Tagged Questions
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0answers
11 views
How to calculate the transition matrix of a stochastic process? [on hold]
When I was learning about stochastic processes, the transition matrix was already given. But when I started to use a stochastic process, I've no idea how to calculate the probability of transitioning ...
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1answer
23 views
Using characteristic function to deduce convergence of Bernoulli random variables
Let $Y_1, Y_2,...$ be a sequence of independent Bernoulli(0.5) random variables and $X_n =
\sum_{i=1}^{n} Y_i 2^{-i}$ I need to use the characteristic function to deduce that $X_n$ converges in ...
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votes
1answer
36 views
Large deviation of Bernoulli random variables and applying Chernoff bound
Let $X_1,X_2,..,X_n$ be i.i.d Bernoulli random variables with $P(X_1)=0.005$ and let $S_n:=X_1+...+X_n$. I need to:
Evaluate the exact value of $P(S_{100} \geq 4)$
Use the Chernoff bound to estimate ...
1
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0answers
21 views
Potentials in Probability Theory
Could someone give an intuitive interpretation of potentials in the field of probability theory. How do they link to the theory of stochastic processes. And maybe link this with SEP. References are ...
1
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1answer
28 views
Waiting time in an immigration-birth process
Could someone please verify that none of the four given choices are correct? Isn't the correct answer $$\frac 1{(\lambda + 4\beta)^2} + \frac 1{(\lambda + 5\beta)^2} + ... +\frac 1{(\lambda + ...
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1answer
23 views
Girsanov theorem and change of measure
Under the risk neutral measure Q, the stock price S follows a process $dS_t=rS_t dt+ σS_t dW_t^Q$, $W_t^Q$ is a standard brownian motion. Another measure is introduced with which I am not familiar ...
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1answer
41 views
does a simple random walk eventually hit every point?
Let $M_n=\sum_{k=0}^{n}X_k$ be a simple random walk starting at $0$, where $P(X_n=1)=P(X_n=-1)=\frac{1}{2}$.
What is the probability the random walk hits the point $z\in\mathbb{Z}$?
I have a feeling ...
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1answer
29 views
Ito vs Stratonovich SDE with irregular time-dependence in coefficients
Suppose I am interested in the Stratonovich SDE
$$ dX_t = b(t,X_t) dt + \sigma(t,X_t) \circ dB_t $$
If the coefficients are smooth enough in time and space, I can show this is equivalent to the Ito ...
1
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1answer
32 views
Is a Bernoulli process a Markov chain?
For a Bernoulli process, the outcome of a future trial is independent of the outcome of past trials. I.e., the future behaviour of a Bernoulli process is independent of its past, i.e. a Bernoulli ...
1
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1answer
76 views
an issue with expectation
in book's Bernt.Øks SDE i read that book and i have some serious issues :( page 21 Example 7.4.2 )
Consider n-dimensional Brownian motion $W=(W_1, \ldots ,W_n)$
starting at $a=(a_1,\ldots,a_n) \in ...
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0answers
25 views
Stochastic processes - irreducible Markov chains.
This is a problem from Introduction to Stochastic Processes by Lawler.
Suppose $X_n$ is an irreducible Markov chain on the state space $\{1, \dots, N\}$. Show that there exists $C < \infty$ and ...
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0answers
17 views
Markov processes reflected at the supremum [on hold]
Let $X_t$ be a temporally homogeneous Markov process on $R$, and let $M_t=\sup_{0\le s\le t}X_s$. Then is $M-X$ also a Markov process?
0
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1answer
38 views
Positive conditional expectation
We have a log-normal variable $x = e^{\mu +\sigma w}$, where $w$ is standard normal.
We want to compute $E[(x - K)^{+}\mid z]$. I'm not sure if I can write it as $E[(x ̃ - K) 1_{\{x ̃- K>0\} }]$, ...
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0answers
30 views
question about the sequential continuity of the set of probability measures
I have a question about the sequential continuity of the set of probability measures. Let $\Omega$ be the space of continuous functions defined in $[0,1]$ taking values in $\mathbb{R}$. Let ...
0
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1answer
53 views
Given a function and how do you determine the pdf of the left side given the pdf of the right side variables?
Given a function and how do you determine the pdf of the left side given the pdf of the right side variables? Specifically what is the pdf of W, given the equation
$$
W = I^2 R$$
with $I$ and $R$ are ...
