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 $t-s$.
6
votes
0answers
484 views
Quadratic variation of Brownian motion and almost-sure convergence
Say that $W(t)$ is a Brownian motion. The quadratic variation $[W,W](t)$ is defined in terms of a partition $\Pi = \{0 = t_0 < t_1 < \cdots < t_n = t\}$ by
$$
\begin{split}
[W,W](t) &= ...
4
votes
0answers
148 views
Using a laplace type expansion to get bounds on an integral arising in the study of Brownian motion
Let $ 0 < r < 1$, fix $x > 1$ and consider the integral
$$ 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
0answers
28 views
lower bound of expectation of stochastic differential equation
I'm looking for a lower bound on the expected value of a smooth, non-negative, increasing function $\mathbb{E}f(X_t)$, $f(0)=0$ of the solution to a stochastic differential equation $X_t = x + ...
3
votes
0answers
110 views
Hölder Continuity of Fractional Brownian Motion
I would like to prove the following theorem:
Let $H\in (0,1)$. The fractional Brownian motion $B_H$ admits a version whose sample paths are $a.s.$ Hölder continuous of order strict less than $H$.
...
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, ...
3
votes
0answers
132 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 ...
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. ...
2
votes
0answers
42 views
Negative moments of a functional of Wiener process
At the moment I am reading D. Nualart's The Malliavin Calculus and Related Topics. The problem I am trying to solve is the following: Show that the random variable $\int_0^1 s^2\arctan W_s\, ds$ ...
2
votes
0answers
57 views
Independence of Brownian Motion with respect to a stopping time
Let $B_t$ be a brownian motion, $B_0=0$, and $\gamma \in \mathbb{R}$.
Now, let's build the following stopping time:
\begin{equation}
T = \inf \{ t \geq 0 : |B_t + \gamma t| = 1 \}.
\end{equation}
If ...
2
votes
0answers
77 views
A problem with regard to Wiener process
Let $W$ be a Wiener process and $U_x$ is the amount of time spent below $x$ during time interval $(0,1)$. Hence $U_x=\int\limits_0^1I_{\{W(t)<x\}}dt$. My question is: what is the probability ...
2
votes
0answers
77 views
Integral representation of fractional Brownian motion
Let $H\in$ $]0,1[$. A fractional Brownian motion $\left(B_H(t)\right)_{t\geq 0}$ can be represented as $${1\over C(H)}\int_\mathbb{R}\left((t-s)_+^{H-{1\over2}}-(-s)_+^{H-{1\over2}}\right)dB(s)$$ ...
2
votes
0answers
103 views
Correlated diffusion processes and covariance matrix
I'm really noob in maths topics so I hope you will excuse me if I use terms which aren't correct.
I would like to simulate $n$ dimensional diffusion processes with $n$ noises.
Each process has its ...
2
votes
0answers
116 views
Show that $O_t$ is a Gaussian Process
Let $B_t$ be a Brownian motion process. Let $$O_t = e^{-\alpha t} \int^t_0 e^{\alpha s} dB_s$$ Find $\mathsf{E}[O_t]$ and show that $O_t$ is a Gaussian process.
I think ...
2
votes
0answers
99 views
Ruin probability
Let $X_t$ be a solution of the stochastic differential equation
$$ dX_t= -\frac{c-1}{2 X_t}dt+ dB_t, \, \qquad X_0=x_0$$
where $c$ is a real constant and $B_t$ is a Brownian motion. Can you give me ...
1
vote
0answers
38 views
Rate of increase of maximum process of Brownian Motion
Suppose $M_t=\sup_{0\leq s\leq t}\{B_s\}$, where $\{B_t\}_0^{\infty}$ is a standard Brownian Motion. I would like to know if it is true that $M_t e^{-t}$ converges to 0 almost surely?
Thanks!
1
vote
0answers
44 views
Expected value of brownian motion for all positive paths
I've got this question but I can't figure it out.
Derive the expected value of $B(t_1)$ of all paths that are positive $t_1$ and calculate the expectation for $t_1=1$ and variance$=1$?
Thanks
1
vote
0answers
52 views
Quadratic variation process of $G$–Brownian motion
I would like to prove the inequality
$$\hat{\mathbb{E}}\left[\left(\int^T_0 \eta_t d \langle B \rangle_t \right)^2\right] \leq C \hat{\mathbb{E}}\left[ \int^T_0 \eta^2_t dt \right],$$
where $\langle B ...
1
vote
0answers
44 views
Two Questions about Brownian Motion
How do you show $B_T\in\mathcal{F}_T$ for T is a stopping time?
Note the filtration is generated by the Brownian motion (and not necessarily completed, in particular, ...
1
vote
0answers
32 views
Independence of Brownian motion-related stopping times
Let $(B_t,\mathcal{F}_t)_{t \geq 0}$ a Brownian motion on a probability space $(\Omega,\mathcal{A},\mathbb{P})$. For $a \in \mathbb{R}$ define a stopping time $\tau_a$ by $$\tau_a := \tau(a) := ...
1
vote
0answers
27 views
Simulating of GBM
I have a question regarding the simulation of a GBM. I have found similar questions here but nothing which takes reference to my specific problem:
Given a GBM of the form
$dS(t) = \mu S(t) dt + ...
1
vote
0answers
188 views
Show that this semimartingale is a local martingale
Let $B_t$ be a standard Wiener motion, $I_t=\int_0^t|B_s|^2\!\text{ds}\ $and $S_t=\max_{0\leq s\leq t}B_s$. Let also $F:\mathbb{R}^2_+\times\mathbb{R}\times\mathbb{R}_+\rightarrow\mathbb{R}$ a ...
1
vote
0answers
83 views
Is this geometric Brownian Motion?
The SDE for GBM is usually specified as:
$$dX(t) = X(t)[\mu dt + \sigma dW(t)]$$
If we model diffusion as stochastic, is the following still GBM?
$$dX(t) = X(t)[\mu dt + \sigma_t dW(t)]$$
...
1
vote
0answers
38 views
Fractional Brownian motion, selfsimilar
Let $0<H<1$. A real-valued Gaussian process $\left(B_H(t)\right)_{t\geq 0}$ is called fractional Brownian motion (fBm) if $\ \mathbb{E}[B_H(t)]=0$ and
...
1
vote
0answers
36 views
What is the intuitive meaning of $K_1, K_2, K_3$ in regards to the conditional density formula derivation in Brownian motion.
In my text, there is a passage that says:
"Suppose we require the conditional distribution of $X(s)$ given that $X(t) = B$, where $s < t$. The conditional density is:
$$
\begin{align*}
f_{s\mid ...
1
vote
0answers
119 views
Integral with respect to Wiener process.
Suppose that $\sigma(t,T)$ is a deterministic process, where $t$ varies and $T$ is a constant. We also have that $t \in [0,T]$. Also $W(t)$ is a Wiener process.
My First Question
What is ...
1
vote
0answers
127 views
Stochastic integral: Interchanging the order of expectation and integration
Let $B$ be a standard Brownian motion and
$$
X_t=\int_0^t f_s ds+\int_0^t g_s dB_s,
$$
where, $|f|$ and $|g|$ are both bounded, almost surely, by some positive constant $M$.
Is it true that
$$
...
1
vote
0answers
37 views
prove that two r.v.s share the same law
I have a question in my homework about Brownian motion. Does someone have a idea about the following question?
Let $X=B^+$ or $|B|$ where $B$ is a standard BM, $p>1$ be a real number and $q$ its ...
1
vote
0answers
156 views
Defining Brownian motion through Kolmogorov's extension theorem
In section 2.2. of Oksendal's book on Stochasic differential equations, he defines Brownian motion by specifying a family of probability measures $\nu_{t_1, \ldots, t_k}(F_1, \ldots, F_k)$ that ...
0
votes
0answers
48 views
Drift equation / Girsanov's Theorem
Define $$\frac{dS_t}{S_t} := \mu \, dt + \sigma^B \, dB_t +\sigma^W \, dW_t,$$ $$dS_t^{(0)} := S_t^{(0)}r \, dt,$$ where the constants $\mu, \sigma^B,\sigma^W$ and $r$ all positive and $\mathcal{F}_t$ ...
0
votes
0answers
35 views
Intuitive meaning of the generator of a Brownian motion $L=\frac{d}{\lambda(dx)}\frac{d}{dx}$
For a standard Brownian motion $B_t$, the generator is
$$
L_B=\frac12 \frac{d}{dx} \frac{d}{dx},
$$
we say that $B_t$ is a diffusion with canonical scale the Euclidean space, and speed measure the ...
0
votes
0answers
43 views
Quadratic Variation of a Brownian Martingale
I would like to show that the quadratic variation of the square integrable martingale $(W_t)^2-t$ where W is a Wiener process is $\int_{i=0}^t (W_s)^2ds$ . Any hints?
thank you
0
votes
0answers
137 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
$$ Y(t) = e^{\sigma X(t)-0.5\sigma^2t} $$
$$ Y(t) = e^{0.5t}\Bigg(1 - ...
0
votes
0answers
135 views
Analysis of Brownian Motion
The following tasks consider transformation an analysis of Brownian Motion.
For the proces $ Y(t) = -\theta \mu t + \sigma X(t) $ design an algebraic substitution to $ X(t) $ that removes the drift ...
0
votes
0answers
117 views
Geometric Brownian Motion
Consider asset price $S$ that evolves according to Geomtric Brownian Motion with constant $\mu$ and $\sigma$
$$dS = \mu Sdt + \sigma SdX$$
Show by the application of Itô's Lemma to function $\log S$ ...
0
votes
0answers
44 views
how to recognize stochastic process among wiener process, log-normal, normal, and mean-reversion
Wiener process (brownian motion), normal distribution, log-normal, and mean-reversion are 4 most frequently used stochastic processes in modelling.
i wonder, given 30-50 sample points, is there a ...
0
votes
0answers
74 views
How to prove this inequality involving integration with respect to Brownian motion?
If $B_t$ is the Brownian Motion, I have to verify that
$$E\left\lvert\int_s^t G(t,w)\,dB_t\right\rvert^6\leq 15^2\cdot (t-s)^2\cdot\int_s^t E\lvert G(t,w)\rvert^6\,dt$$
0
votes
0answers
41 views
Probability of a brownian motion leaving some area
Let $B_t$, $t\geq0$ be a standard $n$-dimensional Brownian motion, that is $B_t(\omega)\in\mathbb{R}^n$ and let $\Lambda\subset\mathbb{R}^n$ be some ball such that the Brownian motion starts within ...
0
votes
0answers
85 views
Explanation of the Girsanov's transformation
The Girsanov's theorem is making me all confused. In my course literature they explain it by some simple discrete examples of coin-tossing etc. Saying that $Z$ is the ratio of $\frac{P^a(A)}{P(A)}$ ...
0
votes
0answers
72 views
Conditional Expectation Problem.
I'm looking at the formula derivation process as shown below.
I can't seem to understand how $E[X(t_3)|X(t_1)X(t_2)]$ turns to $X(t_2)$. I understand how the first equation turns into the second ...
0
votes
0answers
48 views
Curious of how a process in an answer(already given) is derived re brownian motion.
The question and the answer is given below. Can someone explain how the process A changed to B? I know there are multiple steps that are not shown in the answer below, but I'm just curious about the ...
0
votes
0answers
58 views
integral related to Gaussian random variable and Brownian Motion
This integral arises from some work I am doing related to Brownian motion.
The integral of interest is the following
$\int^{\infty}_{t=0}\int^{b}_{x=-\infty}\frac{1}{\sqrt{2\pi ...
0
votes
0answers
118 views
Girsanov Transformation Example
Is this the correct use of Girsanov's transformation where $B_{n}$ is a discrete Brownian motion?
For example computing:
$E[(B_{n}+2n)^{2}]$
Set: $\widetilde{B_{n}}=B_{n}+2n$
And ...
0
votes
0answers
104 views
Levy's characterization of Brownian motion
Consider two processes:
$$A(T) = A(t)e^{(r-\frac12 \sigma_A^2)(T-t)+\sigma_A (W_A(T)-W_A(t))}$$
$$B(T) = B(t)e^{(r-\frac12 \sigma_B^2)(T-t)+\sigma_B (W_B(T)-W_B(t))}$$
where $W$ is a Wiener process ...
0
votes
0answers
63 views
Probability of Bankruptcy
Suppose that you plan to start a new business that provides expedited mail service. You expect
that the market for fast expedited mails are quite stable and the demand can be modeled as a
drifted ...
0
votes
0answers
41 views
Condition on $f$ for $e^{\int_0^t f(B_s)ds}$ to be of finite variation
Let $B$ be a standard Brownian motion, and,
$$
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 ...
0
votes
0answers
51 views
Controlling auto-correlated 1D Brownian motion
I have 1D Brownian motion process $x(t)$, and ability to control it.
The control allows to shift the $x$ by $D$ at any time.
I need the controlled process to be zero-mean, and to use the control ...
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 ...
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
0
votes
0answers
178 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 ...
