1
vote
2answers
58 views

Uniformly integrable martingale

I have the following martingale. $M_n=\exp\left(aB_n-\frac{1}{2}a^2n\right)$ for $n\geq0$ and $a\neq0$, $B_n$ is a BM. I have to show that for $a>0$, $M_n\rightarrow0$ in probability. Is $M_n$ ...
0
votes
2answers
74 views

Doob's decomposition of a brownian motion.

Let $B_n$ be a discrete Brownian motion. I need to find the Doob decomposition for ($B_n^2$). Can someone help me please. Thank you in advance.
2
votes
0answers
40 views

A right-inverse of Brownian motion local time at zero has stationary independent increments

Let $L_0^t$ be the local time for a standard Brownian motion at $0$ and define $$X_t=\sup\{s\ge0:L_0^s\le t\}, t\ge0. $$ I would like to show that $(X_t)$ has stationary independent increments. That ...
0
votes
2answers
56 views

Find the generator of Markov Process

Homework question: Consider the Markov process $X_t=B_t-t^2+t$ where $B_t$ is the Brownian motion. Find the generator $Q$ of this process. I am completely confused how to find the generator for ...
1
vote
1answer
44 views

Convergence to integral: $\sum_{k=1}^{k_n}f\left(B_{t_{k-1}^{(n)}}\right)\left(B_{t_{k-1}^{(n)}}-B_{t_{k}^{(n)}}\right)^2 \to_p \int_0^Tf(B_t)dt$

The problem goes: Let $(B_t)$ be a standard Brownian motion, and $f:\mathbb{R}\to\mathbb{R}$ be continuous. Show that if $T>0$ and $(P_n)$ is a sequence of partitions of $[0,T]$: ...
3
votes
0answers
44 views

Conditional expectation and coupled set of ODEs

How to find a coupled set of ODEs and initial conditions for the deterministic functions $a$ and $b$ such that $$\mathbb{E}\left[e^{-\int_{t}^{T} W^2(u)du} | \mathcal{F(t)}\right] = e^{-a(T-t) - ...
5
votes
2answers
105 views

Estimating the maximum of a Brownian motion over the unit interval

Let $\left(B_t\right)_{t \in \left[0,\infty\right)}$ be a standard Brownian motion over the probability space $\left(\Omega, \mathcal{A}, P\right)$. For each $x \in \left(0, \infty\right)$, give an ...
2
votes
0answers
99 views

Almost sure non differentiability of Brownian Motion

Problem: Let $t>0$, show that the standard Brownian motion is almost surely not differentiable a $t$ Now, through a Borel Cantelli argument I proved that, almost surely $$\limsup_{\epsilon ...
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
0answers
24 views

Brownian motion minimisation problem

Let $B_t$ be a Brownian motion, let $\sigma > 0$ be fixed and let $X_t$ be a process with fixed beginning value $x_0$ that satisfies $dXt = u_tdt + \sigma X_tdB_t.$ Solve ...
2
votes
0answers
49 views

Can anyone explain me this proof about a Brownian Motion?

Prove that the process $W_t=(1+t)U_{t/(1+t)}$ on $[0,\infty)$ is a Brownian motion. $\text{(b)}$ Clearly $Y_0=U_0=0$, and inherits continuity of sample paths from $U_t$ (and hence from $W_t$). ...
1
vote
1answer
202 views

Distribution of Brownian Bridge

PROBLEM $U_t = B_t - tB_1$, $B_t$ is a Brownian motion on $[0,1]$. What is a Brownian Bridge and give the twodimensional distributions of the vector $(U_s, U_t)$. I think that a Brownian ...
0
votes
2answers
343 views

Variance of stochastic integral of brownian motion

How do i compute this integral? $ Var [\int_0^T W(t)dW(t)] $ I know the following $E [\int_0^T W(t)dW(t)]$ is 0 but i'm not sure how to apporch the above
3
votes
1answer
65 views

The first two moments of $\int_0^1 B_s^2 \, ds$

I was trying to solve the following problem from Continuous Martingales and Brownian Motion by Daniel Revuz and Marc Yor, but got my solution back as the answer for variance was wrong. I have already ...
1
vote
0answers
77 views

Expected value of brownian motion for all positive paths

I've got this question but I can't figure it out. Derive the expected value of $B(t_1)$ of all paths that are positive $t_1$ and calculate the expectation for $t_1=1$ and variance$=1$? Thanks
1
vote
1answer
471 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 ...
3
votes
1answer
53 views

two r.v sharing the same law

I have a question: Let $X=B^{+}$ or $X=|B|$ where $B$ is the standard Brownian motion. Set $$J_p=\sup_{t\geq 0}(X_t-t^{\frac{p}{2}})$$ where $p>1$ and $q$ its conjugate ...
1
vote
1answer
285 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 ...
1
vote
1answer
457 views

Expectations of multiplied Wiener processes.

I wish to evaluate the following: $E[W(t-1)W(t)^2]$ $E[W(t)^3]$ where $t > 1$, $W$ is a standard Brownian motion and we are at $\mathscr{F}_0$ now. I know that $E[W(t-1)W(t)] = \min{(t-1,t)} ...
1
vote
1answer
31 views

a homework question about Levy air

I have a question in my homework: Let $X_t$ and $Y_t$ be two Brownian motions issue de $0$ and define $$S_t=\int_0^tX_s\,dY_s-\int_0^tY_s\,dX_s$$ Show that $$E[e^{i\lambda S_t}]=E[\cos(\lambda ...
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
369 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) - ...
2
votes
1answer
188 views

About Brownian motion

Let $T$ be the last time before $1$ a Brownian motion visits $0$. Explain why $$X(t)=B(t+T)-B(T)=B(t+T)$$ is not a Brownian motion. This problem is from Introduction to Stochastic Calculus with ...
1
vote
0answers
187 views

Integral with respect to Wiener process.

Suppose that $\sigma(t,T)$ is a deterministic process, where $t$ varies and $T$ is a constant. We also have that $t \in [0,T]$. Also $W(t)$ is a Wiener process. My First Question What is ...
3
votes
1answer
442 views

Solving SDE: $dX(t) = udt + \sigma X(t)dB(t)$

Solve the SDE: $dX(t) = udt + \sigma X(t)dB(t)$ Provided Question The SDE is $dX(t) = udt + \sigma X(t)dB(t)$. Find $X(t)$, where $X(t)$ is some stochastic process and $B(t)$ is a Wiener process. ...
1
vote
1answer
32 views

Integrating $d(e^{-ut}X(t))$, where $X(t)$ is stochastic.

Given that $\sigma e^{-ut}dB(t) = d(e^{-ut}X(t))$, where $X(t)$ is a stochastic process and $B(t)$ is a Wiener process, we have that: $$ \int_0^t d(e^{-ut}X(s)) = X(0) + \sigma \int_0^t e^{-us}dB(s) ...
1
vote
1answer
89 views

Derive an SDE for $B^2(t)$, where $B(t)$ is standard Brownian Motion

Original Question: Derive an SDE for $B^2(t)$, where $B(t)$ is standard Brownian Motion. Attempt at an answer: Apply Ito's calculus over $f(t,b):= B^2(t)$. $$df(t,b) = \frac{\partial ...
1
vote
1answer
468 views

Min of two stopping times is also a stopping time.

Preface: I'm having trouble with the correct solution. The Original Question: Given that $\mathscr{F}_t$ is a filtration that satisfies all the usual conditions, and given ...
0
votes
1answer
42 views

Breaking up Wiener processes with indicator functions?

Consider a Wiener process $W_t$ which is adapted to $\mathscr{F}_t$, where this filtration has all of the standard properties. I'm also working with a stock-standard probability space here. I want to ...
2
votes
1answer
80 views

A nonmeasurable set on $\mathbb{R}^{\left[0,\infty\right)}$.

Let $\left\{ X_{t}\right\} _{t\geq o}$ the canonical version of Brownian motion, i.e., if we consider $\Omega:=\mathbb{R}^{\left[0,\infty\right)}$ the set of the real valued functions on ...
0
votes
1answer
79 views

Probability that a Brownian Particle will Exit a Certain Part of the Slit Open Unit Disk

The exact statement of the problem is: For each z in the slit open unit disk, find the probability that a Brownian particle will exit the region through the circular part of the boundary. So this is ...
4
votes
1answer
295 views

Stochastic integral inequality

Let $W_t$ be a Brownian motion with $m$ independent components on $(\Omega,F,P)$. Let $G(\omega,t)=[g_{ij}(\omega,t)]_{1\leq i\leq n,1\leq j\leq m}$ in $V^{n\times m}[S,T]$ such that ...
4
votes
1answer
352 views

Solutions to stochastic differential equations

I'm in a bit of trouble with my homework and was wondering if anyone could help me find the solutions to these two stochastic differential equations. Would really appreciate it! Thanks in advance! :) ...