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stochastic process: how can probability space be the same?

"What am I doing wrong"? Answering that first, what you're doing wrong is changing your source of randomness to fit the random variable in question. You shouldn't do that. In experiment ...
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Is the sum of two Brownian motions always a martingale, even if the two are possibly correlated?

I interpret the question to mean: If $\{B(t)\}$ and $\{W(t)\}$ are two Brownian motions defined on the same probability space, is $\{B(t)+W(t)\}$ necessarily a Martingale even if $B$ and $W$ are ...
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Book suggestions (Stochastics, Brownian Motion etc.)

I would suggest the following books: J. M. Steele, Stochastic Calculus and Financial Applications (*) R. van Handel, Stochastic Calculus, Filtering, and Stochastic Control (*) J. R. Norris, Markov ...
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If $\frac{dX_{t}}{X_{t}} = dL_{t}$, where $L_{t}$ is a local martingale, then is $X_{t}$ a local martingale?

There is essentially only one process $X$ satisfying $dX_t = X_t dL_t$, namely $X_t := C\exp(L_t - \frac 12 \langle L,L\rangle_t)$ where $C \in \mathbb{R}$ can be arbitrary. More precisely, if $Y$ is ...
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Ito integral over an indicator function

As the comment says, the question makes little sense for a geometric Brownian motion. So let $P$ be just a general Ito process with $P_t = \int_0^t a_s ds + \int_0^t b_s dW_s$. You want to write the ...
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Is it possible that the first arrival time of a poisson process is infinite?

A homogeneous Poisson process with intensity $\lambda > 0$ almost surely has a finite first arrival time, since $$\lim_{t \to \infty} \Pr[T \le t] = \lim_{t \to \infty} 1 - e^{-\lambda t} = 1.$$ ...
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$(a)$ Calculate $P(X_5>0|W_3=1)$ given that $\{W_t\}$ is an Simple Brownian Motion and $\{X_t\}$ a stochastic process

Hint \begin{align*} \mathbb P\{X_5>0\mid W_3=1\}&=\mathbb P\{W_5^2>5\mid W_3=1\}\\ &=\mathbb P\{(W_5-W_3)^2+2(W_5-W_3)W_3+W_3^2>5\mid W_3=1\}\\ &=\mathbb P\{(W_5-W_3)^2+2(W_5-W_3)+...
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Does local Hölder-continuity suffice to get an upper bound for Hausdorff dimension - or does it need to be global? ((Mistake in the literature?))

The issue is the definition of local Holder continuity. In general, we say that a property $P$ of metric spaces is satisfied locally in a metric space $M$, if every point in $M$ has an open ...
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Quadratic variation of linear combination of two processes

Since either $W = 0$ or $W = 1$, $Z_t := W X_t + (1-W)Y_t$ will either equal $X_t$ or $Y_t$ for all $t$. Hence $[Z]_t = W [X]_t + (1-W)[Y]_t$. This could be verified by checking that $Z_t^2 - [Z]_t$ ...
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You are right. (1) is wrong. Here is a counter-example: let $(Y_t)$ be Brownian motion and $\tau=\inf \{t \geq 0: Y_t=1\}$. Then $Y_{\tau}=1$. If we take $s \leq t$ the LHS of (1) is $1$ and RHS is $... • 2,226 1 vote Accepted How$\{B(t): t \leq T\}$is$\mathcal{F}^+-$measurable in Brownian Motion, with$T$a stopping time Following up on my comment, we will show that given$A=[B_T \in I]$for any interval$I$, we have$A \cap [T\le t] \in \mathcal{F}^+(t)$for all$t$. To take this to a Borel set of$\mathbb{R}$rather ... • 1,702 1 vote Accepted Brownian motion and martingale Yes, this can be proved in the same way. As noted in the comments, when$f\$ is harmonic the second term vanishes.
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