0
votes
0answers
30 views

Brownian Motion first hitting time distribution

I have a question concerning the distribution of the first hitting time of Brownian Motion $\tau_x = \inf_{t\geq 0}\{W_t=x\}$, where $W_t$ is Brownian motion. Using some calculus, I found out that the ...
0
votes
1answer
18 views

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 ...
1
vote
1answer
33 views

Property of Brownian Motion's paths

We are considering a Brownian Motion $(B_t)_t$ with values in $\mathbb{R} $ starting from $x$ defined on the stochastic basis: $$(\Omega,\mathcal{E},(\mathcal{F}_t)_t,\mathbb{P}^x)$$ Then, let's ...
2
votes
1answer
42 views

The law of the iterated logarithm for BM and boundedness of stopping times

My question is regarding the usefulness of the law of the iterated logarithm, and its connection to stopping times. In many answers of this forum, I understand that some people often claim that some ...
2
votes
1answer
56 views

Hitting time for Brownian Motion Surplus Process

I'm trying to solve this question for a continuous surplus process. The surplus process is $$U_s=U_0+s-B_s$$ where $B_t$ is a Brownian motion representing payouts, $U_0$ is starting capital, $s$ is ...
2
votes
0answers
49 views

Stopping times, open sets and Brownian Motion

Let $B_t$ be a brownian motion started at 0. I am trying to prove that $\tau$, defined as: $$ \tau = \inf\{t > 0 \mbox{ }|\mbox{ } \left|B_t\right| \geq \frac{1}{1+t} \} $$ is a stopping time with ...
3
votes
1answer
51 views

Extended (or augmented) stopping times

I am trying to prove that $\tau$, defined as: $$ \tau = inf\{t > 0 \mbox{ }|\mbox{ } B_t < t-1 \} $$ is a stopping time with respect to the filtration $(\mathscr{F}_{t+}^B)_{t\geq 0}$ where ...
0
votes
0answers
65 views

brownian motion and stopping time

I have an exercise about Brownian motion which I don't understand completely. Let $(B_s)_{s\geq0}$ be a standard real Brownian motion. For $t > 0$, we define the random times $g_t ...
3
votes
1answer
90 views

Expectation of brownian motion at hitting time

Am i correct in my derivation? I want to calculate $\mathbb{E}B_{\tau_a}$. From the definition of the hitting time i get $B_{\tau_a}=a$, so $$\mathbb{E}B_{\tau_a}=\mathbb{E}a=a$$ I am new to the ...
2
votes
1answer
81 views

Show that this is a stopping time

Show that $\sigma=\inf \{ t\ge 0 : |B_t|= \log t \}$ is a stopping time with respect to $(\mathcal F_t^B)_{t\ge0}$. I've been trying to put the set $\{\sigma\le t\}$ equal to a countable union and ...
1
vote
1answer
58 views

Strong Markov Property Brownian Motion Question

If $\tau$ is a stopping time and $\omega(t)$ is Brownian Motion then the Strong Markov Theorem states that $Z(t)=\omega(t+\tau) -\omega(\tau)$ conditioned on $\{\tau <\infty\}$ is distributed as ...
1
vote
0answers
27 views

Probability of winding number in 2D Brownian motion

Let $B_t$ be a 2D Brownian Motion with $B_0 = (1,0)$. Now, express $B_t$ in polars, that is, $B_t = (r(t), \theta(t))$. Let $\tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}$. What is $\mathbb{P}[\tau ...
1
vote
1answer
53 views

question on Brownian Motion stopping time and end state

I came across this equation in my lecture notes, which states: $P(T_a < t , W_t \ge a) = P(W_t \ge a)$ where $T_a = \min\{t \ge 0, W_t \ge a\}$. I'm really confused by this equation: as far as I ...
1
vote
0answers
51 views

Bounding the expectation of monotone function of stopping times of Brownian motion

Let $X_t$ be a standard Brownian motion and let $Y_t:=X_t + \epsilon B_t$ where $B_t$ is an independent standard Brownian motion and $\epsilon>0$ is small. Let f be a monotone increasing function. ...
1
vote
0answers
107 views

Brownian motion hitting probability

Let $B_t$ be a brownian motion and $g(t)$ a function of the time $t$. $B_0=0$. Let $\Phi$ be the c.d.f. of a normal distribution. At time $t$, the probability that $B_t > g(t)$ equals ...
2
votes
2answers
233 views

First hitting time for a brownian motion with a exponential boundary

Let $B_t$ be the standard Brownian Motion. Is the distribution/density of the first hitting time of $B_t$ for an exponential decaying boundary known? Trying to be more formal, if ...
2
votes
1answer
183 views

Conditional hitting time distribution of a Brownian motion

This problem cropped up in some research I am doing. I imagine it is standard, but I cannot seem to find the answer. Let $W_t$ be a standard Brownian motion. Suppose there are four values $a < 0 ...
2
votes
1answer
195 views

Doob's stopping time theorem with unbounded stopping time

Let $(X_t)_{t\geq0}$ be Brownan motion on $\mathbb R$, and $\tau$ is a stopping time adapted with the natural filtration generated by the Brownian motion. If $X_0=0$, $E(e^{\tau/2})<+\infty$. ...
4
votes
1answer
258 views

Stopping time and Brownian motion (specific example)

Let $B$ be a Brownian motion. I want to show that $$ \inf\{t\geq0 \mid B(t)=\max_{x\in [0,1]}B(s)\} $$ is not a stopping time w.r.t. the standard filtration. How can one intuitively see that this ...
3
votes
1answer
174 views

Optional sampling exercise

I came across the following exercise in Stochastic Calculus: Let $B=(B_t)_{t\geq0}$ be a standard Brownian motion. Let also $M$ be the following process: $M_t=B^4_t-6t(B^2_t-\dfrac{t}{3})$ for ...
3
votes
1answer
214 views

Stopping time on Wiener Process

Let $W_t$ be a Wiener process and for $a\geq0$ $$\tau_a:=\inf \left\{ t\geq0: |W_t|=\sqrt{at+7} \right\}.$$ Is $\tau_a<\infty$ almost everywhere? What about $E(\tau_a)$ then?