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We have $U_{a,b}^T(Y) = U_{a,b}^{h(T)}(X)$. By construction, the function $h$ maps $[0,T]$ onto $[0,h(T)]$. More precisely, each $t' \in [0,h(T)]$ can be written $h(h^\leftarrow(t'))$, where $h^\lefta …

Partial answer The question may be whether the process $f$ is progressively measurable under the assumptions. If yes, we can conclude as follows. Fix $t \ge 0$. For each $\omega$, $f(\cdot,\omega)$ is …

Partial answer First, an heuristic argument. When we condition by events with low probability, the main is given by behaviour the less improbable situation. Here we condition by $S_1 := \max_{0 \le s …

I think that I have a counterexample. Let $(X,Y)$ be a Brownian motion in $\mathbb{R}^2$. Then $M = \int_0^\cdot X_s \mathrm{d}Y_s$ is a martingale, in the natural filtration of $(X,Y)$, in its own fi …

Almost surely, we can write for every $t \ge 0$, $M_t=M_0+\beta_{\langle M,M \rangle_t}$, where $\beta$ is some Brownian motion. By Kahane's theorem, almost surely, for every $s \ge 0$, $\limsup_{\del …

As I mentionned in my comment, there is an ambiguity in the statement of you question. Anyway, if $B$ is a standard one dimensional Brownian motion, if $\lambda$ is a real number, then $(B_t-\lambda t …

Partial answer. Continuous (local) martingales are time-changed Brownian motion. A way to obtain funny local martingales which also are Markov processes is to start from a Brownian motion $B$ and its …

If $X$ and $Y$ are independent random variables taking values in arbitrary spaces $E$ and $F$, if $Z = f(X,Y)$ for any measurable map $f : E \times F \to G$, then the family of distributions of the ra …

For every Gaussian centered process $X$, if one takes an independent copy $X'$, then the process $X+X'$ has the same distribution as $\sqrt{2}X$ (to see that, compare their finite dimensional marginal …

I use capital letters for random variables and small letters for possible values. Let $W$ be a brownian motion, defined on the canonical space $\mathcal{C}(\mathbb{R}_+)$ endowed with the Wiener measu …

An example: Matzinger studied the revocery of the environment from the second factor of a RWRE. https://arxiv.org/pdf/1110.6853.pdf https://matzi.math.gatech.edu/overview.pdf