What Stochastic Calculi Other Than Ito And Stratonovich Exist? When learning about stochastic calculus, you typically encounter Ito and Stratonovich calculi, usually in that order. There are many differences between the two (Ito processes have better martingale and Markov properties, while Stratonovich processes obey the chain rule from ordinary calculus), but at the fundamental level, these differences stem from how the integrals of each calculus is defined:


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*The Ito calculus is just integration using the forward Euler scheme: $$dX_t=a(t,X_t)dt + b(t,X_t)dW_t \Rightarrow$$ $$ X_t-X_0=\lim_{\Delta t\to 0}\Big(\sum_{n} a(t_n,X_{t_n}) (t_{n+1}-t_n) + \sum_{n} b(t_n,X_{t_n}) (W_{t_{n+1}}-W_{t_n}) \Big)$$

*The Stratonovich calculus is just integration using the Trapezoidal rule: $$dX_t=a(t,X_t)dt + b(t,X_t)\circ dW_t \Rightarrow$$ $$ X_t-X_0=\lim_{\Delta t\to 0}\Big(\sum_{n} \frac{a(t_{n+1},X_{t_{n+1}})+a(t_{n},X_{t_{n}})}{2} (t_{n+1}-t_n) + \sum_{n} \frac{b(t_{n+1},X_{t_{n+1}})+ b(t_n,X_{t_n})}{2} (W_{t_{n+1}}-W_{t_n}) \Big)$$
(The above are just rough sketches, especially the sum bounds. I denote Brownian motion by $W_t$, and $\Delta t=t_{n+1}-t_n$ is assumed constant above even though the actual time mesh is unimportant)
So...what happens if I choose another integration method, like Simpson's rule? Runge-Kutta? (I remember from Kloeden and Platen that R-K is not possible for some reason) Backward Euler? Et cetera? Is it even possible to do so? Will I end up with something reducible to Ito or Stratonovich, does it lead to "garbage" calculi (i.e. nothing of interest), or is there some other useful calculus out there? 
 A: Actually, the answer to this lies in quite a bit more advanced topic called rough path theory (beware: PDF). 
A rough path is a way of "enhancing" a $\alpha$-Hölder continuous path with some extra information. 
A rough path is an ordered pair, $\textbf{X}=(X, \mathbb{X})$ where $X\colon [0,T]\to V$ where $V$ is some Banach space (typically $\Bbb{R}$) and a second order process $\Bbb{X}\colon [0,T]^2\to V\otimes V$. The pair must satisfy $\Bbb{X}_{s,t}-\Bbb{X}_{s,u}-\Bbb{X}_{u,t}=X_{s,u}\otimes X_{u,t}$ 
The second order process defines the following integral: 
$$\int_s^t X_{s,r}\otimes dX_r=\colon\Bbb{X}_{s,t}$$
Rough paths can be thought of as a generalization of Ito and Stratonovich calculus. We can have the Ito rough path, $(B,\Bbb{B}^{Ito})$ and the Stratonovich rough path, $(B,\Bbb{B}^{Strat})$.
So your question boils down to, "how many different types of Brownian motion rough paths are there?". And the answer is given in the PDF I linked you. The answer is given in example 4.13 (page 59). If you have the process $B$, which is $\alpha$-Hölder continuous, with an enhancement, $\Bbb{B}$, then you can always add on another $2\alpha$-Hölder continuous function. 
Meaning you can invent any stochastic calculi you want just by adding functions. For example Stratonovich calculus is just Ito Calculus adding a term: $\Bbb{B}^{Strat}_{s,t}=\Bbb{B}^{Ito}_{s,t}+\frac12(t-s)I$.
I believe but cannot find proof that ALL such Brownian rough paths are functions added onto the Ito rough path. 
EDIT: I asked my adviser and it is true that all Brownian rough paths are increments of functions added onto the Ito rough path and this is indeed true and not so bad to prove. Essentially, just take the difference between $\Bbb{B}^{Ito}$ and your rough path, and the triple difference should be $0$, which as a theorem implies the difference is the difference of two variables, or the increment of a function.
So this is indeed an IFF.
