Questions tagged [stochastic-processes]

Question 1

Consider a filtered probability space $(\Omega,\mathcal F,\mathbb F=\{\mathcal F_t\}_{t\in[0,T]},\mathbb P)$ where the filtration satisfies the standard assumptions. Let $X=(X_t)_{t\in[0,T]}$ be a ...

Question 2

I am learning about interacting particle systems on lattices, and I am trying to clarify the logical order between two things: Detailed balance. For two configurations $C$ and $C'$, detailed balance ...

Question 3

My main research interest is PDE. I'm not very familiar with probability theory. Recently, I am reading a textbook related to probability interpretation of PDE. The part about Wiener process (Brownian ...

Question 4

I have the weight process $\omega_t$, and the asset price process $S_t$, both of them are Ito processes: $$\frac{d\omega_t}{\omega_t}=\mu^1_t\,dt+\sigma^1_tdW^1_t,\qquad \frac{dS_t}{S_t}=\mu^2_t\,dt+\...

Question 5

I am reading Le Gall's GTM book on Brownian motion and am confused about the proof of the generalized Ito formula in Chapter 9. Let me introduce the notations from that book: Let $X=M+V$ be a ...

Question 6

Let $W$ be a Wiener process. Define its set of zeros as $C_{\omega}\equiv \{t\in \mathbb{R}_+: W_t(\omega)=0\}$. I know this set is closed, therefore, its complement $\mathbb{R}_+ \backslash C_{\omega}...

Question 7

Let $M = (M_t)_{t\in\mathbb{N}}$ be a uniformly integrable martingale starting at $M_0 = 0$. By optional sampling for any stopping time $T$ almost surely \begin{equation*} \mathbb{E}[M_T|\mathscr{F}_0]...

Question 8

Let $X_t$ be a stochastic process satisfying, $$ dX_t = \mu(t)X_t dt + \sigma(t)X_t dW_t $$ where $W_t$ is a Brownian motion. Is there a way to solve the following for $y$? $$ y'(t) a(t) + y(t) b(t) + ...

Question 9

I've been reading about the Black-Scholes formula on wikipedia and investopedia and it seems like a lot. My understanding is very limited, but naively it makes sense to me one would be able to assign ...

Question 10

I’m studying basic queueing theory, in particular a single–server queue with one arrival stream and one server (a G/G/1 type setup). Let A be the interarrival time with mean 𝔼(A), B be the service ...

Question 11

Suppose we know that $P(X=1)=p, P(X=-1)=q=1-p$ We have a martingales a) $M_n = (\frac{q}{p})^{S_n}$, b) $M_n = S_n - n(p-q)$ where $S_n =\sum_{i=0}^n X_i$,( $X_i$ iid). How to show (in a simple way) ...

Question 12

I have a Question regarding the chapter 5.5.(Girsanov Theorem) of the Book "Brownian Motion,Martingales, and Stochastic Calculus" from le Gall. There is stated in Prop 5.21, that if D is a ...

Question 13

Let $(S, \mathcal{S})$ be some measurable state space, $\Omega := S^{\mathbb{N}_0}$, $X_i$ the coordinate maps, $\mathcal{A} := \sigma(X_0, X_1, \dots)$ and $\theta: \Omega \rightarrow \Omega, (x_0, ...

Question 14

In Schilling's "Brownian Motion", it is argued in Remark 21.24 that if the stochastic process $X^x$ is the solution to an SDE with initial value $x\in\mathbb{R}$, then it depends measurably ...

Question 15

I have two sequences of independent stochastic processes $X^n$ and $Y^n$ that are known to converge weakly to the Brownian bridges $\mathcal X$ and $\mathcal Y$, respectively. For a continuous ...

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Questions tagged [stochastic-processes]

Joint measurability of the following random field

Stationary state in interacting particle systems

Some questions about Wiener process (Brownian motion).

How to deal with Ito processes product with time delay?

Measurability of stochastic integrals appearing in local times

Beginning time of i-th excursions of Wiener process is a stopping time?

Optional sampling for a martingale at a hitting time

Is there a way to solve linear 1st order ODEs with time-varying coefficients driven by geometric Brownian motion?

How does the Black-Scholes equation handle drift? [duplicate]

Why do we need \rho (utilization)<1 in queuing theory?

Martingales convergence a.s.

Stochastic exponential and martingales

Family of i.i.d. random variables is ergodic under shift

Solution to SDE as measurable function of initial value

Convergence of transform of stochastic processes to transform of Brownian bridges

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