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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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$L^2$ martingales relation to martingales?

Are all continuous time $L^2$-martingales i.e martingales such that $\mathbb{E}X_{t}^2<\infty$ for all $t$ also martingales in the sense that $\mathbb{E}\mid X_{t} \mid<\infty$ for all $t$, which is the common definition
Second
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If $X_t$ is right-continuous and adapted process, is $X_{t-}$ always predictable?

If $X_t$ is right-continuous and adapted process, is $X_{t-}$ always predictable? Or there are other conditions that must be satisfied? I've become a bit unsure due to the following example in "Introduction to Stochastic Calculus with…
learningmath
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Poisson Process - Finding conditional expectation and conditional variance

Consider a Poisson process $X={X(t);t\ge0}$ of rate $λ=5$. Here $X(t)$ is the number of customers arrived up to the time$=t$. Suppose that $X(1)=5$ (so 5 customers arrived by the end of the first hour). Find the conditional expectation and…
Drake
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Calculating absolute value of Brownian motion

I am trying to calculate $\mathbb{E}\mid B_{t} \, \mid$ in terms of $t$, but am really stuck. Any toughts?
Persi
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Show that $(|W_{t}|^{2-d})_{t \in (0, \infty)}$ is a supermartingale for $d \geqslant 3$

Let $W_{t} = (W_{t}^{1}, \ldots, W_{t}^{d}), t \geqslant 0$ be a $d$-dimensional Wiener process, e.i. $W^{1}, \ldots, W^{d}$ are independent 1-dimensional Wiener processes. Let $ d \geqslant 3 $. Show that $(|W_{t}|^{2-d})_{t \in (0, \infty)}$ is…
Olga98
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How to read transition probability matrix for Markov chain

Suppose that whether or not it rains today depends on previous weather conditions through the last two days. So if $RR$ (rained yesterday and today), then it will rain tomorrow with probability $0.7$. For $SR$ (rained today but not yesterday), it…
Alti
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a.s convergence of $\frac{W_{t}}{N_{t}}$, $\frac{W_{t}}{\sqrt{N_{t}}}$, $\frac{|W_{t}|}{\sqrt{t \ln \ln t^{2}}}$

Let $W$ be a standard Wiener process in $\mathbb{R}$, and let $N$ be a Poisson process with parameter $\lambda$. Decide if those limits exist a.e and calculate them: $\lim_{t \rightarrow \infty}\frac{W_{t}}{N_{t}}$ $\lim_{t \rightarrow…
Olga98
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Why is $Q^\frac{1}{2}$ a symmetric nonnegative definite?

Im new in Studying stochastic and currently reading to gain something. This question is utilized in The following link If $Q$ is an operator on a Hilbert space $U$, $(e_n)$ is an ONB of $U$ consisting of eigenvectors of $Q$, then $(Q^{1/2}e_n)$ is…
raijin
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$\sigma$-Algebra of T-past

Hi i have following defnition $T:\Omega\rightarrow I$ $\sigma_T=\{A\in\Omega|A \cap\{{T \leq t\}}\in\sigma_t\, \text{for all t}\in I \}$ I do not understand the sense of the definition. Why there has to be the condition $A\in\Omega|A \cap\{{T \leq…
tim123
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Whether the integrated Brownian motion a Gaussian process?

We can define the integration of Brownian motion $W_t$ wrt t as $$Z_t=\int_0^t W_sds.$$ We have know that $Z_t\sim N(0, t^3/3)$, but I am still not so sure that $Z_t$ is a Gaussian process or not. I don't have clue to follow by definition of a…
Gary
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Conditional expectation of $\sin S_n$ given $S_n^2$

Consider the following exercise from Lawler's Stochastic Calculus: Suppose $X_1 , X_2 , \dots$ are independent random variables with $$ P\{X_j = 1\} = P\{X_j = −1\} = \frac12. $$ Let $S_n = X_1 + \dots + X_n$. Find $$ E(\sin S_n |…
J. D.
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Concerning the derivation of the Fokker-Planck equation, as in Reichl?

My question is; how does expansion of the conditional probability density $P_{1|1}(y_1,t_1|y_2,t_1+\tau)$ get put into the form of(6.23) of Reichl$^1$? The term with the minus sign at the front, seems a bit mysterious, could some explanation of it…
user151522
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Bernoulli process, probability of time of success

With a Bernoulli process success probability $p$ and failure probability $q = 1 - p$, the probability for $T_k$ time of the $k^{th}$ success is $$P(T_k \leq n) = \sum_{j=k}^n \binom{n}{j}p^jq^{n-j}$$ My question is how does this work for $k=1$?…
Vahan
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Let $X_t=\int_0^t \sigma (t)dB_t$. What is the law of $X_t$?

If $X_t=\int_0^t \sigma (s)dB_s$ is it possible to have information on the law of $X_t$ ? For example, if $h$ is very small, then $$X_{t+h}-X_t\approx \sigma (t)(B_{t+s}-B_t)\sim\mathcal N(0,t\sigma (t)^2).$$ Can we do better ? I often heard that…
user657324
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Bernoulli process, expected value of conditional

Consider a Bernoulli process where at each step or time there is a success, 1, or failure, 0. $N_k$ denotes the number of successes at the $k^{th}$ step. The probability of success is $p$ and the probability of failure is $q$, where $p + q = 1$. The…
Vahan
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