Tagged Questions
-1
votes
1answer
97 views
Show that $M$ is a martingale
Let $B$ be typical Brownian motion with $\mu >0$ and $x \in \mathbb{R}$. $X(t):=x+B(t)+\mu t$, for each $t\geqslant 0$, Brownian motion with velocity $\mu$ that starts at $x$. For $r \in ...
1
vote
1answer
58 views
Using a Brownian martingale to compute the second moment of a hitting time
Prove $ W_t=B_t^4 -6B_t^2t+3t^2$ is a martingale, and compute $E(T^2)$ where $T=\inf(t\ge0,B_t=-a, B_t=b)$ if $a=b$.
Ok, if $0\lt t\lt s$, $W_t$ is a martingale if $E(W_s|[B_r]_{r\le t})=W_t$
So ...
1
vote
1answer
47 views
Backward martingale property of quadratic variation
Let $\pi_n$ denotes a refining sequence of partitions of a finite closed interval (refining means $\pi_n\subset\pi_{n+1}).$ And we denote $\pi_n B = \sum_{t_i\in \pi_n}(B_{t_{i+1}}-B_{t_i})^2$, where ...
0
votes
1answer
110 views
Checking for Martingales on Stochastic processes
I am confused about how to check whether a process is Martingale. I know, I have to check for clear drift but a bit confused about to approach this problems. I need to apply Ito's first i think. For ...
0
votes
0answers
136 views
Are these processes martingales?
Determine and prove if the following processes $ Y(t) $ are martingales. Assume that $ X(t) $ is the standard Brownian Motion
$$ Y(t) = e^{\sigma X(t)-0.5\sigma^2t} $$
$$ Y(t) = e^{0.5t}\Bigg(1 - ...
1
vote
1answer
120 views
d-dimensional Brownian motion and martingales
I was solving questions from the Martingales chapter in "Stochastic Processes" by Richard Bass. There was a question regarding d- dimensional Brownian motions(BM):
Let $(W_t^1,...,W_t^d)$ be a d ...
0
votes
1answer
77 views
How to show that $X_t = \frac{1}{\left| B_t -x\right|}\mathbb{1}_{\left\{ B_t \neq x\right\}}$ (“inverse brownian”) is a martingale?
Consider
$$X_t = \frac{1}{\left| B_t -x\right|}\mathbb{1}_{\left\{ B_t \neq x\right\}}$$
where $ \left(B_{t }\right)_{t \geq 0}$ is a $ \mathcal F_t$- brownian motion in $\mathbb R ^3$, null at ...
-3
votes
1answer
161 views
How to prove the martingale?
How to prove that the integral $\int_{0}^{+\infty}\upsilon e^{-ru}S_{u}dW_{u}^{Q}$ is a martingale under Q where $S_{t}$ is a martingale under Q and $\mathbb{E}^{Q}[\int_{0}^{+\infty}|\upsilon ...
1
vote
1answer
166 views
How do you show this is a martingale?
How do you show the following process is a martingale? My notes say it is a martingale by I can't work it out.
$$
E[e^{\sigma B(t) - \frac{\sigma ^2 t}{2}} | \mathscr{F}(s)]
$$
I tried to multiply ...
3
votes
2answers
258 views
Show that this process is a martingale
Let $B_t$ be a Brownian motion and $M_t=\max_{0\leq s\leq t}B_s$. Show that: $$(M_t-B_t)^4-6t(M_t-B_t)^2+3t^2$$
is a martingale for $t\geq0$.
1
vote
1answer
123 views
Martingale representation theorem
Trying to figure out how to solve problems on the 'form':
Find a real number $z$ and a square integrable, adapted process $\psi(s,w)$ such that
$$G(w) = z + \int \psi(s,w)\,dB_s(w)$$
for som ...
3
votes
1answer
119 views
Optional sampling exercise
I came across the following exercise in Stochastic Calculus:
Let $B=(B_t)_{t\geq0}$ be a standard Brownian motion. Let also $M$ be the following process:
$M_t=B^4_t-6t(B^2_t-\dfrac{t}{3})$ for ...
1
vote
1answer
105 views
Wiener martingale
Is $\frac{W^{2}(t)}{t}$ a martingale w.r.t. the usual filtration? $t>0$ and $W(t)$ is the Wiener process.
What I have so far:
Define $Z(t)=\frac{W^2(t)}{t}$.
By Ito we get
...
2
votes
2answers
61 views
Certain transformation of Brownian motion is a submartingale
I have a question about a proof in Protter. Let $B$ Brownian motion and $u$ a harmonic (subharmonic) function. Then $u(B)$ is a local martingale (submartingale). I was able to show the case of local ...
1
vote
1answer
403 views
Questions and Solutions in Brownian Motion and Stochastic Calculus?
I am currently studying Brownian Motion and Stochastic Calculus. I believe the best way to understand any subject well is to do as many questions as possible. Unfortunately, I haven't been able to ...
3
votes
1answer
190 views
$\mathcal{F_t}$-martingales with Itô's formula?
I need a little help with a problem. I am given some stochastic processes and supposed to show that they are $\mathcal{F_t}-$martingales. The first one is this, and they all look similar:
...
0
votes
1answer
228 views
$d$-Dimensional Brownian Motion Martingales
Let $d > 1$ and let $W_t$ denote a standard $d$-dimensional Brownian motion starting at $x\neq 0$. Let $M_t = \log|W_t|$ for $d = 2$, and $M_t= |W_t|^{2-d}$ for $d > 2$. Show that $M_t$ is a ...
1
vote
1answer
334 views
Convergence of the exponential martingale
How can we show that this martingale $$ e^{aW_{t} - \frac{1}{2}a^2t}$$ converges to $0$ as $ t \rightarrow \infty$ using law of iterated logarithm, for $a \neq 0$.
12
votes
3answers
1k views
The Laplace transform of the first hitting time of Brownian motion
Let $B_t$ be the standard Brownian motion process, $a > 0$, and let $H_a = \inf \{ t : B_t > a \}$ be a stopping time. I want to show that the Laplace transform of $H_a$ is
...
