Tagged Questions
1
vote
1answer
15 views
How do you make dependent Brownian motions independent?
Can someone explain to me how to take 2 correlated Brownian motions and make them independent? I can't seem to grasp this process.
Just assuming $dB_1(t)dB_2(t) = \rho dt $
From what was explained ...
3
votes
0answers
27 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 + ...
1
vote
1answer
37 views
How is Brownian motion predictable?
Could someone please explain how Brownian motion is predictable? My understanding is that a predictable process is one that depends on information up to time t say but not t itself, therefore W_t has ...
0
votes
0answers
47 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 $\mathcal{F}_t$ ...
1
vote
1answer
64 views
Approximation of stochastic integral
Let $f \in C^2_C(\mathbb{R})$ and $$X_t = X_0 + \int_0^t \sigma(s) \, dB_s + \int_0^t b(s) \, ds$$ (1-dim.) Itô process where $\sigma,b: [0,\infty) \times \Omega \to \mathbb{R}$ progressively ...
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
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 ...
5
votes
1answer
127 views
Ito's Lemma and Brownian Motion
Show by using Ito's Lemma, for $k \geq 2$ the following result hold.
$$E[W(t)^k] = \frac{1}{2} k(k-1)\int_0^t E[W(s)^{k-2}]ds$$
where $W(t) = N(0,t)$ is standard Brownian motion.
I think ...
1
vote
1answer
49 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 ...
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
1answer
53 views
Some preliminaries for the canonical construction of a Brownian Motion, help needed.
I have a lecture in stochastic analysis and I was given some facts, which are completely new to me and I do not really understand hot to understand/proof them. I would very happy if somebody could ...
1
vote
1answer
95 views
Distribution of integral with respect to Brownian motion
Let $B$ be a Brownian motion and define the complex sequence $(X(n))_{n\in \mathbb Z}$ as
$$ X(n) := \int^\pi_{-\pi} e^{inx} dB(\pi + x)$$
What is the distribution of $X(n), n\in \mathbb Z$?
0
votes
2answers
39 views
Identity for exponential of Brownian motion using scaling relation
Let $B$ be a Brownian motion and $s\wedge t := \min\{s,t\}$, $s\vee t := \max\{s,t\}$.
I stumbled over the following identity:
$$ \mathbb E[\exp(B(s\wedge t) + B(s\vee t))]
\\=\mathbb ...
4
votes
1answer
126 views
Stopping time and Brownian motion (specific example)
Let $B$ be a Brownian motion. I want to show that
$$ \inf\{t\geq0 \mid B(t)=\max_{x\in [0,1]}B(s)\} $$
is not a stopping time w.r.t. the standard filtration.
How can one intuitively see that this ...
0
votes
1answer
47 views
Expectation of a stochastic exponential
In class a while ago we used the following simplification:
$$ \mathbb E \left[ \exp\left(\langle \boldsymbol a,\mathbf W_t\rangle \right) \right] \quad =\quad \exp\left(\frac12 |\boldsymbol a|^2 ...
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
$$ Y(t) = e^{\sigma X(t)-0.5\sigma^2t} $$
$$ Y(t) = e^{0.5t}\Bigg(1 - ...
0
votes
0answers
135 views
Analysis of Brownian Motion
The following tasks consider transformation an analysis of Brownian Motion.
For the proces $ Y(t) = -\theta \mu t + \sigma X(t) $ design an algebraic substitution to $ X(t) $ that removes the drift ...
0
votes
0answers
115 views
Geometric Brownian Motion
Consider asset price $S$ that evolves according to Geomtric Brownian Motion with constant $\mu$ and $\sigma$
$$dS = \mu Sdt + \sigma SdX$$
Show by the application of Itô's Lemma to function $\log S$ ...
2
votes
1answer
65 views
Distribution of the integral of a diffusion process
Suppose $X(t)$ is a diffusion process with $E[X(t)]=0$ and variances $\sigma^2_t$ concave in time. If $X$ is also a Brownian motion, then the distribution of $\int_0^T X(t) dt$ is known to be ...
1
vote
1answer
66 views
Convergence to Brownian motion integral
Let $X_i$ be i.i.d with $\mathbb{E}(X_i) = 0$ and $Var(X_i) =1, \, S_n = \sum_{i=1}^n X_i$. I would like to show that $\sum_{i=1}^n \frac{f(S_i/\sqrt{n})}{n}$ converges to $\int_0^1 f(B_t)dt$ in ...
3
votes
1answer
108 views
linear combination of two Wiener processes
I have a question concerning the linear combination of two Wiener processes (please see http://en.wikipedia.org/wiki/Wiener_process for a definition). Let $W$ and $\tilde{W}$ be two Wiener processes ...
3
votes
1answer
34 views
two r.v sharing the same law
I have a question: Let $X=B^{+}$ or $X=|B|$ where $B$ is the standard Brownian motion.
Set
$$J_p=\sup_{t\geq 0}(X_t-t^{\frac{p}{2}})$$
where $p>1$ and $q$ its conjugate ...
5
votes
1answer
208 views
Computing the limit of the expectation of a function of a stochastic process (phew!)
I state my problem in a few lines then describe what I have already done.
I have a quite simple stochastic differential equation (SDE):
$dx=-2x \, dt+\sqrt{1-x^2} \, dW$ with $W$ a brownian.
I ...
5
votes
1answer
61 views
How to show that the following process is a submartingale
Suppose we have a filtration $(\mathcal{F}_t)$ satisfying the usual conditions. Let $W$ be a Brownian Motion with respect to that filtration. We define the two processes
$X_t:=W^2_t$ and ...
3
votes
1answer
99 views
convergence ito integral
It is easy to calculate the integral $\int_0^T B_t \, dB_t=\frac{1}{2}B_T^2-\frac{1}{2}T$
That means I showed that $\int_0^T S_n \, ...
6
votes
1answer
271 views
Expectation of an integral w.r.t. Brownian Motion
I know the following statement:
if $f$ is a deterministic function and continuous, i.e. $f\in C^0([0,T],\mathbb{R})$, then $\int f(s)dW_s$ is normally distributed with mean zero and variance $\int ...
1
vote
1answer
52 views
Fractional Brownian motion as integral, mean zero
Let $H\in$ $]0,1[$. A fractional Brownian motion $\left(B_H(t)\right)_{t\geq 0}$ can be represented as $$X(t)={1\over ...
4
votes
1answer
139 views
Brownian Motion Covariance: max instead of min
It is known that $\operatorname{Cov}(B_t,B_s)=\min(t,s)$ where $B$ is Brownian motion.
Can one think of an Ito process or integral (preferrably plain Gaussian process) $W$ such that ...
2
votes
2answers
97 views
Relation between $\text{d}M_t$ and $\text{d}B_t$ when $M_t=\max_{0\leq s\leq t}B_s$
Let $B_t$ be a standard Wiener motion. What can we say about $\text{d}M_t$ and $\text{d}B_t$ when $M_t=\max_{0\leq s\leq t}B_s$? Is there a relation?
0
votes
2answers
193 views
Conditional Expectation of integral of Wiener process
Let $W_t$ be a standard Wiener process. How can we calculate: $$\mathbb{E}\left[\int_0^t|W_r|^2\text{d}r \ |\ \mathcal{F}_s\right]$$
where $(\mathcal{F}_s)_{s\geq0}$ is the natural filtration?
2
votes
1answer
69 views
Applying Ergodic Theorem on fractional Brownian motion
For a fractional Brownian motion $B_H$ consider the sequence for $p>0$
$$Y_{n,p}={1\over n}\sum\limits_{i=1}^n \left|B_H(i)-B_H(i-1)\right|^p.$$
By the Ergodic Theorem it is ...
3
votes
2answers
259 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 $t\geq0$.
3
votes
1answer
132 views
Show that $M_t$ is a Standard Brownian Motion
Let $M=(M_t)_{t\geq0}$ with $$M_t=\int_0^{\log\sqrt{1+2t}}e^s\text{d}B_s$$
where $(B_t)_{t\geq0}$ is a Standard Brownian Motion.
Show that $M$ is also a Standard Brownian Motion and compute ...
3
votes
0answers
109 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 $H$.
...
1
vote
0answers
187 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
37 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
1answer
26 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
$$E[e^{i\lambda S_t}]=E[\cos(\lambda ...
5
votes
1answer
157 views
Expected Value of Brownian motion using ito isometry
Find
$$
E\ \left[\left(\int_{0}^T e^{s+W_s}dW_s \right)^2\right],
$$
where $(W_s)$ is a Brownian motion.
I tried to use Ito isometry to solve this question, but still not yet to find the right ...
2
votes
0answers
76 views
Integral representation of fractional Brownian motion
Let $H\in$ $]0,1[$. A fractional Brownian motion $\left(B_H(t)\right)_{t\geq 0}$ can be represented as $${1\over C(H)}\int_\mathbb{R}\left((t-s)_+^{H-{1\over2}}-(-s)_+^{H-{1\over2}}\right)dB(s)$$ ...
2
votes
1answer
70 views
Long Range Dependence, Fractional Brownian Motion
A stationary sequence $(X_n)_{n\in\mathbb{N}}$ exhibits long-range dependence if the autocovariance function $\rho(n):=\mathrm{cov}(X_k,X_{k+n})$ satisfy $$\lim\limits_{n\to\infty}{\rho(n) \over ...
6
votes
3answers
222 views
Expected value of average of Brownian motion
For a standard one-dimensional Brownian motion $W(t)$, calculate:
$$E\bigg[\Big(\frac{1}{T}\int\limits_0^TW_t\, dt\Big)^2\bigg]$$
Note: I am not able to figure out how to approach this problem. All ...
2
votes
1answer
327 views
Transition density and distribution: (Ornstein–Uhlenbeck process)
Let $\left(X_{t},\, t\geq0\right)$ be the weak solution to the SDE
below with $\alpha,\,\beta,\,\gamma$ constants:
$$
dX_{t}=(-\alpha X_{t}+\gamma)dt+\beta dB_{t}\quad\forall t\geq0,\, X_{0}=x_{0}
$$
...
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 ...
2
votes
1answer
235 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
$$A:=E[e^{- \frac12 \sigma^2 (T-t) - ...
3
votes
1answer
120 views
Optional sampling exercise
I came across the following exercise in Stochastic Calculus:
Let $B=(B_t)_{t\geq0}$ be a standard Brownian motion. Let also $M$ be the following process:
$M_t=B^4_t-6t(B^2_t-\dfrac{t}{3})$ for ...
1
vote
0answers
119 views
Integral with respect to Wiener process.
Suppose that $\sigma(t,T)$ is a deterministic process, where $t$ varies and $T$ is a constant. We also have that $t \in [0,T]$. Also $W(t)$ is a Wiener process.
My First Question
What is ...
0
votes
0answers
41 views
Condition on $f$ for $e^{\int_0^t f(B_s)ds}$ to be of finite variation
Let $B$ be a standard Brownian motion, and,
$$
X_t=e^{\int_0^t f(B_s)ds},
$$
for some function $f$.
What are the condition on $f$ for $X_t$ to be of finite variation?
Let $Y_t=\int_0^t f(B_s)ds$, if ...
2
votes
1answer
107 views
Quadratic variation of $X_t=\int_0^t B_s \, ds$
Let $B$ be a standard brownian motion and
$$
X_t=\int_0^t B_s \, ds.
$$
What is the quadratic variation $[X]_t$ of $X$?
I see $dX_t$ as an sde with drift term $B_t$.
1
vote
0answers
126 views
Stochastic integral: Interchanging the order of expectation and integration
Let $B$ be a standard Brownian motion and
$$
X_t=\int_0^t f_s ds+\int_0^t g_s dB_s,
$$
where, $|f|$ and $|g|$ are both bounded, almost surely, by some positive constant $M$.
Is it true that
$$
...
0
votes
1answer
44 views
Condition for existence of a stochastic differential equation
With $B$ a standard Brownian motion, write
$$
dX_t=f_tdt+g_tdB_t.
$$
What are the conditions on $\left(f\right)_{t\ge 0}$ and $\left(g\right)_{t\ge 0}$ for $X_t$ to exists?
I think ...
