Question related to Brownian motion, a stochastic process denoted $W_t$, $t\geq 0$, with independent increments, such that $W(t)-W(s)$ is normally distributed, with $0$ mean and variance $t-s$.
0
votes
2answers
321 views
Lottery “Sum” forecasting
I was wondering if anyone can provide some mathematical insights to forecasting the "SUM" in this link as a time series. It is an oscillatory, range bound and poisson distribution. How can Monte Carlo ...
0
votes
1answer
62 views
A Brownian motion starting from 0, it becomes -1 once reach -1, what is its expectation?
$X_t$ is a Brownian Motion, it reaches becomes -1 forever once it reaches -1.
Mathematically, $T = \inf\{t: W_t = -1\}$ is a stopping time.
When $t<T$, $X_t = W_t$ ,while $t\geq T$, $X_t = -1$.
...
6
votes
1answer
475 views
Hitting time of Brownian Motion with a drift
Let $X_t =x+bt+\sqrt{2}W_t$, where $W_t$ is a standard Brownian motion.
Let $T=\inf\{t: |X_t|=1\}$. I am trying to find $\mathbb{E}[T]$ for the case $b\neq0$.
Firstly, I am going to apply Girsanov to ...
1
vote
2answers
226 views
Independent increments in squared brownian motion
I'm trying to prove that the squared Brownian motion $(W_t^2)$ doesn't have independent increments, I tried using the covariance and it doesn't quite work, can anybody give me any pointer to how to ...
3
votes
1answer
192 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:
...
1
vote
2answers
184 views
Derivation of an alternative representation of the Ornstein-Uhlenbeck process
The Ornstein-Uhlenbeck process can be defined as:
$X_t = e^{-\lambda t} \left( X_0 + \int_0^t e^{\lambda s} dB_s \right)$
where $\lambda > 0$ and $\{ B_t \}_{t \geq 0}$ is the standard Brownian ...
0
votes
1answer
75 views
Independence of $B(t)-B(s)$ with respect to $\mathcal{F}_s$
Assume that $X,\{Y_\alpha\}$ (where $\alpha \in A$, $A$ can be uncountable) are
random variables. If $X$ and $Y_\alpha$ are independent for all $\alpha \in A$, i.e., $\sigma(X)$ and ...
3
votes
0answers
91 views
A question regarding the strong Markov property
In our lecture on Brownian motion & stochastic calculus we proved:
If $ X $ is a canonical RCLL process having the strong Markov property and $ \tau $ is a stopping time with $ \tau < + \infty, ...
1
vote
1answer
88 views
Integrable dominating function for stopped Brownian motion [duplicate]
Possible Duplicate:
Dominated convergence problems with Wald's identity for the Brownian Motion
Let $(B_t)_{t \geq 0}$ be a Brownian motion, $T$ a stopping time with $E(T)<\infty$ ...
5
votes
1answer
317 views
Dominated convergence problems with Wald's identity for the Brownian Motion
In the course of proving Wald's second identity $E(B^2_T)=E(T)$, where $(B_t)_{t\geq0}$ is the Brownian motion and $T$ is a stopping time with $E(T)<\infty$, I got stuck with the following problem. ...
2
votes
1answer
98 views
$\mathbb{E}[e^{B_t}|F_s]$: expectation of some Brownian motion
I am supposed to find the following ($B_t$ is a Brownian motion, and $\mathcal{F}_s$ the generated filtration):
$$\mathbb{E}[e^{B_t}|\mathcal{F}_s]=?$$ I tried this: shifting by $s$ to the left to ...
9
votes
3answers
758 views
Can I apply the Girsanov theorem to an Ornstein-Uhlenbeck process?
Let $W_t$ be a standard Brownian motion, and $X_t$ a measurable adapted process. Girsanov's theorem says that under certain conditions, the Brownian motion with drift $Y_t = W_t - \int_0^t X_s\,ds$ ...
0
votes
1answer
74 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
331 views
Variance of product of Brownian motions
Let $\{B_{t}\}_{t\geq0}$ be Brownian motion. What is the variance
of $B_{t}B_{s}$?
3
votes
2answers
156 views
What's the solution of Stock price based on GBM model?
Stock price has a classic model based on GBM:
$$dS = \mu S dt + \sigma S dW$$
based on this call options values could be solve -- Black-Scholes formula.
But, what is the solution for the Stock ...
0
votes
1answer
229 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 ...
3
votes
0answers
134 views
Question about an exercise in Revuz/Yor
I'm solving exercise 2.28 in Revuz/Yor. I was able to prove 1). Unfortunately at 2) I got stuck. I have to show:
Let $B$ be a d-dimensional Brownian motion and $A\in \mathcal{A}:=\cap_t ...
5
votes
3answers
741 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 ...
1
vote
1answer
389 views
Independent increments of Brownian Motion
Suppose we have the $(W_t)$ Brownian Motion and the filtration $F=(\mathcal{F}_t)$, where $\mathcal{F}_t:=\sigma(W_s;s\le t)$. I know that for any $n\in \mathbb{N}$ and $0\le ...
14
votes
2answers
448 views
Brownian bridge expression for a Brownian motion
Let $B_t$ be a standard Brownian motion in $\mathbb R$, then the Brownian bridge on $[0,1]$ is defined as
$$
Y_t = a(1-t)+bt+(1-t)\int\limits_0^t\frac{\mathrm dB_s}{1-s}
$$
for $0\leq t<1$. Here ...
0
votes
0answers
32 views
Model stochastic processes
I'd like to model something similar to Brownian motion, where particles behave according to a microscopic local law. I'd like to examine the effective macroscopic behaviour.
What sort of mathematical ...
1
vote
1answer
166 views
Continuity of the Ito integral
Let $B_t$ be a 1-dimensional Brownian motion. I am following "Stochastic Differential Equations" by Bernt Øksendal. On the page 32 (it is displayed in the link I've put) there is a proof of existence ...
1
vote
2answers
124 views
Distributions of Stochastic Processes
I learned that Brownian motion is any stochastic process $(W_{t})_{t\geq0}$ that satisfies four well-known properties. But I still don't understand how these four properties uniquely determine ...
3
votes
0answers
167 views
Expected time spent in the set
An exercise 2.14 from Bernt Øksendal's "Stochastic Differential Equations":
Let $B_t$ be $n$-dimensional Brownian motion and let $K\subset \mathbb R^n$ have zero $n$-dimensional Lebesgue measure. ...
3
votes
1answer
204 views
Definition of the Brownian motion
The way I understood the definition of a Brownian motion $B_t$ in $\mathbb R$ is that it consists of two parts:
We first define the finite-dimensional distributions
$$
...
2
votes
1answer
141 views
Stopping time on Wiener Process
Let $W_t$ be a Wiener process and for $a\geq0$
$$\tau_a:=\inf \left\{ t\geq0: |W_t|=\sqrt{at+7} \right\}.$$
Is $\tau_a<\infty$ almost everywhere? What about $E(\tau_a)$ then?
4
votes
2answers
92 views
maximum of a brownian motion and its integral
Let $W_{t}$ be a brownian motion and
$$
W^{*}_{t} = \max_{s<t} W_{s}
$$
Then can you please explain why we have this:
$$
(W^{*}_{t} - W_{t})dW^{*}_{t} = 0
$$
3
votes
1answer
253 views
minimum of hitting time of a brownian motion
Let $Y$ be an exponential random variable with rate parameter $\lambda$. Let $T_{a}$ be the first hitting time of a Brownian Motion. I want to find
$$
P(\min(T_{a}, T_{-a}) < Y)
$$
In order to ...
1
vote
1answer
335 views
Convergence of the exponential martingale
How can we show that this martingale $$ e^{aW_{t} - \frac{1}{2}a^2t}$$ converges to $0$ as $ t \rightarrow \infty$ using law of iterated logarithm, for $a \neq 0$.
4
votes
3answers
812 views
expected value of brownian motion
How can you find this expected value?
$$
\mathbb{E}[|W_{t}^2 - t|]
$$
where $W_{t}$ is a brownian motion.
10
votes
2answers
229 views
Problem about partial sum of exponential random variable
Let $X_1, X_2, \dots,X_n, X_{n+1}$ be independent random variable of exponential distribution, and the mean is 1.
Let $S_i = X_1 + \dots + X_i$
I want to know ...
1
vote
1answer
93 views
Interpretation of Notation of Laplacian and Brownian Motion
Let $f: \mathbb{R}^d \to \mathbb{R}$ be a twice differentiable function. In particular, $\Delta f$ is well defined. Let $W := (W_t)_{t \geq 0}$ be a $d$-dimensional standard Brownian Motion. ...
2
votes
1answer
291 views
Difference between a Brownian Motion and the root of its square
Let $W_{t}$ be a Wiener Process (a Brownian Motion starting at $W_{0} = 0$). What is the difference between $W_{t}$ and $\sqrt{W_{t}^{2}}$?
Using the Ito formula (in differential notation), ...
0
votes
0answers
495 views
Sum of two geometric brownian motion
Is sum of two geometric brownian motion a markov process or a diffusion process? It is given that the two Weiner processes are independent.
Thanks
2
votes
1answer
115 views
Length of Wiener Sausage
I am deriving a formula for a volume of Wiener sausage in one dimension.
$$\mathbb{E}[\operatorname{vol}(W(t))] = 2r+\sqrt{\frac{8t}{\pi}}$$
where $W(t) = \bigcup_{s\leq ...
2
votes
1answer
376 views
Absolute value of Brownian motion
I need to show that $$R_t=\frac{1}{|B_t|}$$ is bounded in $\mathcal{L^2}$ for $(t \ge 1)$, where $B_t$ is a 3-dimensional standard Brownian motion.
I am trying to find a bound for ...
0
votes
0answers
179 views
Brownian Bridge. Law of a process
Let $(B_t , 0 ≤ t ≤ 1)$ be a standard Brownian motion in 1
dimension. We let $(Z_t^y = yt + (B_t − tB_1 ), 0 ≤ t ≤ 1)$ for any $y \in R$ and call it the Brownian bridge from $0$ to $y$. Let $W_0^y$ be ...
0
votes
0answers
243 views
Min and Max of Geometric Brownian motion
I am trying to derive the distribution of $M_X(t) = \max\limits_{0\leq s\leq t}X(s)$ and $m_X(t) = \min\limits_{0\leq s\leq t}X(s)$, where $dX(t)=\mu X(t) dt+\sigma X(t)dB(t)$ and $B(t)$ is standard ...
2
votes
1answer
238 views
Expectation of an integral of the minimum of a Brownian motion and a constant
I would like to compute the expectation of the following expectation
$\mathbb{E}[\int_a^\infty e^{-rt}\min(x_t,c)\,dt]\,$
where a, r, c are constants, $dx_t = \mu x_t dt + \sigma x_t dW_t$ is a ...
4
votes
0answers
148 views
Using a laplace type expansion to get bounds on an integral arising in the study of Brownian motion
Let $ 0 < r < 1$, fix $x > 1$ and consider the integral
$$ I_{r}(x) = \int_{1}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^r}.$$
In the investigation of ...
3
votes
1answer
86 views
$\varlimsup_{t\rightarrow\infty} \frac{B_t}{\sqrt{t}}>0$
I am trying to prove the following statement about the standard Brownian Motion:
$\varlimsup_{t\rightarrow\infty} \frac{B_t}{\sqrt{t}}>0$.
I know that it is trivial to prove the above statement by ...
5
votes
1answer
405 views
Quadratic Variation of Brownian Motion
Quadratic Variation of a Brownian motion $B$ over the interval $[0,t]$ is defined as the limit in probability of any sequence of partitions $\Pi_n([0,t])=\{0=t^n_0<\cdots<t^n_{k(n)}=t\}$ of the ...
1
vote
1answer
231 views
confusion about the multi-dimensional Brownian motion
I am confused on the multi-dimensional Brownian motion.
$B_t$ is a standard Brownian motion based on a filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \geq 0}, \mathbb{P})$ if
...
4
votes
1answer
177 views
Stochastic integral inequality
Let $W_t$ be a Brownian motion with $m$ independent components on $(\Omega,F,P)$.
Let $G(\omega,t)=[g_{ij}(\omega,t)]_{1\leq i\leq n,1\leq j\leq m}$ in $V^{n\times m}[S,T]$ such that
...
2
votes
0answers
99 views
Ruin probability
Let $X_t$ be a solution of the stochastic differential equation
$$ dX_t= -\frac{c-1}{2 X_t}dt+ dB_t, \, \qquad X_0=x_0$$
where $c$ is a real constant and $B_t$ is a Brownian motion. Can you give me ...
8
votes
2answers
394 views
Sobolev meets Wiener
Even though the Wiener process (Brownian motion) is continuous, it has no derivative at any point.
Does it at least have weak derivatives?
4
votes
1answer
632 views
How to derive the Ornstein-Uhlenbeck Stochastic Integral Equation?
I have a question regarding the Ornstein -Uhlenbeck process. We have a simplified version with Stochastic Integral Equation: $X_t=-a\int^t_0 X_s\,ds +B_t$. B is the Brownian motion.
And its analytic ...
4
votes
1answer
248 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! :)
...
2
votes
2answers
218 views
Wiener Process $dB^2=dt$
Why is $dB^2=dt$? Every online source I've come across lists this as an exercise or just states it, but why isn't this ever explicitly proved? I know that $dB=\sqrt{dt}Z$, but I don't know what ...
3
votes
1answer
171 views
Are hitting times of Brownian motion independent?
Suppose that $B_t$ is a standard Brownian motion. And $T_a$, $T_b$ are the hitting time whereas $a<0$, $b>0$. Then are these two random variables independent?
