Questions tagged [stochastic-processes]

Question 1

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 2

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 3

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 4

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 5

I want to model a system in terms of probability of failure. If I use a stochastic differential equation that is bounded [0,1], can I assume that it models probability failure? I know that failure ...

Question 6

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 7

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 8

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 ...

Question 9

Let $S$ be an increasing Levy process. Let $Z$ be another Levy process in $R^d$, independent of $S$. A standard result is $X\equiv Z_{S}$ is also Levy. Now, to characterize the distribution of $X$, we ...

Question 10

First, consider a symmetric random walk $X_n := Y_1 + \dots + Y_n$, with $P(Y_k = \pm 1) = \frac{1}{2}$ for all $k \in \mathbb{N}$. For $c > 0$ define the stopping time $T_c := \min \{n \geq 0 \,|\,...

Question 11

Let $X$ be a Brownian semimartingale $$ X_t = X_0+\int_0^t\mu_s\,ds+\int_0^t\sigma_s\,dW_s $$ with $\mu$ and $\sigma$ bounded and cadlag, $W$ Brownian motion and $$ \sigma_s = \sigma_0+\int_0^t\mu^{\...

Question 12

Let $\mathbb{H}_1,\mathbb{H}_2$ be two vector spaces over $\mathbb{R}$, and assume that we have a miltilinear function $f:\mathbb{H}_1\times \mathbb{H}_1\times \mathbb{H}_2\times\mathbb{H}_2\to \...

Question 13

One sometimes studies the operator $$ f(s) \mapsto \int_0^1 {\rm cov}(X_\tau X_t) f(\tau) \, d\tau, $$ where $ X $ is a stochastic process (not necessarily ergodic). What is the meaning of this ...

Question 14

I'm a physics student currently reading "Econophysics and Physical Economics by Peter Richmond, J¨urgen Mimkes, and Stefan Hutzler" for the first time and this is my first touch with the ...

Question 15

I am asked to prove the following: Given $X=(X_t)$ a continuous process and $f : \mathbb{R} \to \mathbb{R}$ measurable. If $(\mathcal{F}, P)$ is a complete $\sigma$-algebra, then $\sup_{s \in [0,t]} f(...

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Questions tagged [stochastic-processes]

How does the Black-Scholes equation handle drift?

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

Martingales convergence a.s.

Stochastic exponential and martingales

If a Stochastic Differential Equation is bounded [0,1], can we assume that models probability failure; [closed]

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

decomposition of levy process under subordination

Convergence of Random Walk and Bounded Martingals

Representation of stopping time

A multilinear version of Cauchy-Schwarz inequality

Hilbert Schmidt operator and random process

Beginner question about Option pricing, Risk and Rational portfolios

Running supremum and complete $\sigma$-algebra

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