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Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
27
votes
What is a cumulant really?
A nice question, with probably many possible answers. I'll give it a shot. I think three phenomena should be noted.
i) The cumulant function is the Laplace transform of the probability distribution. U …
2
votes
0
answers
436
views
Hitting time of a specific Markov chain using martingale approach (or otherwise)
Let $0 < c < 1$. Consider the Markov chain $(X_i)$ on $\{0, 1, \dots, n\}$, with transition probabilities
$$ P(k,k+1) = \left(1 - \tfrac {k}{n} \right)(1-c), \quad k = 0, \dots, n-1, $$
$$ P(k,k-1) …
1
vote
1
answer
219
views
Uniqueness of invariant measure for equivalent transition probabilities
Suppose $P(x,dy)$ and $Q(x,dy)$ are two Markov transition kernels on a topological space $E$ equipped with Borel $\sigma$-algebra $\mathcal B(E)$. Suppose for every $x \in E$, $P(x,\cdot)$ and $Q(x, \ …
1
vote
Limit of first passage time
The key thing is that your diffusion is Feller under the stated conditions, so $x \mapsto \mathbf E_x f(X_t)$ is continuous. Therefore $x \mapsto \int_0^{\infty} \mathbf E_x f(X_t) e^{-rt} \ d t$ is c …
3
votes
Accepted
Computing transition operators for Markov processes
1) Explicit expressions for transition densities
In the case of linear systems with additive noise of the form
$d X(t) = (A X_t + b) \, d t + \sigma \, d B_t$
it is possible to obtain an explicit …
4
votes
Accepted
Ergodic and mixing processes
Yes, the results you quote are general statements on mixing and ergodicity, which can be translated to stochastic process as follows.
In the source you mention (and many other sources), mixing is def …
0
votes
On the existence and uniqueness of solution to SPDE with nonlinear growth coefficients
flawed, see Martin Hairer's comment below.
Step I) Perform substitution $u_t(x) = \exp(\psi_t(x))$. The SPDE for $\psi$ becomes, after dividing by $u_t(x)$,
\begin{equation}
\frac{\partial}{\partial …
1
vote
Size of KL-divergence neighbourhoods
Perhaps more a question for math.stackexchange.com.
Unless your probability space $(\Omega, \mathcal F)$ is trivial (i.e. $\mathcal F = \{ \emptyset, \Omega \})$, the sets $\mathbf P$ and $\mathbf Q …
0
votes
Drawing random variates from a partially described probability distribution
It would help if you know the marginal of at least one variable $x_i$.
Suppose this is the case, i.e. without loss of generality suppose that you know $p(x_1=0)$.
Then $p(x_1=i, x_2=j) = \left\{ \beg …
1
vote
Continuity of caglad process
I don't know if continuity of the density functions gives you anything. For example, think about a pure jump process with smooth density function of the jump distribution.
You may be familiar with Ko …
0
votes
Accepted
Finding the Levy triplet of a Levy process
You are looking for the Lévy symbol of a 2-dimensional process $(N,D)$, so your symbol will be a function on $\mathbb R^2$. It will be $\phi_{N,D}(u_1, u_2) = \phi_N(u_1) + \phi_D(u_2)$, by the follow …
2
votes
Approximation of stochastic differential equations
As formulated, there are some difficulties for the local Lipschitz case.
For quadratic $f(x)=x^2$, consider the corresponding ODE for $x$ (i.e. the SDE with $\sigma = 0$). Then there exists a locally …
3
votes
Accepted
Reference on continuous-time finite state filtering
This question is related to the topic of stochastic filtering theory. See e.g. the following monographs
* Bucy, Joseph - Filtering for stochastic processes with applications to guidance
* Bain, Crisan …