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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^ …

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 …

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

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 $X_t=Z$ 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 $t\in [0,1]$.

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 …

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

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 …

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 …

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.

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

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 …

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 …

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 …

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 …