1
vote
0answers
22 views

Question regarding Notes on Strong Markov Property

I wrote the following notes from a lecture a couple of weeks ago and I don't understand a particular line. Suppose $B_t$ is a Brownian Motion Now look at $B^x_t = x + B_t$ which is a BM starting at ...
0
votes
0answers
24 views

Maximal principle for elliptic or linear integro-differential operator

Consider $L$ the operator forming as $$ Lg= -g^{'}(x)+(g(x+1)-g(x)) $$. $h$ on $[0,\infty)$ satisfies the following integro-differential equation $$ Lh \geq 0 $$ with boundary condition: $$ ...
0
votes
1answer
50 views

Markov processes and semimartingales

Semimartingales and Markov processes are two fundamental families in probability theory. There are many specific processes that belongs to the intersection of those two families, e.g. Levy processes. ...
1
vote
0answers
22 views

Finite state Markov chains and expectations

Let $X_t$ be a finite state Markov chain with generator matrix $Q$. For a give function $f(x)>0$ define: $$ u(t, i) = \mathbb{E}\left[\int_t^T f(X_s) \mathrm{d} s| X_t = i\right] $$ Are there any ...
1
vote
2answers
230 views

Diffusion process. Distribution vs transition probability.

I need confirmation on the following problem: Take a SDE of the form: \begin{equation} dX_t=a(X_t,t)dt+b(X_t,t)dW_t \end{equation} where all the conditions, such that the solution $X_t$ is defined ...
11
votes
1answer
397 views

Motivation of Feynman-Kac formula and its relation to Kolmogorov backward/forward equations?

Kolmogorov backward/forward equations are pdes, derived for the semigroups constructed from the Markov transition kernels. Feynman-Kac formula is also a pde corresponding to a stochastic process ...
3
votes
1answer
445 views

Multidimensional infinitesimal generator of a jump-diffusion

Let $X=\{X_t\}_{t\geq0}$ be an $n$-dimensional Markov process, defined by the SDE $$dX_t = \mu(t, X_t) \, dt + \sigma(t,X_t) \, dB_t+\beta(t-,X_{t-}) \, dN_t,$$ where $\mu, \sigma$ and $\beta$ are ...
2
votes
1answer
174 views

Prove that Brownian Motion absorbed at the origin is Markov

I'm trying to prove that Brownian motion absorbed at the origin is a Markov process with respect to the original filtration $\{\mathcal{F}_{t}\}$. To be more specific, let $(B_{t},\mathcal{F}_{t})_{t ...