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3 votes

Penalty shootout

Of course, one can write an expression for the probability in question as a certain sum over the set $\{-1,0,1\}^N$. What kind of expression for this probability do you want? Given the three ...
Iosif Pinelis's user avatar
2 votes

Smoothness of resolvent of the infinitesimal generator of an Ito diffusion acting on bounded continuous function

It is sufficient that $\inf_x \sigma(x)>0$ and $\sup_x \sigma(x)<\infty$, and $f$ does not have to be monotone. In this case, denoting $\mathcal L f(x) = \frac{1}{2}\sigma(x)f''(x)+\mu(x)f'(x)$, ...
zhoraster's user avatar
  • 1,523
2 votes

Exact probability distribution for an alternating renewal counting process

We assume that the counter is not shut down at time $t = 0$. Let $T_1, T_2, \ldots$ be independent, $\exp(\Phi)$-distributed random variables. Then $S_1 := T_1$ is the time when photon 1 is detected, $...
Dieter Kadelka's user avatar
1 vote
Accepted

Ornstein Uhlenbeck process with discontinuous drift

If you consider the process $Y_t=|X_t|,$ Ito's formula gives $$ dY_t=\sqrt{2}dB_t+\frac{dt}{Y_t}-l(\theta_t)Y_t\,dt, $$ where $\theta_t$ is an argument of $X_t$, $l(\theta_t)\in \{1,2,3\}$ is given ...
Kostya_I's user avatar
  • 8,927
3 votes

Best textbooks/resources for "advanced" probability theory?

My favourite introduction to stochastic processes and stochastic calculus is the book Stochastic Calculus by Paolo Baldi. It is very clear yet precise at the same time, and comes with hundreds of ...
4 votes
Accepted

Is this predictable process left-continuous?

In the following example, all random variables are constant with respect to $\omega$, so $\Omega$ could be taken to be a point mass. For each $n$, let $$X_t^n = \begin{cases} 0, & t \le 1-1/n \\ 1,...
Nate Eldredge's user avatar
2 votes
Accepted

Is it true that $F(X_0, \cdot) = X_0 + \int_0^T \sigma(s, X_0) \, \mathrm d B_s$ a.s.?

Yes. We fix some notation. Write $Y := F(X_0, \cdot)$, $Z := X_0 + \int_0^T \sigma(s, X_0) \, dB_s$, and consider the process $W_t := B_t - B_0$ together with its completed natural filtration $\...
Nate River's user avatar
  • 4,861
3 votes
Accepted

Does $X_t$ with $t>0$ admit a density?

I believe the answer is yes. By independence of $X_0$ from the Brownian increments, by the claim here we can write $X_t = F(X_0, \omega)$, where $F(x, \omega) := x + \int_0^t\sigma(s, x) \, dB_s$. By ...
Nate River's user avatar
  • 4,861
5 votes

The equivalence of stochastic quantization and path integral quantization

Some older sources than https://arxiv.org/pdf/hep-th/0312315 that prove the equivalence between stochastic quantisation and regular QFT (Euclidean path integrals) include: Equivalence of stochastic ...
Carlo Beenakker's user avatar
1 vote
Accepted

Conditional expectation w.r.t filtration of Brownian motion as a continuous map of its paths

The answer to the question as stated is no. Take the SDE coefficients $\alpha$ and $\beta$ to be deterministic, then $X_t$ is $\mathcal F_t$ measurable, so that $\mathbb E[X_t | \mathcal F_t] = X_t$ ...
Nate River's user avatar
  • 4,861
3 votes

Constructing Wiener process on a given probability space

$\newcommand\om\omega\newcommand\Om\Omega\newcommand\F{\mathcal F}$Let $(\Om,\F,P)$ be a probability space. A necessary and sufficient condition for $(\Om,\F,P)$ to support a sequence of independent ...
Iosif Pinelis's user avatar

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