A definition of Stochastic integral that is both a martingale and preserves chain rule?

Question

my question is straightforward: is there any definition of Stochastic integral (so I assume at least some kind of Riemann sum compatibility and linearity) that is both a local martingale and preserves the chain rule? Or in case this is negative if there's a proven result that there can't be any definition of this kind?

I know that Ito integral preserves the local martingale property, but the chain rule is ``messed up''; then there's Stratonovich definition which preserves chain rule, but the integral is not a local martingale.

Answer 1

Nice question, but I think the answer is no. The integral that preserves ordinary calculus and be a martingale would have to be the Ito and Stratonovich integral at the same time. As we know: $$ \underbrace{\int_0^tW_s\circ \,dW_s}_{\textstyle \text{Stratonovich}}=\underbrace{\int_0^tW_s\,dW_s}_{\textstyle \text{Ito (Martingale)}}+\frac{t}{2}=\frac{W_t^2}{2}\,. $$ The last term ensures that the Stratonovich integral follows ordinary calculus. Since $x^2/2$ is convex it is also clear that the Stratonovich integral is a continuous supermartingale. Therefore:

• The uniqueness of the Doob-Meyer decomposition forbids the existens of another martingale $M_t$ such that the Stratonovich integral would equal $M_t$.

Conversely, if you want an integral that satisfies ordinary calculus, the integral of $W_t$ w.r.t. itself must equal $W^2_t/2$ which is a strict supermartingale and not a martingale.