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By the law of Large numbers, $X_t/t \to b$ almost surely as $t \to +\infty$, hence $X_t \to +\infty$ almost surely as $t \to +\infty$. Therefore $X_\infty = +\infty$ almost surely under $P$ and also u …

The process $M$ defined by $M_t = \exp(2B_t-2t)$ is a non-negative continuous martingale starting at $1$ and converging to $0$. Given $a \ge 0$, the values of $M$ until $\tau_a := \inf\{t \ge a : B_t …

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 …

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 …