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The phrase "postulating" might be a bit misleading. Often, the iterated integrals (the second order process) are defined in some canonical way. Besides, all rough path lifts differ by the increment of …

I found an answer - the answer is yes. Rogers and Williams page 358.

This is of course true and follows from the interpolation inequality. (See Friz-Victoir Proposition 5.5)

Let $X_t=Z$ for all $t$ where $Z=W^2+1$ for your favorite random variable $W$. Then $X_t$ is a continuous martingale and $G(t)=t\mathbb P(X_1\geq t)=t$ for $t\in [0,1]$.

I am adding another answer as there has been a condition added. Let $Y_t=e^{B(t)-\frac{1}{2}t}$. It is well known that $Y$ is a martingale and $Y_t\geq 0$ for all $t$. Therefore letting $X_t=22+Y_t$ g …

Let $\mu_0$ be the law of a Brownian motion $B$. Let $\mu$ be any measure equivalent to $\mu_0$. Then by a converse version of Girsanov there exists a progressively measurable $F$ whose sample paths a …

Here is a partial answer. If $f\in C^2$ then we have by Itô's lemma on $f$ that for all $u$ a.s. that $$\int_0^u f'(B_r) dB_r+\frac12\int_0^u f''(B_r) dr=\int_0^u g(B_r) dB_r+\frac12\int_0^u h(B_r) dr …

By Doob-Dynkin we have that $A_t=f(t,\{B_s\}_{0\leq s\leq t})$. If $\mu_0$ is the law of Brownian motion then under the measure $\mu=\exp\left(\int_0^T A_sdB_s-1/2\int_0^T A_s^2 ds\right)\mu_0$ we hav …

There's a few results. First of all there is the classical Sanov's Theorem. One other result is about Gaussian measures. For a centered Gaussian measure $\mu_0$ on Banach space $\mathcal B$ we can def …

You have seen all kinds of integration theories before — Itô, Stratonovich, and I'm sure plenty others. Rough paths takes a step back and asks what we want from an integration theory. And so long as w …

Using the argument of http://users.sussex.ac.uk/~md326/MAP.pdf or https://arxiv.org/abs/2209.04523 We have that if $\mu_0$ is a centered Gaussian measure then its Onsager-Machlup function is $\operato …

You can definitely have Gaussian measures on Frechet spaces. However see Bogachev's Gaussian Measures Theorem 3.6.5 to see that you always have an embedded full measure Banach space.

Look at Nualart's The Malliavin calculus and related topics, Theorem 4.1.2

See also a preprint https://arxiv.org/abs/1306.2382 and a paper https://repository.lsu.edu/josa/vol5/iss2/3/ which use a kind of "Wick rotation." In the Chatterjee preprint, a Cauchy random variable c …

The following might help. Let $\mu, \nu$ be two probability measures on $\mathbb R^d$. For simplicity, let $a$ be Lipschitz with constant $L$ and $\sigma\equiv 1$. Then $$|X^\mu(t)-X^\nu(t)|\leq |x_0^ …