# Tagged Questions

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### Expectations of certain Brownian motion equations

$B_t$ is Brownian motion. It is assumed that motion starts at $0$. I do not understand how the highlighted equalities hold true. Is the first one equivalent to ...
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### Is there an example that shows that the optional stopping theorem fails for finite (unbounded) stopping times?

Is there a martingale $M=(M_t)_{t\geq 0}$ and finite stopping times $S,T$ with $S \leq T$ a.s. such that $\mathrm{E}(|M_T|)<\infty$, but $M_S \neq \mathrm{E}(M_T|\mathcal{F}_S)$ a.s.? I found a ...
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### convergence of Ito integral

Suppose there is a deterministic process $\phi$ in $L^2(R)$. Need to prove that $\int_0^n \phi_u dW_u$ converges in $L^2(P)$ to some $X\in L^2(P)$ as $n\rightarrow\infty$. Also need to show that ...
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### Brownian motion transition density question

Let $Y_t = M_t - W_t$ where $M_t$ is the running maximum of brownian motion and $W_t$ is brownian motion. I want to show that $P^0[Y_{t+s} \in dy| Y_t = x] = p(s,x,y)+p(s,x,-y)$ where $p$ is the ...
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### Sample continuity of Brownian motion

I wanted to know if the Brownian motion and the fractional Brownian motion are almost surely sample continuous or not? Many thanks.
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### Tail of hitting times for Brownian motion on the circle

For $y\in \mathbb R/\mathbb Z$ and $\varphi\in C([0,\infty);\mathbb R/\mathbb Z)$ let $T_{y}(\varphi) \ := \ \inf\{t>0: \varphi_t = y \} \ \ \$ (first time the path $\varphi$ hits $y$) ...
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### Brownian Motion with Optional Stopping Theorem (OST)

Let $(B_t)_{t \geq 0}$ be a standard Brownian Motion and let $T:=\inf\{t \geq 0: B_t=at-b\}$ for some positive constant $a,b>0$. Calculate $\mathbb{E}[T]$. How do i begin it?
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### Expectation Geometric Brownian Motion

Can someone help show me a simple way to show: $$\mathbb{E}(S_t)= S_0e^{\mu t}$$ for $$S_t = S_0\exp\left( \left(\mu - \frac{\sigma^2}{2} \right)t + \sigma W_t\right)$$ from this page: ...
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### Distribution of Sum of Two Brownian Motions

How do we find the distribution of the sum of two Brownian Motions? The questions was asked here: Distribution of Brownian motion, and was answered with We can write ...
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### How is Brownian motion predictable?

Could someone please explain how Brownian motion is predictable? My understanding is that a predictable process is one that depends on information up to time t say but not t itself, therefore W_t has ...
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### Given the SDE: $dX_t=dB_t+b(X_t) dt$ with $(x,b(x)) \leq 0, \forall x \in \mathbb{R}^n$, prove that $E[|X_t|^2] \leq nt+E[|X_0|^2]$

I'm working on this problem: Given a solution $X_t$ to the SDE $$dX_t=dB_t+b(X_t) dt$$ where $B_t$ is an $n$-dimensional Brownian motion, and $b:\mathbb{R}^n \to \mathbb{R}^n$ a Lipschitz ...
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### Condition on $f$ for $e^{\int_0^t f(B_s)ds}$ to be of finite variation

Let $B$ be a standard Brownian motion, and, $$X_t=e^{\int_0^t f(B_s)ds},$$ for some function $f$. What are the condition on $f$ for $X_t$ to be of finite variation? Let $Y_t=\int_0^t f(B_s)ds$, if ...
Let $B$ be a standard Brownian motion and $$X_t=\int_0^t f_s ds+\int_0^t g_s dB_s,$$ where, $|f|$ and $|g|$ are both bounded, almost surely, by some positive constant $M$. Is it true that $$... 1answer 68 views ### Condition for existence of a stochastic differential equation With B a standard Brownian motion, write$$ dX_t=f_tdt+g_tdB_t. $$What are the conditions on \left(f\right)_{t\ge 0} and \left(g\right)_{t\ge 0} for X_t to exists? I think ... 1answer 272 views ### Integration of Wiener process: \int_{t_1}^{t_2} dB(s) We all know that \int_0^t dB(s) = B(t), where B(t) is a standard Brownian Motion. However, is the following identity true? Also, why or why not? \boxed{ \displaystyle \ \ \int_{t_1}^{t_2} ... 2answers 99 views ### Second order linear partial differential equation: \partial_t u(t,x)+\frac12 \partial_{x,x} u(t,x)+u(t,x)v(x)=0 Is there a way to solve$$ \partial_t u(t,x)+\frac12 \partial_{x,x}u(t,x)+u(t,x)v(x)=0? $$This appeared as a condition for$$ X_t=u(t,B_t)e^{\int_0^tv(B_s)ds} $$to be a martingale. With B a ... 1answer 876 views ### Applying Ito formula to the Brownian bridge Let B be a standard Brownian motion and$$ W_t=(1-t)\int_0^t \frac{1}{1-s}dB_s $$be a Brownian bridge. Calculate dW_t. To apply Ito formula define$$ f(t,B_t)=(1-t) \int_0^t\frac{1}{1-s}dB_s  ...
I have two possibly related questions. Let $\tau:=\min\{t\geq0:B_t=1\}$, where $B_t$ is a standard Brownian motion. I am supposed to derive the fact that $\mathbf{E}\tau=\infty$ by applying some ...