Then \( \bs{Y} = \{Y_n: n \in \N\}\) is a Markov process in discrete time. WebMarkov processes are continuous time Markov models based on Eqn. [1][2], The probabilities of weather conditions (modeled as either rainy or sunny), given the weather on the preceding day, Conversely, suppose that \( \bs{X} = \{X_n: n \in \N\} \) has independent increments. Reinforcement Learning via Markov Decision Process If you want to delve even deeper, try the free information theory course on Khan Academy (and consider other online course sites too). The current state For example, the entry at row 1 and column 2 records the probability of moving from state 1 to state 2. The one step transition kernel \( P \) is given by \[ P[(x, y), A \times B] = I(y, A) Q(x, y, B); \quad x, \, y \in S, \; A, \, B \in \mathscr{S} \], Note first that for \( n \in \N \), \( \sigma\{Y_k: k \le n\} = \sigma\{(X_k, X_{k+1}): k \le n\} = \mathscr{F}_{n+1} \) so the natural filtration associated with the process \( \bs{Y} \) is \( \{\mathscr{F}_{n+1}: n \in \N\} \). We need to decide what proportion of salmons to catch in a year in a specific area maximizing the longer term return. Hence \( Q_s * Q_t \) is the distribution of \( \left[X_s - X_0\right] + \left[X_{s+t} - X_s\right] = X_{s+t} - X_0 \). From now on, we will usually assume that our Markov processes are homogeneous. If the Markov chain includes N states, the matrix will be N x N, with the entry (I, J) representing the chance of migrating from the state I to state J. Stay Connected with a larger ecosystem of data science and ML Professionals, It surprised us all, including the people who are working on these things (LLMs). With the explanation out of the way, let's explore some of the real world applications where theycome in handy. Real Applications of Markov's Inequality From the additive property of expected value and the stationary property, \[ m_0(t + s) = \E(X_{t+s} - X_0) = \E[(X_{t + s} - X_s) + (X_s - X_0)] = \E(X_{t+s} - X_s) + \E(X_s - X_0) = m_0(t) + m_0(s) \], From the additive property of variance for. This means that for \( f \in \mathscr{C}_0 \) and \( t \in [0, \infty) \), \[ \|P_{t+s} f - P_t f \| = \sup\{\left|P_{t+s}f(x) - P_t f(x)\right|: x \in S\} \to 0 \text{ as } s \to 0 \]. Markov Decision Process Definition, Working, and The goal is to decide on the actions to play or quit maximizing total rewards. PageRank assigns a value to a page depending on the number of backlinks referring to it. The book is also freely available for download. To use the PageRank algorithm, we assume the web to be a directed graph, with web pages acting as nodes and hyperlinks acting as edges. Also, the state space \( (S, \mathscr{S}) \) has a natural reference measure measure \( \lambda \), namely counting measure in the discrete case and Lebesgue measure in the continuous case. Markov chains are an essential component of stochastic systems. Expressing a problem as an MDP is the first step towards solving it through techniques like dynamic programming or other techniques of RL. The time set \( T \) is either \( \N \) (discrete time) or \( [0, \infty) \) (continuous time). processes If \(t \in T\) then (assuming that the expected value exists), \[ P_t f(x) = \int_S P_t(x, dy) f(y) = \E\left[f(X_t) \mid X_0 = x\right], \quad x \in S \]. Such real world problems show the usefulness and power of this framework. Moreover, \( P_t \) is a contraction operator on \( \mathscr{B} \), since \( \left\|P_t f\right\| \le \|f\| \) for \( f \in \mathscr{B} \). Agriculture: how much to plant based on weather and soil state. Let \( t \mapsto X_t(x) \) denote the unique solution with \( X_0(x) = x \) for \( x \in \R \). Finally for general \( f \in \mathscr{B} \) by considering positive and negative parts. It's easiest to state the distributions in differential form. 16: Markov Processes - Statistics LibreTexts Recall that \[ g_t(n) = e^{-t} \frac{t^n}{n! Suppose that \( \bs{X} = \{X_t: t \in T\} \) is a random process with \( S \subseteq \R\) as the set of states. University of Texas at Tyler Scholar Works at UT Tyler WebIntroduction to MDPs. A Markov chain is a stochastic model that describes a sequence of possible events or transitions from one state to another of a system. Then jump ahead to the study of discrete-time Markov chains. Markov Such state transitions are represented by arrows from the action node to the state nodes. A positive measure \( \mu \) on \( (S, \mathscr{S}) \) is invariant for \( \bs{X}\) if \( \mu P_t = \mu \) for every \( t \in T \). Reinforcement Learning Formulation via Markov Decision Process (MDP) The basic elements of a reinforcement learning problem are: Environment: The outside world with which the agent interacts. Generative AI is booming and we should not be shocked. If in addition, \( \sigma_0^2 = \var(X_0) \in (0, \infty) \) and \( \sigma_1^2 = \var(X_1) \in (0, \infty) \) then \( v(t) = \sigma_0^2 + (\sigma_1^2 - \sigma_0^2) t \) for \( t \in T \). According to the figure, a bull week is followed by another bull week 90% of the time, a bear week 7.5% of the time, and a stagnant week the other 2.5% of the time. Continuous-time Markov chain is a type of stochastic litigation where continuity makes it different from the Markov series. Recall again that \( P_s(x, \cdot) \) is the conditional distribution of \( X_s \) given \( X_0 = x \) for \( x \in S \). In essence, your words are analyzed and incorporated into the app's Markov chain probabilities. Suppose that \( \lambda \) is the reference measure on \( (S, \mathscr{S}) \) and that \( \bs{X} = \{X_t: t \in T\} \) is a Markov process on \( S \) and with transition densities \( \{p_t: t \in T\} \). In particular, the right operator \( P_t \) is defined on \( \mathscr{B} \), the vector space of bounded, linear functions \( f: S \to \R \), and in fact is a linear operator on \( \mathscr{B} \). If \( \mu_s \) is the distribution of \( X_s \) then \( X_{s+t} \) has distribution \( \mu_{s+t} = \mu_s P_t \). The probability distribution now is all about calculating the likelihood that the following word will be like or love if the preceding word is I., In our example, the word like comes in two of the three phrases after I, but the word love appears just once. So we will often assume that a Feller Markov process has sample paths that are right continuous have left limits, since we know there is a version with these properties. Are you looking for a complete repository of Python libraries used in data science,check out here. The time space \( (T, \mathscr{T}) \) has a natural measure; counting measure \( \# \) in the discrete case, and Lebesgue in the continuous case. States: these can refer to for example grid maps in robotics, or for example door open and door closed. Let \( \tau_t = \tau + t \) and let \( Y_t = \left(X_{\tau_t}, \tau_t\right) \) for \( t \in T \). Notice that the rows of P sum to 1: this is because P is a stochastic matrix.[3]. And the word love is always followed by the word cycling.. One of the interesting implications of Markov chain theory is that as the length of the chain increases (i.e. Suppose that \( \bs{X} = \{X_t: t \in T\} \) is a Markov process with state space \( (S, \mathscr{S}) \) and that \( (t_0, t_1, t_2, \ldots) \) is a sequence in \( T \) with \( 0 = t_0 \lt t_1 \lt t_2 \lt \cdots \). The Markov and homogenous properties follow from the fact that \( X_{t+s}(x) = X_t(X_s(x)) \) for \( s, \, t \in [0, \infty) \) and \( x \in S \). For an overview of Markov chains in general state space, see Markov chains on a measurable state space. So any process that has the states, actions, transition probabilities Listed here are a few simple examples where MDP Hence if \( \mu \) is a probability measure that is invariant for \( \bs{X} \), and \( X_0 \) has distribution \( \mu \), then \( X_t \) has distribution \( \mu \) for every \( t \in T \) so that the process \( \bs{X} \) is identically distributed. The probability distribution is concerned with assessing the likelihood of transitioning from one state to another, in our instance from one word to another. If we sample a Markov process at an increasing sequence of points in time, we get another Markov process in discrete time. Examples of the Markov Decision Process MDPs have contributed significantly across several application domains, such as computer science, electrical engineering, manufacturing, operations research, finance and economics, telecommunications, and so on. For example, if \( t \in T \) with \( t \gt 0 \), then conditioning on \( X_0 \) gives \[ \P(X_0 \in A, X_t \in B) = \int_A \P(X_t \in B \mid X_0 = x) \mu_0(dx) = \int_A P_t(x, B) \mu(dx) = \int_A \int_B P_t(x, dy) \mu_0(dx) \] for \( A, \, B \in \mathscr{S} \). The Markov chains were used to forecast the election outcomes in Ghana in 2016. Generating points along line with specifying the origin of point generation in QGIS. Consider the random walk on \( \R \) with steps that have the standard normal distribution. As with the regular Markov property, the strong Markov property depends on the underlying filtration \( \mathfrak{F} \). Introduction to Markov models and Markov Chains - The AI dream Also, everyday certain portion of patients in the hospital recovers and released. A process \( \bs{X} = \{X_n: n \in \N\} \) has independent increments if and only if there exists a sequence of independent, real-valued random variables \( (U_0, U_1, \ldots) \) such that \[ X_n = \sum_{i=0}^n U_i \] In addition, \( \bs{X} \) has stationary increments if and only if \( (U_1, U_2, \ldots) \) are identically distributed. It is Memoryless due to this characteristic of the Markov Chain. The Wiener process is named after Norbert Wiener, who demonstrated its mathematical existence, but it is also known as the Brownian motion process or simply Brownian motion due to its historical significance as a model for Brownian movement in liquids (Image will be Uploaded Soon) In summary, an MDP is useful when you want to plan an efficient sequence of actions in which your actions can be not always 100% effective. If \( \bs{X} \) has stationary increments in the sense of our definition, then the process \( \bs{Y} = \{Y_t = X_t - X_0: t \in T\} \) has stationary increments in the more restricted sense. Otherwise, the state vectors will oscillate over time without converging. Let \( Y_n = X_{t_n} \) for \( n \in \N \). The compact sets are simply the finite sets, and the reference measure is \( \# \), counting measure. You keep going, noting that Day 2 was also sunny, but Day 3 was cloudy, then Day 4 was rainy, which led into a thunderstorm on Day 5, followed by sunny and clear skies on Day 6. In continuous time, however, two serious problems remain. The proofs are simple using the independent and stationary increments properties. Suppose that \( s, \, t \in T \). Again, this result is only interesting in continuous time \( T = [0, \infty) \). It's easy to describe processes with stationary independent increments in discrete time. Note that the duration is captured as part of the current state and therefore the Markov property is still preserved. Condition (a) means that \( P_t \) is an operator on the vector space \( \mathscr{C}_0 \), in addition to being an operator on the larger space \( \mathscr{B} \). Enterprises look for tech enablers that can bring in the domain expertise for particular use cases, Analytics India Magazine Pvt Ltd & AIM Media House LLC 2023. Basically, he invented the Markov chain,hencethe naming. For example, if today is sunny, then: Now repeat this for every possible weather condition. So action = {0, min(100 s, number of requests)}. = Journal of Physics: Conference Series PAPER OPEN Markov The next state of the board depends on the current state, and the next roll of the dice. This result is very important for constructing Markov processes. Using the transition probabilities, the steady-state probabilities indicate that 62.5% of weeks will be in a bull market, 31.25% of weeks will be in a bear market and 6.25% of weeks will be stagnant, since: A thorough development and many examples can be found in the on-line monograph Meyn & Tweedie 2005.[7]. Suppose now that \( \bs{X} = \{X_t: t \in T\} \) is a stochastic process on \( (\Omega, \mathscr{F}, \P) \) with state space \( S \) and time space \( T \). That is, the state at time \( t + s \) depends only on the state at time \( s \) and the time increment \( t \). Then the transition density is \[ p_t(x, y) = g_t(y - x), \quad x, \, y \in S \]. Intuitively, \( \mathscr{F}_t \) is the collection of event up to time \( t \in T \). Any chance you can fix the links? Suppose \( \bs{X} = \{X_t: t \in T\} \) is a Markov process with transition operators \( \bs{P} = \{P_t: t \in T\} \), and that \( (t_1, \ldots, t_n) \in T^n \) with \( 0 \lt t_1 \lt \cdots \lt t_n \). , Markov chains are used in a variety of situations because they can be designed to model many real-world processes. Action: Each day the hospital gets requests of number of patients to admit based on a Poisson random variable. It only takes a minute to sign up. The last phrase means that for every \( \epsilon \gt 0 \), there exists a compact set \( C \subseteq S \) such that \( \left|f(x)\right| \lt \epsilon \) if \( x \notin C \). Rewards are generated depending only on the (current state, action) pair. You may have heard the term "Markov chain" before, but unless you've taken a few classes on probability theory or computer science algorithms, you probably don't know what they are, how they work, and why they're so important. It provides a way to model the dependencies of current information (e.g. Some of the statements are not completely rigorous and some of the proofs are omitted or are sketches, because we want to emphasize the main ideas without getting bogged down in technicalities. Suppose that \( \bs{X} = \{X_n: n \in \N\} \) is a stochastic process with state space \( (S, \mathscr{S}) \) and that \(\bs{X}\) satisfies the recurrence relation \[ X_{n+1} = g(X_n), \quad n \in \N \] where \( g: S \to S \) is measurable. Notice, the arrows exiting a state always sums up to exactly 1, similarly the entries in each row in the transition matrix must add up to exactly 1 - representing probability distribution. The Markov and time homogeneous properties simply follow from the trivial fact that \( g^{m+n}(X_0) = g^n[g^m(X_0)] \), so that \( X_{m+n} = g^n(X_m) \). What should I follow, if two altimeters show different altitudes? Our goal in this discussion is to explore these connections. It uses GTP3 and Markov Chain to generate text and random the text but still tends to be meaningful. However, this is not always the case. Markov chain is a random process with Markov characteristics, which exists in the discrete index set and state space in probability theory and mathematical statistics. (Most of the time, anyway.). The mean and variance functions for a Lvy process are particularly simple. In both cases, \( T \) is given the Borel \( \sigma \)-algebra \( \mathscr{T} \), the \( \sigma \)-algebra generated by the open sets. It has at least one absorbing state. Ideally you'd be more granular, opting for an hour-by-hour analysis instead of a day-by-day analysis, but this is just an example to illustrate the concept, so bear with me! Actions: For simplicity assumes there are only two actions; fish and not_to_fish. Briefly speaking, a random variable is a Markov process if the transition probability, from state at time to another state , depends only on the current state . That is, which is independent of the states before . In addition, the sequence of random variables generated by a Markov process is subsequently called a Markov chain. Continuous-time Markov chain is a type of stochastic litigation where continuity makes it different from the Markov series. That is, \( g_s * g_t = g_{s+t} \). The probability distribution of taking actions At from a state St is called policy (At | St). But if a large proportion of salmons are caught then the yield of the next year will be lower. They explain states, actions and probabilities which are fine. Every entry in the vector indicates the likelihood of starting in that condition. : Conf. At any given time stamp t, the process is as follows. At any round if participants failed to answer correctly then s/he looses all the rewards earned so far. The actions can only be dependent on the current state and not on any previous state or previous actions (Markov property). Recall that the commutative property generally does not hold for the product operation on kernels. State Transitions: Transitions are deterministic. The probability of The second uses the fact that \( \bs{X} \) has the strong Markov property relative to \( \mathfrak{G} \), and the third follows since \( \bs{X_\tau} \) measurable with respect to \( \mathscr{F}_\tau \). There is a bot on Reddit that generates random and meaningful text messages. The weather on day 0 (today) is known to be sunny. The Borel \( \sigma \)-algebra \( \mathscr{T}_\infty \) is used on \( T_\infty \), which again is just the power set in the discrete case. With the strong Markov and homogeneous properties, the process \( \{X_{\tau + t}: t \in T\} \) given \( X_\tau = x \) is equivalent in distribution to the process \( \{X_t: t \in T\} \) given \( X_0 = x \). Therefore the action is a number between 0 to (100 s) where s is the current state i.e. Simply put, Subreddit Simulator takes in a massive chunk of ALL the comments and titles made across Reddit's numerous communities, then analyzes the word-by-word makeup of each sentence. Can it find patterns amoung infinite amounts of data? When is Markov's Inequality useful? There are certainly more general Markov processes, but most of the important processes that occur in applications are Feller processes, and a number of nice properties flow from the assumptions. For \( t \in T \), let \[ P_t(x, A) = \P(X_t \in A \mid X_0 = x), \quad x \in S, \, A \in \mathscr{S} \] Then \( P_t \) is a probability kernel on \( (S, \mathscr{S}) \), known as the transition kernel of \( \bs{X} \) for time \( t \). Consider a random walk on the number line where, at each step, the position (call it x) may change by +1 (to the right) or 1 (to the left) with probabilities: For example, if the constant, c, equals 1, the probabilities of a move to the left at positions x = 2,1,0,1,2 are given by Next, \begin{align*} \P[Y_{n+1} \in A \times B \mid Y_n = (x, y)] & = \P[(X_{n+1}, X_{n+2}) \in A \times B \mid (X_n, X_{n+1}) = (x, y)] \\ & = \P(X_{n+1} \in A, X_{n+2} \in B \mid X_n = x, X_{n+1} = y) = \P(y \in A, X_{n+2} \in B \mid X_n = x, X_{n + 1} = y) \\ & = I(y, A) Q(x, y, B) \end{align*}. [ 32] proposed a method combining Monte Carlo simulations and directional sampling to analyse object reliability sensitivity. Hence \( \bs{Y} \) is a Markov process. WebExamples in Markov Decision Processes is an essential source of reference for mathematicians and all those who apply the optimal control theory to practical purposes. Recall also that usually there is a natural reference measure \( \lambda \) on \( (S, \mathscr{S}) \). If \( s, \, t \in T \) then \( p_s p_t = p_{s+t} \). Reward = (number of cars expected to pass in the next time step) * exp( * duration of the traffic light red in the other direction). So the only possible source of randomness is in the initial state. 3 Both actions and rewards can be probabilistic. That is, \[ P_t(x, A) = \P(X_t \in A \mid X_0 = x) = \int_A p_t(x, y) \lambda(dy), \quad x \in S, \, A \in \mathscr{S} \] The next theorem gives the Chapman-Kolmogorov equation, named for Sydney Chapman and Andrei Kolmogorov, the fundamental relationship between the probability kernels, and the reason for the name transition kernel. If \( Q \) has probability density function \( g \) with respect to the reference measure \( \lambda \), then the one-step transition density is \[ p(x, y) = g(y - x), \quad x, \, y \in S \]. 4 Continuing in this manner gives the general result. The random process \( \bs{X} \) is a Markov process if and only if \[ \E[f(X_{s+t}) \mid \mathscr{F}_s] = \E[f(X_{s+t}) \mid X_s] \] for every \( s, \, t \in T \) and every \( f \in \mathscr{B} \). Suppose in addition that \( (U_1, U_2, \ldots) \) are identically distributed. Absorbing Markov Chain. Moreover, by the stationary property, \[ \E[f(X_{s+t}) \mid X_s = x] = \int_S f(x + y) Q_t(dy), \quad x \in S \]. It can't know for sure what you meant to type next, but it's correct more often than not. undirected graphical models) to data science. If \( s, \, t \in T \) and \( f \in \mathscr{B} \) then \[ \E[f(X_{s+t}) \mid \mathscr{F}_s] = \E\left(\E[f(X_{s+t}) \mid \mathscr{G}_s] \mid \mathscr{F}_s\right)= \E\left(\E[f(X_{s+t}) \mid X_s] \mid \mathscr{F}_s\right) = \E[f(X_{s+t}) \mid X_s] \] The first equality is a basic property of conditional expected value. Figure 2: An example of the Markov decision process. WebThe Monte Carlo Markov chain simulation algorithm [ 31] was developed to optimise maintenance policy and resulted in a 10% reduction in total costs for every mile of track. Ser. Markov Explanation - Doctor Nerve Condition (b) actually implies a stronger form of continuity in time. Our first result in this discussion is that a non-homogeneous Markov process can be turned into a homogenous Markov process, but only at the expense of enlarging the state space. The strong Markov property for our stochastic process \( \bs{X} = \{X_t: t \in T\} \) states that the future is independent of the past, given the present, when the present time is a stopping time. The stock market is a volatile system with a high degree of unpredictability. Thus suppose that \( \bs{U} = (U_0, U_1, \ldots) \) is a sequence of independent, real-valued random variables, with \( (U_1, U_2, \ldots) \) identically distributed with common distribution \( Q \). WebThe concept of a Markov chain was developed by a Russian Mathematician Andrei A. Markov (1856-1922). Chapter 3 of the book Reinforcement Learning An Introduction by Sutton and Barto [1] provides an excellent introduction to MDP. If you've never used Reddit, we encourage you to at least check out this fascinating experiment called /r/SubredditSimulator. Suppose that \( \bs{X} = \{X_t: t \in T\} \) is a non-homogeneous Markov process with state space \( (S, \mathscr{S}) \). If you want to predict what the weather might be like in one week, you can explore the various probabilities over the next seven days and see which ones are most likely. But this forces \( X_0 = 0 \) with probability 1, and as usual with Markov processes, it's best to keep the initial distribution unspecified. This one for example: https://www.youtube.com/watch?v=ip4iSMRW5X4. Joel Lee was formerly the Editor in Chief of MakeUseOf from 2018 to 2021. When \( S \) has an LCCB topology and \( \mathscr{S} \) is the Borel \( \sigma \)-algebra, the measure \( \lambda \) wil usually be a Borel measure satisfying \( \lambda(C) \lt \infty \) if \( C \subseteq S \) is compact. Action either changes the traffic light color or not. 6 For example, in Google Keyboard, there's a setting called Share snippets that asks to "share snippets of what and how you type in Google apps to improve Google Keyboard". WebA Markov analysis looks at a sequence of events, and analyzes the tendency of one event to be followed by another. not on a list of previous states). So we usually don't want filtrations that are too much finer than the natural one. By definition and the substitution rule, \begin{align*} \P[Y_{s + t} \in A \times B \mid Y_s = (x, r)] & = \P\left(X_{\tau_{s + t}} \in A, \tau_{s + t} \in B \mid X_{\tau_s} = x, \tau_s = r\right) \\ & = \P \left(X_{\tau + s + t} \in A, \tau + s + t \in B \mid X_{\tau + s} = x, \tau + s = r\right) \\ & = \P(X_{r + t} \in A, r + t \in B \mid X_r = x, \tau + s = r) \end{align*} But \( \tau \) is independent of \( \bs{X} \), so the last term is \[ \P(X_{r + t} \in A, r + t \in B \mid X_r = x) = \P(X_{r+t} \in A \mid X_r = x) \bs{1}(r + t \in B) \] The important point is that the last expression does not depend on \( s \), so \( \bs{Y} \) is homogeneous. However, you can certainly benefit from understanding how they work. He has a B.S. The agent needs to find optimal action on a given state that will maximize this total rewards. another, is this true? We do know of such a process, namely the Poisson process with rate 1. The goal of the agent is to maximize the total rewards (Rt) collected over a period of time. For the remainder of this discussion, assume that \( \bs X = \{X_t: t \in T\} \) has stationary, independent increments, and let \( Q_t \) denote the distribution of \( X_t - X_0 \) for \( t \in T \). You do this over the entire 30-year data set (which would be just shy of 11,000 days) and calculate the probabilities of what tomorrow's weather will be like based on today's weather. These examples and corresponding transition graphs can help developing the skills to express problem using MDP. It is a very useful framework to model problems that maximizes longer term return by taking sequence of actions. To formalize this, we wish to calculate the likelihood of travelling from state I to state J over M steps. That is, \[ p_t(x, z) = \int_S p_s(x, y) p_t(y, z) \lambda(dy), \quad x, \, z \in S \]. Hence \[ \E[f(X_{\tau+t}) \mid \mathscr{F}_\tau] = \E\left(\E[f(X_{\tau+t}) \mid \mathscr{G}_\tau] \mid \mathscr{F}_\tau\right)= \E\left(\E[f(X_{\tau+t}) \mid X_\tau] \mid \mathscr{F}_\tau\right) = \E[f(X_{\tau+t}) \mid X_\tau] \] The first equality is a basic property of conditional expected value. is at least one Pn with all non-zero entries). A game of snakes and ladders or any other game whose moves are determined entirely by dice is a Markov chain, indeed, an absorbing Markov chain. A. Markov began the study of an important new type of chance process. Do you know of any other cool uses for Markov chains? I would call it planning, not predicting like regression for example. Discrete-time Markov chain (or discrete-time discrete-state Markov process) 2. So combining this with the remark above, note that if \( \bs{P} \) is a Feller semigroup of transition operators, then \( f \mapsto P_t f \) is continuous on \( \mathscr{C}_0 \) for fixed \( t \in T \), and \( t \mapsto P_t f \) is continuous on \( T \) for fixed \( f \in \mathscr{C}_0 \). Connect and share knowledge within a single location that is structured and easy to search. WebExamples in Markov Decision Processes is an essential source of reference for mathematicians and all those who apply the optimal control theory to practical purposes. Thus, the finer the filtration, the larger the collection of stopping times. This Markov process is known as a random walk (although unfortunately, the term random walk is used in a number of other contexts as well). If you are a new student of probability you may want to just browse this section, to get the basic ideas and notation, but skipping over the proofs and technical details.
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