Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [stochastic-processes]

For questions about stochastic processes, for example random walks and Brownian motion.

A stochastic process is a collection of random variables representing the evolution of a system of random variables over time. A typical example is a .

A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Use stochastic calculus (Ito's lemma) to compute the expectation

Calculate $E[\cos(X)e^X]$, where $X\sim N(0,\sigma^2)$. Use stochastic calculus instead of integrating w.r.t the normal density. During the discussion with friends, we believe that we should use Brownian motion $B_t$ to represent X and then somehow…
user16859
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Doob-like inequality

I am not sure that this is a proper question. I am looking for the verification of the proof rather than for an answer. There is one inequality I use pretty often and I would like to be sure that it is correct. Namely, if $X_n$ is a non-negative…
SBF
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When is a Markov process independent-increment?

An independent-increment stochastic process must be Markov. I am now wondering about the reverse case. Why do some Markov processes fail to be independent-increment? What are some examples of Markov processes that are not independent-increment? Is…
Tim
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Derivation of Differential Chapman-Kolmogorov equation, Kramers-Moyal expansion

I'm stuck with the derivation of the differential Chapman-Kolmogorov equation provided in Gardiner 1985, section 3.4. This is supposed to be some middle ground between the master equation and the Fokker-Planck equation since it allows for jumps to…
S.Surace
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Time changes of the Itō integral

Let $(B_t^1, \ldots, B_t^d)$ be a standard $d$-dimensional Brownian motion, and $H_t^j$, $j=1, \ldots d$ be continuous processes adapted to the filtration $\{\mathcal{F}_t\}$. Let $$Z_t = \sum_{j=1}^d \int_0^t H_s^j dB_s^j$$ $$\langle Z \rangle_t =…
user98123
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Independence Lemma, is it non-trivial?

I'm reading Steven E. Shreve's "Stochastic Calculus for Finance II, Continuous-Time models", and a bit confused on the Independence Lemma (Lemma 2.3.4). The lemma says: Lemma 2.3.4 (Independence). Let $(\Omega,\mathscr{F},\mathbb{P})$ be a…
athos
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Is Brownian bridge a Markov process

As in the title, question is whether a Brownian bridge: $X_{t} = B_{t} - tB_{1}$ is a Markov process. I could sort of prove it by the markov property, but not sure whether it's sufficient. Does anyone have any good ideas? Thanks.
Math
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Infinitesimal generator of time-dependent Markov diffusion

If one talks about homogeneous Markov diffusion $$ \mathrm dX_t = \mu(X_t)\mathrm dt+\sigma(X_t)\mathrm dw_t $$ with $\mu,\sigma$ sufficiently differentiable and of appropriate dimensions, there is nice equation for a function $m_f(x,t) = \mathsf…
SBF
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Stochastic integral with a Poisson process

I have a Poisson process $X_t$ for $t\ge0$. How I can find a process $b_t$ such that $$\exp ({\alpha X_t})=1+\int_0^t b_{s^{-}}dX_s$$ where $\alpha\in\mathbb{R}$ and what would be the expectation of $\exp ({\alpha X_t})$. The last question is how…
user5644
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How to characterize recurrent and transient states of Markov chain

According to Wikipedia with a little rephrasing: A state $i$ is transient if and only if $P(T_i < \infty) <1$, recurrent if and only if $P(T_i < \infty) =1$, where $T_i$ is the first hitting time to i, i.e. $T_i=\inf\{n \in \mathbb{N} \cup…
Tim
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Convergence of Brownian integral

Let $B$ be a Brownian motion. I'm trying to show that $$\left(\int_0^te^{B_s}ds\right)^\frac{1}{\sqrt{t}}$$ convergences in distribution as $t \to \infty$. As a hint, we are told to consider the limit as $p\to\infty$ of…
Ben Derrett
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question on a variant of an alternating renewal process (interarrival times are per gender)

This question asks about a variant of an alternating renewal process. I am sitting at a cafe watching men and women walk by. The interarrival time $X$ between successive men is iid with distribution $F$, while the interarrival time $Y$ between…
Ron Rivest
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How does the sample space $\Omega$ look like in a stochastic process?

So, I try to understand the confusing terminology of stochastics - especially that of stochastic processes. Let $(\Omega,\Sigma,P)$ be a probability space and $(Z,\Sigma')$ a measure space with $\Sigma\Sigma'$-measureable functions $X_t$, i.e.…
S. Den
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Running maximum absolute value of Wiener process

In Wikipedia a formula is given for the distribution of $$M_t = \max_{0\leq s \leq t} W_s$$ even conditioned on $W_t$. I wonder if there is also a simple expression for (note the absolute value) $$\tilde M_t = \max_{0\leq s \leq t} |W_s|$$ maybe…
Fabian
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Sample function in a stochastic process

In the Wikipedia article about stochastic processes, a sample function is defined by specifying some $\omega\in\Omega$, as $X(\cdot,\omega):T\to S$, which then is a non-random function. My (conceptual) question is: with $\{X(t,\cdot)\}_{t\in T}$…
DominikS
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