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  "body": "\n## Summary\n\nA stochastic process is a family $(X_t)_{t \\in T}$ of [[random variables|0DKRR22AEQ]] on one [[probability space|0ZYBZ34RT2]] $(\\Omega, \\mathcal{F}, P)$, all with values in the same measurable space $(S, \\mathcal{S})$.\nThe set $T$ is the index set, often a set of times such as $\\{0, 1, 2, \\dots\\}$ or $[0, \\infty)$. The set $S$ is the state space.\nFor each outcome $\\omega \\in \\Omega$, the function $t \\mapsto X_t(\\omega)$ is a sample path of the process.\n\n## Remarks\n\nThe process is in discrete time if $T$ is countable, such as $\\{0, 1, 2, \\dots\\}$, and in continuous time if $T$ is an interval.\nThe finite-dimensional distributions are the laws of the vectors $(X_{t_1}, \\dots, X_{t_n})$ for finite sets $\\{t_1, \\dots, t_n\\} \\subseteq T$. The Kolmogorov extension theorem builds a process from a consistent family of such laws.\nImportant examples are the Bernoulli process, random walks, Markov chains, the Poisson process and the Wiener process.\n\n## Explanations\n\n### L1 human\n\nA stochastic process is a random quantity that changes with time.\nExamples are the price of a share day by day and the number of customers in a queue.\nAt each time it is a random variable, and one run of the whole process gives a random path.\n",
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