Let $(\Omega,\mathcal{F},\mathbb{F},P)$ be a filtered probability space and $X=(X_t)_{t\geq 0}$ be a Brownian motion with respect to the natural filtration $(\mathcal{F}_t)_{t\geq 0}$ generated by $X$. Define $$Y_t=X_t+\mu t$$ for $t\geq 0$ and a measure $Q_t$ on $\mathcal{F}_t$ by $$Q_t(A)=E[exp(\mu X_t-\mu^2\frac{t}{2}),A]$$ for $t\geq 0$. I have a proof that $Q\circ Y^{-1}=P\circ X^{-1}$ holds, but have a […]

I am reading the following statement and cannot justify the upper limit. Any help would be greatly appreciated. Thanks. $$ E\left[\exp\left(2\int_{0}^{T}B_{s}^{2}\,{\rm d}s\right)\right] < \infty \quad\mbox{if and only if}\quad T < {\pi \over {4}} $$

Let us consider the standard Brownian motion and the natural filtration $(\mathcal{F}_t^B)$. It is known that $(\mathcal{F}_t^B)$ is not right-continuous at $t=0$. But what about $t>0$? Is it true that $(\mathcal{F}_t^B)$ is not right-continuous at $t>0$? If so, could you please explain why it is not right-continuous?

With a standard brownian motion $B_t$, I’m trying to find the distribution of the “range”: $$R_{t} = \sup_{0 \leq s \leq t} B_s – \inf_{0 \leq s \leq t} B_s = \overline{M_t}-\underline{M_t}$$ The reflection principle gives, for $a > 0$, $P(\overline{M_t} \geq a) = 2 P(B_t \geq a)$ (as stated by @A.S. as comment on […]

Let $W$ be a standard Brownian motion. From an earlier proven result I know that $N_t = \exp\left\{a W_t – \frac12 a^2 t \right\}$ defines a martingale on the natural filtration of $W$ for all $a \in \mathbb{R}$. I now want to prove that for every $a>0$ the random variable $$ \sup_{t \geq 0} \left(W_t […]

Let $B_t$ be a standard one-dimensional Brownian motion. Suppose $\lambda > 0$ and let$$T_\lambda = \min\{t : |B_t| = \lambda\}.$$Do there exist $\alpha < \infty$ and $\beta > 0$ (which may depend on $\lambda$) such that for all $t$, we have$$\mathbb{P}\{T_\lambda > t\} \le \alpha e^{-\beta t}?$$

On the first page of Ustunel’s lecture notes, he defines the Wiener measure in the following way: Let $W = C_0([0,1]), \omega \in W, t\in [0,1]$, define $W_t(\omega) = \omega(t)$. If we denote by $\mathcal{B}_t = \sigma\{W_s; s\leq t\}$, then there is one and only one measure $\mu$ on $W$ such that 1) $\mu \{W_0(\omega) […]

I think I have seen once that if the processes $\sigma$ and $W$, a Brownian motion, are independent then one has that $$ E \left[\exp \left(iu\int_t^T \sigma_s \, dW_s\right) \mid \mathcal{F} \right] = E\left[\exp \left(- \frac{1}{2} u^2 \int_t^T \sigma_s^2 \, ds \right) \mid \mathcal{F} \right] $$ for $\mathcal{F}$ the $\sigma$-algebra generated by $(\sigma_s)_{ s \leq […]

Here is the proof of the theorem. I couldn’t understand how to apply Ergodic theorem in this proof. Let $X=(X_t)_{t\geq0}$ be a fractional Brownian motion with self-similar parameter $H\in(0,1)$. We know that when $H = \frac{1}{2}$, fractional Brownian motion is in fact a standard Brownian motion and hence a semimartingale. \paragraph{}Now fix the parameter $H$ […]

Let $B$ be a Brownian motion with natural filtration $(\mathcal{F}_t)_{t\geq 0}$ and let $\mathcal{H}_t$ be the $\sigma$-algebra generated by $\mathcal{F}_t$ and $B_1$. Define $$A_t = B_t-\int_0^{\min(t,1)} \frac{B_1-B_s}{1-s}ds$$ I’m trying to show that $A_t$ is a Brownian motion with respect to $(\mathcal{H_t})_{t\geq0}$. As a first step, I’m attempting to show that $A_t$ is a martingale, but […]

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