Tagged Questions
1
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0answers
22 views
Stochastic process at a fixed time $t_0$.
So basically for a stochastic process $X (t) = \xi t^4 - 5$ where $\xi \sim N(-1;1)$ if I want to find cumulative distribution and probability density at a fixed moment $t_0$ I have to go like this
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1
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1answer
38 views
finding the probability density function of $ dY_t = - Y_t X_t dW_t$
Could someone point me to where I can learn how to derive the stationary distribution for the martingale $Y_t$ which itself has stochastic volatility drive by $X_t$:
\begin{align}
dY_t &= - Y_t\ ...
0
votes
1answer
29 views
The weighted distribution function for combination of two variables
For example, we have two random variables $a$ and $b$. And they have cumulative distribution function $F(x)$ and $H(x)$. We have number $0 < p < 1$.
Suppose, some machine get this random ...
0
votes
0answers
16 views
Girsanov kernel moments
Let $Z_t=e^{\int_0^tq_tdB_t-\frac{1}{2}\int_0^tq^2_tdt}$, where $(q_t)_{t\geq0}$ is a predictable process, and $(B_t)_{t\geq0}$ a $\mathbb{P}$-Brownian motion. In particular, Novikov's condition ...
0
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0answers
28 views
Continuous time Stochastic Process stopping time measurability
Let $\{X_t,\mathcal{F}_t;0\leq t < \infty\}$ be continuous time stochastic processes and $T$ be $\{\mathcal{F}_t\}_{0\leq t < \infty}$ stopping time. How to prove $X_T$ is $\mathcal{F}_T$ ...
0
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0answers
13 views
fGn asymptotic claim correlation
Let $(X_{i})$ be the fractional Gaussian noise for $H\in(0,1)$.
Since it is stationary $\mathbb{E}(X_{i}X_{j})$ only depends on $|j-i|$.
How can I prove for $\rho(|j-i|)=\mathbb{E}(X_{i}X_{j})$ that ...
1
vote
1answer
33 views
Moment generating function of two non-independent Brownian increments
I am writing to ask if it is possible to get closed-form solution to the expression to the following expression:
$\mathbb{E}[e^{\sigma^2(W_t-W_u)(W_s-W_u)}]$ where $W$ is a standard Brownian motion, ...
-1
votes
0answers
19 views
Random process x(t) =C and C is uniform over [-2,3]
I need reassurance that if I do a a few sample realizations of this random process they are all going to look the same. They are going to be an horizontal line with x(t) constant equal to 1/5.
I see ...
-2
votes
0answers
31 views
Transforming a Joint PDF [duplicate]
I have a pdf $f(X,Y)=(\frac{1}{4})^2e^{−\frac{(|x|+|y|)}{2}}$. My goal is to find the joint PDF $f(W,Z)$ taking in consideration this $W=XY$ and $Z=Y/X$.
I know I can not use Jacobian because is a ...
0
votes
0answers
12 views
Submartingale bounds
Let $X_1,X_2,\ldots$ be a submartingale with respect to the filtration generated by it. Is it possible to get any bounds for the probability $\mathbb{P}(X_2 < 0\mid X_1 >0)$ ?
1
vote
1answer
72 views
A Boundary crossing result for discrete brownian bridge
Let $S_n$ be a random walk with gaussian increments with $S_0=0$, i.e. $S_n-S_{n-1}\sim N(0,1), n\geq 1$. Fix $a>0,b\in \mathbb{R}$ and $c<a+bn$. Define the new process
$$
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2
votes
1answer
54 views
Sum of stationary process
Suppose you have two stationary process $A_{t}$ and $B_{t}$. Suppose $Z_{t} = A_{t} + B_{t}$. Show that $Z_{t}$ is stationary. I am unsure how to solve this without knowing if the processes are ...
0
votes
1answer
36 views
Product of stationary stochastic process
Suppose $z_{t} = x_{t}y_{t}$ where $x_{t}$ and $y_{t}$ are 0 mean, independent stationary stochastic process. What is the autocovariance function of $z_{t}$? Show that the spectral density can be ...
1
vote
1answer
36 views
Finding expectation of stochastic process
Suppose $\sigma_{t}^{2} = w + \alpha_{1}y_{t-1}^{2} + \beta_{1}\sigma^{2}_{t-1}$ where $\alpha_{1} + \beta_{1} = 1$ and $y_{t} = \sigma_{t}e_{t}$ and $e_{t}$ is $ N(0,1)$. How do you show that
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1
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1answer
30 views
Inequality related to Doob's martingales
I have the following question on Doob's martingales.
Let $A$ be an integrable $\mathcal F$-measurable random variable on the
stochastic basis $(\Omega, \mathcal F, \mathcal F_t, \mathbb P)$. ...
