1
vote
0answers
73 views

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 $y$) ...
1
vote
1answer
54 views

Brownian Motion with 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?
1
vote
2answers
42 views

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: ...
2
votes
1answer
599 views

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 ...
2
votes
2answers
231 views

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 ...
3
votes
1answer
66 views

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 ...
2
votes
1answer
64 views

Clarke Ocone representation formula

Let $(B_t)_{t}$ a Brownian motion and $F \in L^2(\Omega,\mathcal{F}_T,\mathbb{P})$. Then we know by Itô's representation theorem that there exist a process $X$ such that $$F=\mathbb{E}F+\int_0^T X_s ...
9
votes
2answers
264 views

Area enclosed by 2-dimensional random curve

Consider a 2-dimensional Wiener process $(W_t)_{t \in [0,1]}$. Color every area which is enclosed by the line parametrised by $W_t$ (this means that, when the Wiener process makes a loop and ...
1
vote
1answer
68 views

Covariation of Wiener processes, $\langle W_1,W_2\rangle_t = \rho t$.

I'm wondering why this is true: $\langle W_1,W_2\rangle_t = \rho t$. Where $W_1$ and $W_2$ are standard Brownian Motion. I know that $\langle W_1,W_2\rangle_t = 0.5\big[ \langle W_1 + W_2 \rangle - ...
2
votes
1answer
343 views

Expectation of exponential martingale and indicator function.

Let $W$ be a Wiener process, $r,\sigma \in \mathbb{R}_+$ and $S(T) = S(t)e^{(r-\frac12 \sigma^2)(T-t) + \sigma(W(T)-W(t))}$. I want to evaluate $$A:=E[e^{- \frac12 \sigma^2 (T-t) - ...
0
votes
0answers
53 views

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 ...
1
vote
0answers
511 views

Stochastic integral: Interchanging the order of expectation and integration

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 $$ ...
0
votes
1answer
64 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 ...
1
vote
1answer
256 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} ...
2
votes
2answers
96 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 ...
0
votes
1answer
733 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 $$ f(t,B_t)=(1-t) \int_0^t\frac{1}{1-s}dB_s $$ ...
5
votes
3answers
1k views

On hitting time of Brownian motion and Ito's lemma

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 ...