Conditioning on \( X_s \) gives \[ \P(X_{s+t} \in A) = \E[\P(X_{s+t} \in A \mid X_s)] = \int_S \mu_s(dx) \P(X_{s+t} \in A \mid X_s = x) = \int_S \mu_s(dx) P_t(x, A) = \mu_s P_t(A) \]. The next state of the board depends on the current state, and the next roll of the dice. Then \( \bs{Y} = \{Y_n: n \in \N\} \) is a homogeneous Markov process with state space \( (S \times S, \mathscr{S} \otimes \mathscr{S} \). not on a list of previous states). 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. 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. In continuous time, or with general state spaces, Markov processes can be very strange without additional continuity assumptions. However, they do not always choose the pages in the same order. The idea is that at time \( n \), the walker moves a (directed) distance \( U_n \) on the real line, and these steps are independent and identically distributed. To account for such a scenario, Page and Brin devised the damping factor, which quantifies the likelihood that the surfer abandons the current page and teleports to a new one. {\displaystyle X_{n}} N Of course, the concept depends critically on the filtration. Intuitively, we can tell whether or not \( \tau \le t \) from the information available to us at time \( t \). In a quiz game show there are 10 levels, at each level one question is asked and if answered correctly a certain monetary reward based on the current level is given. but converges to a strictly positive vector only if P is a regular transition matrix (that is, there Conditioning on \( X_s \) gives \[ P_{s+t}(x, A) = \P(X_{s+t} \in A \mid X_0 = x) = \int_S P_s(x, dy) \P(X_{s+t} \in A \mid X_s = y, X_0 = x) \] But by the Markov and time-homogeneous properties, \[ \P(X_{s+t} \in A \mid X_s = y, X_0 = x) = \P(X_t \in A \mid X_0 = y) = P_t(y, A) \] Substituting we have \[ P_{s+t}(x, A) = \int_S P_s(x, dy) P_t(y, A) = (P_s P_t)(x, A) \]. Since time (past, present, future) plays such a fundamental role in Markov processes, it should come as no surprise that random times are important. 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. 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. When the state space is discrete, Markov processes are known as Markov chains. Recall again that \( P_s(x, \cdot) \) is the conditional distribution of \( X_s \) given \( X_0 = x \) for \( x \in S \). (This is always true in discrete time.). Discrete-time Markov process (or discrete-time continuous-state Markov process) 4. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Absorbing Markov Chains If \( \bs{X} = \{X_t: t \in T\} \) is a stochastic process adapted to \( \mathfrak{F} \) and if \( \tau \) is a stopping time relative to \( \mathfrak{F} \), then we would hope that \( X_\tau \) is measurable with respect to \( \mathscr{F}_\tau \) just as \( X_t \) is measurable with respect to \( \mathscr{F}_t \) for deterministic \( t \in T \). 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. denote the mean and variance functions for the centered process \( \{X_t - X_0: t \in T\} \).

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