergence between these limiting cases? If yes, how do they look like (and why)? Could you give a toy example of such a function?

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# Convergence and Divergence of left-point and mid-point Riemann sums

In standard calculus it is a well known fact that left-point and mid-point Riemann sums do become equal in the limit. When it comes to stochastic integration this is no longer the case.
Taking the left-hand sums renders the Ito integral with an extra term, taking the midpoints renders the Stratonovich integral (see for example: Higham, p 531).
While the Ito integral is the usual choice in applied math, the Stratonovich integral is frequently used in physics. Unlike the Itō calculus, Stratonovich integrals are defined such that the chain rule of ordinary calculus holds.

My question
1.) What is/are the deeper reason(s) that we have a convergence in ordinary calculus but have a divergence here?
2.) Are there examples in non-stochastic (or ordinary) calculus where we have a divergence between these limiting cases? If yes, how do they look like (and why)? Could you give a toy example of such a function?