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. but converges to a strictly positive vector only if P is a regular transition matrix (that is, there This shows that the future state (next token) is based on the current state (present token). So this is the most basic rule in the Markov Model. The below diagram shows that there are pairs of tokens where each token in the pair leads to the other one in the same pair. Which ability is most related to insanity: Wisdom, Charisma, Constitution, or Intelligence? This means that \( \P[X_t \in U \mid X_0 = x] \to 1 \) as \( t \downarrow 0 \) for every neighborhood \( U \) of \( x \). From any non-absorbing state in the Markov chain, it is possible to eventually move to some absorbing state (in one or 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. It is not necessary to know when they p Reward = (number of cars expected to pass in the next time step) * exp( * duration of the traffic light red in the other direction). If I know that you have $12 now, then it would be expected that with even odds, you will either have $11 or $13 after the next toss. The result above shows how to obtain the distribution of \( X_t \) from the distribution of \( X_0 \) and the transition kernel \( P_t \) for \( t \in T \). Intuitively, \( \mathscr{F}_t \) is the collection of event up to time \( t \in T \). For this reason, the initial distribution is often unspecified in the study of Markov processesif the process is in state \( x \in S \) at a particular time \( s \in T \), then it doesn't really matter how the process got to state \( x \); the process essentially starts over, independently of the past. Reward: Numerical feedback signal from the environment. This is the Borel \( \sigma \)-algebra for the discrete topology on \( S \), so that every function from \( S \) to another topological space is continuous. Technically, we should say that \( \bs{X} \) is a Markov process relative to the filtration \( \mathfrak{F} \). Solving this pair of simultaneous equations gives the steady state vector: In conclusion, in the long term about 83.3% of days are sunny. Then \( \bs{Y} = \{Y_n: n \in \N\}\) is a Markov process in discrete time. The transition kernels satisfy \(P_s P_t = P_{s+t} \). To anticipate the likelihood of future states happening, elevate your transition matrix P to the Mth power. A 50 percent chance that tomorrow will be sunny again. A Markov chain is a stochastic process that meets the Markov property, which states that while the present is known, the past and future are independent. The best answers are voted up and rise to the top, Not the answer you're looking for? {\displaystyle {\dfrac {1}{6}},{\dfrac {1}{4}},{\dfrac {1}{2}},{\dfrac {3}{4}},{\dfrac {5}{6}}} It then follows that \( P_t \) is a continuous operator on \( \mathscr{B} \) for \( t \in T \). Given these two dependencies, the starting state of the Markov chain may be calculated by taking the product of P x I. If you've never used Reddit, we encourage you to at least check out this fascinating experiment called /r/SubredditSimulator. Let \( t \mapsto X_t(x) \) denote the unique solution with \( X_0(x) = x \) for \( x \in \R \). So action = {0, min(100 s, number of requests)}. Actions: For simplicity assumes there are only two actions; fish and not_to_fish. There is a bot on Reddit that generates random and meaningful text messages. Such state transitions are represented by arrows from the action node to the state nodes. The random process \( \bs{X} \) is a strong Markov process if \[ \E[f(X_{\tau + t}) \mid \mathscr{F}_\tau] = \E[f(X_{\tau + t}) \mid X_\tau] \] for every \(t \in T \), stopping time \( \tau \), and \( f \in \mathscr{B} \). That is, for \( n \in \N \) \[ \P(X_{n+2} \in A \mid \mathscr{F}_{n+1}) = \P(X_{n+2} \in A \mid X_n, X_{n+1}), \quad A \in \mathscr{S} \] where \( \{\mathscr{F}_n: n \in \N\} \) is the natural filtration associated with the process \( \bs{X} \). For example, if we roll a die and want to know the probability of the result being a 5 or greater we have that . For example, if today is sunny, then: Now repeat this for every possible weather condition. The term discrete state space means that \( S \) is countable with \( \mathscr{S} = \mathscr{P}(S) \), the collection of all subsets of \( S \). A Markov process is a random process in which the future is independent of the past, given the present. 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. Bootstrap percentiles are used to calculate confidence ranges for these forecasts. A lesser but significant proportion of the time, the surfer will abandon the current page and select a random page from the web to teleport to. The operator on the right is given next. represents the number of dollars you have after n tosses, with Continuing in this manner gives the general result. If one could help instantiate the homogeneous Markov chains using a very simple real-world example and then change one condition to make it an unhomogeneous one, I would appreciate it very much. In the first case, \( T \) is given the discrete topology and in the second case \( T \) is given the usual Euclidean topology. That is, \( \mathscr{F}_0 \) contains all of the null events (and hence also all of the almost certain events), and therefore so does \( \mathscr{F}_t \) for all \( t \in T \). These examples and corresponding transition graphs can help developing the skills to express problem using MDP. For a real-valued stochastic process \( \bs X = \{X_t: t \in T\} \), let \( m \) and \( v \) denote the mean and variance functions, so that \[ m(t) = \E(X_t), \; v(t) = \var(X_t); \quad t \in T \] assuming of course that the these exist. 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 \( X_{m+n} = g^n(X_m) \). Suppose that \( \bs{P} = \{P_t: t \in T\} \) is a Feller semigroup of transition operators. 5 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 \( t \in T \). Why refined oil is cheaper than cold press oil? Then \[ \P\left(Y_{k+n} \in A \mid \mathscr{G}_k\right) = \P\left(X_{t_{n+k}} \in A \mid \mathscr{G}_k\right) = \P\left(X_{t_{n+k}} \in A \mid X_{t_k}\right) = \P\left(Y_{n+k} \in A \mid Y_k\right) \]. If \( \mu_0 = \E(X_0) \in \R \) and \( \mu_1 = \E(X_1) \in \R \) then \( m(t) = \mu_0 + (\mu_1 - \mu_0) t \) for \( t \in T \). A hospital has a certain number of beds. How is white allowed to castle 0-0-0 in this position? Run the simulation of standard Brownian motion and note the behavior of the process. A finite-state machine can be used as a representation of a Markov chain. {\displaystyle X_{t}} Phys. Moreover, \( g_t \to g_0 \) as \( t \downarrow 0 \). , sunny days can transition into cloudy days) and those transitions are based on probabilities. If the individual moves to State 2, the length of time spent there is State: Current situation of the agent. A Markov chain is a stochastic model that describes a sequence of possible events or transitions from one state to another of a system. A gambler If \( \mu_s \) is the distribution of \( X_s \) then \( X_{s+t} \) has distribution \( \mu_{s+t} = \mu_s P_t \). The usual solution is to add a new death state \( \delta \) to the set of states \( S \), and then to give \( S_\delta = S \cup \{\delta\} \) the \( \sigma \) algebra \( \mathscr{S}_\delta = \mathscr{S} \cup \{A \cup \{\delta\}: A \in \mathscr{S}\} \). 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. WebOne of our prime examples will be the class of birth- and-death processes. 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 \( 0 = t_0 \lt t_1 \lt t_2 \lt \cdots \). Suppose that \( \bs{X} = \{X_t: t \in T\} \) is a homogeneous Markov process with state space \( (S, \mathscr{S}) \) and transition kernels \( \bs{P} = \{P_t: t \in T\} \). There are two kinds of nodes. You have individual states (in this case, weather conditions) where each state can transition into other states (e.g. Chapter 3 of the book Reinforcement Learning An Introduction by Sutton and Barto [1] provides an excellent introduction to MDP. Let \( \mathscr{C} \) denote the collection of bounded, continuous functions \( f: S \to \R \). A typical set of assumptions is that the topology on \( S \) is LCCB: locally compact, Hausdorff, and with a countable base. So the collection of distributions \( \bs{Q} = \{Q_t: t \in T\} \) forms a semigroup, with convolution as the operator. For example, if today is sunny, then: A 50 percent chance that tomorrow will be sunny again. Suppose that for positive \( t \in T \), the distribution \( Q_t \) has probability density function \( g_t \) with respect to the reference measure \( \lambda \). Sourabh has worked as a full-time data scientist for an ISP organisation, experienced in analysing patterns and their implementation in product development. Here is an example in discrete time. In both cases, \( T \) is given the Borel \( \sigma \)-algebra \( \mathscr{T} \), the \( \sigma \)-algebra generated by the open sets. Of course, the concept depends critically on the filtration. A robot playing a computer game or performing a task are often naturally maps to an MDP. 4 It is a very useful framework to model problems that maximizes longer term return by taking sequence of actions. 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 \( A, \, B \in \mathscr{S} \). Recall that a kernel defines two operations: operating on the left with positive measures on \( (S, \mathscr{S}) \) and operating on the right with measurable, real-valued functions. In general, the conditional distribution of one random variable, conditioned on a value of another random variable defines a probability kernel. Your 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 \( T \). Markov chains are a stochastic model that represents a succession of probable events, with predictions or probabilities for the next state based purely on the prior event state, rather than the states before. As always in continuous time, the situation is more complicated and depends on the continuity of the process \( \bs{X} \) and the filtration \( \mathfrak{F} \). The goal of solving an MDP is to find an optimal policy. It is Memoryless due to this characteristic of the Markov Chain. Basically, he invented the Markov chain,hencethe naming. We need to find the optimum portion of salmons to catch to maximize the return over a long time period. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. 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 \( \mathscr{B} \). This means that for \( f \in \mathscr{C}_0 \) and \( t \in [0, \infty) \), \[ \|P_{t+s} f - P_t f \| = \sup\{\left|P_{t+s}f(x) - P_t f(x)\right|: x \in S\} \to 0 \text{ as } s \to 0 \]. The four states are defined as follows, Empty -> no salmons are available; low -> available number of salmons are below a certain threshold t1; medium -> available number of salmons are between t1and t2; high -> available number of salmons are more than t2. Markov processes, named for Andrei Markov, are among the most important of all random processes. These examples and corresponding transition graphs can help developing the Inspection, maintenance and repair: when to replace/inspect based on age, condition, etc. If we sample a Markov process at an increasing sequence of points in time, we get another Markov process in discrete time. Rewards: Number of cars passing the intersection in the next time step minus some sort of discount for the traffic blocked in the other direction. The defining condition, known appropriately enough as the the Markov property, states that the conditional distribution of \( X_{s+t} \) given \( \mathscr{F}_s \) is the same as the conditional distribution of \( X_{s+t} \) just given \( X_s \). X In our situation, we can see that a stock market movement can only take three forms. markov-process graphical-model graph-theory Share Cite Improve this question Follow edited Feb 24, 2019 at 23:42 asked Feb 24, 2019 at [ 32] proposed a method combining Monte Carlo simulations and directional sampling to analyse object reliability sensitivity. Why does a site like About.com get higher priority on search result pages? Thus every subset of \( S \) is measurable, as is every function from \( S \) to another measurable space. It's more complicated than that, of course, but it makes sense. State-space refers to all conceivable combinations of these states. Various spaces of real-valued functions on \( S \) play an important role. With this article, we could understand a bunch of real-life use cases from different fields of life. If \( \bs{X} \) is a strong Markov process relative to \( \mathfrak{G} \) then \( \bs{X} \) is a strong Markov process relative to \( \mathfrak{F} \). But this forces \( X_0 = 0 \) with probability 1, and as usual with Markov processes, it's best to keep the initial distribution unspecified. In this article, we will be discussing a few real-life applications of the Markov chain. In particular, every discrete-time Markov chain is a Feller Markov process. But we can do more. Suppose also that \( \tau \) is a random variable taking values in \( T \), independent of \( \bs{X} \). The same is true in continuous time, given the continuity assumptions that we have on the process \( \bs X \). By the independence property, \( X_s - X_0 \) and \( X_{s+t} - X_s \) are independent. X This Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. If we know the present state \( X_s \), then any additional knowledge of events in the past is irrelevant in terms of predicting the future state \( X_{s + t} \). At each time step we need to decide whether to change the traffic light or not. Then \( \{p_t: t \in [0, \infty)\} \) is the collection of transition densities for a Feller semigroup on \( \N \). So if \( \bs{X} \) is a strong Markov process, then \( \bs{X} \) satisfies the strong Markov property relative to its natural filtration. Markov chains can model the probabilities of claims for insurance, such Oracle claimed that the company started integrating AI within its SCM system before Microsoft, IBM, and SAP. That is, \( P_s P_t = P_t P_s = P_{s+t} \) for \( s, \, t \in T \). Recall again that \( P_s(x, \cdot) \) is the conditional distribution of \( X_s \) given \( X_0 = x \) for \( x \in S \). And the word love is always followed by the word cycling.. Read what the wiki says about Markov chains, Why Enterprises Are Super Hungry for Sustainable Cloud Computing, Oracle Thinks its Ahead of Microsoft, SAP, and IBM in AI SCM, Why LinkedIns Feed Algorithm Needs a Revamp, Council Post: Exploring the Pros and Cons of Generative AI in Speech, Video, 3D and Beyond, Enterprises Die for Domain Expertise Over New Technologies. Substituting \( t = 1 \) we have \( a = \mu_1 - \mu_0 \) and \( b^2 = \sigma_1^2 - \sigma_0^2 \), so the results follow. The compact sets are simply the finite sets, and the reference measure is \( \# \), counting measure. If \( Q \) has probability density function \( g \) with respect to the reference measure \( \lambda \), then the one-step transition density is \[ p(x, y) = g(y - x), \quad x, \, y \in S \]. Combining two results above, if \( X_0 \) has distribution \( \mu_0 \) and \( f: S \to \R \) is measurable, then (again assuming that the expected value exists), \( \mu_0 P_t f = \E[f(X_t)] \) for \( t \in T \). Let us know in a comment down below! Assuming a sequence of independent and identically distributed input signals (for example, symbols from a binary alphabet chosen by coin tosses), if the machine is in state y at time n, then the probability that it moves to state x at time n+1 depends only on the current state. This process is modeled by an absorbing Markov chain with transition matrix = [/ / / / / /]. 2 Water resources: keep the correct water level at reservoirs. 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. As a result, there is a 67 % probability that like will prevail after I, and a 33 % (1/3) probability that love will succeed after I. Similarly, there is a 50% probability that Physics and books would succeed like. Note that if \( S \) is discrete, (a) is automatically satisfied and if \( T \) is discrete, (b) is automatically satisfied. Fair markets believe that market information is dispersed evenly among its participants and that prices vary randomly. , then the sequence WebIntroduction to MDPs. 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. The notion of a Markov chain is an "under the hood" concept, meaning you don't really need to know what they are in order to benefit from them. We also assume that we have a collection \(\mathfrak{F} = \{\mathscr{F}_t: t \in T\}\) of \( \sigma \)-algebras with the properties that \( X_t \) is measurable with respect to \( \mathscr{F}_t \) for \( t \in T \), and the \( \mathscr{F}_s \subseteq \mathscr{F}_t \subseteq \mathscr{F} \) for \( s, \, t \in T \) with \( s \le t \). If \( \bs{X} = \{X_t: t \in T\} \) is a stochastic process on the sample space \( (\Omega, \mathscr{F}) \), and if \( \tau \) is a random time, then naturally we want to consider the state \( X_\tau \) at the random time. 1 But we already know that if \( U, \, V \) are independent variables having Poisson distributions with parameters \( s, \, t \in [0, \infty) \), respectively, then \( U + V \) has the Poisson distribution with parameter \( s + t \). (2 ), where the focus is on the number of individuals in a given state at time t (rather than the transitions The probability distribution of taking actions At from a state St is called policy (At | St). Conditioning on \( X_s \) gives \[ \P(X_{s+t} \in A) = \E[\P(X_{s+t} \in A \mid X_s)] = \int_S \mu_s(dx) \P(X_{s+t} \in A \mid X_s = x) = \int_S \mu_s(dx) P_t(x, A) = \mu_s P_t(A) \]. It can't know for sure what you meant to type next, but it's correct more often than not. The higher the "fixed probability" of arriving at a certain webpage, the higher its PageRank. In an MDP, an agent interacts with an environment by taking actions and seek to maximize the rewards the agent gets from the environment. Do you know of any other cool uses for Markov chains? After examining several years of data, it wasfound that 30% of the people who regularly ride on buses in a given year do not regularly ride the bus in thenext year. For a general state space, the theory is more complicated and technical, as noted above. Can it find patterns amoung infinite amounts of data? 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). As before \(\mathscr{F}_n = \sigma\{X_0, \ldots, X_n\} = \sigma\{U_0, \ldots, U_n\} \) for \( n \in \N \). = But \( P_s \) has density \( p_s \), \( P_t \) has density \( p_t \), and \( P_{s+t} \) has density \( p_{s+t} \). What can this algorithm do for me. A true prediction -- the kind performed by expert meteorologists -- would involve hundreds, or even thousands, of different variables that are constantly changing. MDP allows formalization of sequential decision making where actions from a state not just influences the immediate reward but also the subsequent state. For an overview of Markov chains in general state space, see Markov chains on a measurable state space. A Markov process is a random process indexed by time, and with the property that the future is independent of the past, given the present. 1 A Markovian Decision Process indeed has to do with going from one state to another and is mainly used for planning and decision making. followed by a day of type j. The time space \( (T, \mathscr{T}) \) has a natural measure; counting measure \( \# \) in the discrete case, and Lebesgue in the continuous case. That is, \( g_s * g_t = g_{s+t} \). A Medium publication sharing concepts, ideas and codes. It provides a way to model the dependencies of current information (e.g. If an action takes to empty state then the reward is very low -$200K as it require re-breeding new salmons which takes time and money. Labeling the state space {1=bull, 2=bear, 3=stagnant} the transition matrix for this example is, The distribution over states can be written as a stochastic row vector x with the relation x(n+1)=x(n)P. So if at time n the system is in state x(n), then three time periods later, at time n+3 the distribution is, In particular, if at time n the system is in state 2(bear), then at time n+3 the distribution is. It is composed of states, transition scheme between states, and emission of outputs (discrete or continuous). If \( s, \, t \in T \) then \( p_s p_t = p_{s+t} \). The measurability of \( x \mapsto \P(X_t \in A \mid X_0 = x) \) for \( A \in \mathscr{S} \) is built into the definition of conditional probability. 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 more incoming links, the more valuable it is. { This essentially deterministic process can be extended to a very important class of Markov processes by the addition of a stochastic term related to Brownian motion. Rewards: Fishing at certain state generates rewards, lets assume the rewards of fishing at state low, medium and high are $5K, $50K and $100k respectively. Suppose that \( \tau \) is a finite stopping time for \( \mathfrak{F} \) and that \( t \in T \) and \( f \in \mathscr{B} \). Zhang et al. denotes the number of kernels which have popped up to time t, the problem can be defined as finding the number of kernels that will pop in some later time. N Have you ever wondered how those name generators worked? For \( t \in T \), let \( m_0(t) = \E(X_t - X_0) = m(t) - \mu_0 \) and \( v_0(t) = \var(X_t - X_0) = v(t) - \sigma_0^2\). X Clearly \( \bs{X} \) is uniquely determined by the initial state, and in fact \( X_n = g^n(X_0) \) for \( n \in \N \) where \( g^n \) is the \( n \)-fold composition power of \( g \). The probability here is a the probability of giving correct answer in that level. The time set \( T \) is either \( \N \) (discrete time) or \( [0, \infty) \) (continuous time). As noted in the introduction, Markov processes can be viewed as stochastic counterparts of deterministic recurrence relations (discrete time) and differential equations (continuous time). It has at least one absorbing state. One of the interesting implications of Markov chain theory is that as the length of the chain increases (i.e. The matrix P represents the weather model in which a sunny day is 90% likely to be followed by another sunny day, and a rainy day is 50% likely to be followed by another rainy day. Since, MDP is about making future decisions by taking action at present, yes! That is, if we let \( P = P_1 \) then \( P_n = P^n \) for \( n \in \N \). 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 \( \{\mathscr{F}_{n+1}: n \in \N\} \). Indeed, the PageRank algorithm is a modified (read: more advanced) form of the Markov chain algorithm. (P)i j is the probability that, if a given day is of type i, it will be And this is the basis of how Google ranks webpages. 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. Suppose first that \( \bs{U} = (U_0, U_1, \ldots) \) is a sequence of independent, real-valued random variables, and define \( X_n = \sum_{i=0}^n U_i \) for \( n \in \N \). For \( t \in (0, \infty) \), let \( g_t \) denote the probability density function of the normal distribution with mean 0 and variance \( t \), and let \( p_t(x, y) = g_t(y - x) \) for \( x, \, y \in \R \). State Transitions: Fishing in a state has higher a probability to move to a state with lower number of salmons. Similarly, not_to_fish action has higher probability to move to a state with higher number of salmons (excepts for the state high). Again, in discrete time, if \( P f = f \) then \( P^n f = f \) for all \( n \in \N \), so \( f \) is harmonic for \( \bs{X} \). Recall that one basic way to describe a stochastic process is to give its finite dimensional distributions, that is, the distribution of \( \left(X_{t_1}, X_{t_2}, \ldots, X_{t_n}\right) \) for every \( n \in \N_+ \) and every \( (t_1, t_2, \ldots, t_n) \in T^n \). However, this will generally not be the case unless \( \bs{X} \) is progressively measurable relative to \( \mathfrak{F} \), which means that \( \bs{X}: \Omega \times T_t \to S \) is measurable with respect to \( \mathscr{F}_t \otimes \mathscr{T}_t \) and \( \mathscr{S} \) where \( T_t = \{s \in T: s \le t\} \) and \( \mathscr{T}_t \) the corresponding Borel \( \sigma \)-algebra. Because the user can teleport to any web page, each page has a chance of being picked by the nth page. So here's a crash course -- everything you need to know about Markov chains condensed down into a single, digestible article. Because it turns out that users tend to arrive there as they surf the web. A 20 percent chance Then \( \bs{X} \) is a homogeneous Markov process with one-step transition operator \( P \) given by \( P f = f \circ g \) for a measurable function \( f: S \to \R \). Ser. You may have agonized over the naming of your characters (at least at one point or another) -- and when you just couldn't seem to think of a name you like, you probably resorted to an online name generator. Then \( \bs{Y} = \{Y_t: t \in T\} \) is a homogeneous Markov process with state space \( (S \times T, \mathscr{S} \otimes \mathscr{T}) \). When \( T = [0, \infty) \) or when the state space is a general space, continuity assumptions usually need to be imposed in order to rule out various types of weird behavior that would otherwise complicate the theory. This is not as big of a loss of generality as you might think. All of the unique words from the preceding statements, namely I, like, love, Physics, Cycling, and Books, might construct the various states. This is why keyboard apps ask if they can collect data on your typing habits. The proofs are simple using the independent and stationary increments properties. This follows from induction and repeated use of the Markov property. Then \(\{p_t: t \in [0, \infty)\} \) is the collection of transition densities of a Feller semigroup on \( \R \). That is, \( g_s * g_t = g_{s+t} \). But many other real world problems can be solved through this framework too. If we sample a homogeneous Markov process at multiples of a fixed, positive time, we get a homogenous Markov process in discrete time. Condition (b) actually implies a stronger form of continuity in time. 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 \( x \in S \).

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