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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 …
Martin Sleziak's user avatar
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 …
Joris Bierkens's user avatar
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 …
Joris Bierkens's user avatar
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 …
Joris Bierkens's user avatar
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 …
Joris Bierkens's user avatar
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 …
Joris Bierkens's user avatar
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 …
Joris Bierkens's user avatar
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 …
Joris Bierkens's user avatar
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 …
Joris Bierkens's user avatar
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 …
Joris Bierkens's user avatar
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 …
Joris Bierkens's user avatar