Question

Prove the identities $$ \begin{gathered} \int T_{t}\left(y_{2} \mid y_{1}\right) d y_{2}=1 \\ \int T_{t}\left(y_{2} \mid y_{1}\right) P_{1}\left(y_{1}\right) \mathrm{d} y_{1}=P_{1}\left(y_{2}\right) \end{gathered} $$ Conclude from it that \(T_{\text {i regarded as an operator has the eigenvalue } 1 \text {, with left }} \end{array}\) eigenvector \(\psi(y)=1\), and right cigenvector \(P_{1}\). The best known example of a stationary Markov process is the OrnsteinUhlenbeck process" defined by $$ \begin{aligned} P_{1}\left(y_{1}\right) &=\frac{1}{\sqrt{2 \pi}} e^{-t \eta} \\ T_{1}\left(y_{2} \mid y_{1}\right) &=\frac{1}{\sqrt{2 \pi\left(1-c^{-2 t}\right)}} \exp \left[-\frac{\left(y_{2}-y_{1} e^{-t}\right)^{2}}{2\left(1-e^{-2 r}\right)}\right] \end{aligned} $$

Short answer

The normalization condition and the preservation of probabilities over time are satisfied for the Ornstein-Uhlenbeck process, confirming that \(T_t\) is a Markov operator with eigenvalue 1, \(\psi(y) = 1\) as the left eigenvector, and \(P_1\) as the right eigenvector.

Step-by-step solution

Prove the normalization of the transition probability

The first integral represents the normalization condition for the transition probability, which means that the probability of transitioning from state \(y_1\) to any state \(y_2\) must sum (or integrate) to 1. This condition is a fundamental property of probability densities.

Verify the normalization property

Substitute the given \(T_t(y_2 \mid y_1)\) into the integral and compute the integral over \(y_2\). The result should be equal to 1, confirming the normalization of the transition probability.

Prove the preservation of probabilities over time

The second integral represents the conservation of total probability for the process. It states that the probability distribution over time, given by \(P_1(y_2)\), is preserved under the transition probabilities. This means that if you transition probabilities from one time to another, the overall probability distribution does not change.

Apply the transition probability and initial distribution

Insert the explicit expressions for \(T_t(y_2 \mid y_1)\) and \(P_1(y_1)\) given by the Ornstein-Uhlenbeck process into the second integral. By solving the integral, show that it is equal to \(P_1(y_2)\), which confirms the conservation of total probability over time.

Identify the eigenvectors and eigenvalue

The results suggest that \(T_t\) as an operator has the eigenvalue 1, because applying it to the eigenvector \(\psi(y) = 1\) gives the same function back (satisfying the normalization condition). Similarly, applying \(T_t\) to the right eigenvector \(P_1\) produces a function proportional to \(P_1\) itself, hence indicating that \(P_1\) is a right eigenvector.

Conclude the Markov process property

The Ornstein-Uhlenbeck process is a stationary Markov process since the transition probability and the initial distribution satisfy the required conditions for a Markov process, including the normalization and preservation of probabilities over time.

Key concepts

Understanding Stationary Markov Processes

A stationary Markov process is a stochastic process that satisfies two key properties: time homogeneity and the lack of memory. Time homogeneity means that the transition probabilities, which describe the probability of moving from one state to another, are independent of time. In other words, the odds of transitioning between states do not change as time progresses.

The memoryless property, which is a hallmark of Markov processes, dictates that the probability of transitioning to a future state depends only on the current state, not on the sequence of events that preceded it. This characteristic is known as the Markov property. For a process to be stationary, it must also have a steady-state distribution, which remains constant over time, indicating that the statistical properties of the process do not evolve.

In the case of the Ornstein-Uhlenbeck process, it demonstrates the requisite stationarity by maintaining a consistent probability distribution, as shown in the provided exercise solution. This uniformity ensures that the probabilities of transitioning between states remain constant, regardless of how far along the process has proceeded.

Transition Probability and Its Normalization

Transition probability is a fundamental concept in the study of Markov processes. It represents the likelihood of a system shifting from one state to another within a given timeframe. For any Markov process, the sum of the probabilities of moving to any possible future state from the current state has to be one. This condition ensures the probabilities are properly normalized.

Mathematically, this is expressed as follows: The integration of the transition probability over all possible future states, given a current state, equals one.

Normalization of Transition Probability

As detailed in the exercise, the first equation provided portrays this normalization aspect of transition probabilities, ensuring that the model adheres to the fundamental principles of probability. Such normalization allows for the correct assessment of future states without any loss or gain of probability mass over time, crucial for maintaining the process's stationary nature.

Eigenvectors and Eigenvalues in Markov Processes

In the context of Markov processes, eigenvectors and eigenvalues play a pivotal role in understanding the dynamics of transitions over time. Specifically, an eigenvector of a transition operator portrays a state or distribution that, upon application of the transition operator, remains in the same direction in the state space, merely being scaled by a corresponding eigenvalue.

In the Orstein-Uhlenbeck process example, the equation provided demonstrates that using the transition operator on the left eigenvector, the constant function \( \psi(y) = 1 \) results in the same function, indicating that it indeed is an eigenvector associated with the eigenvalue one. Similarly, the initial distribution function \( P_1 \) acts as the right eigenvector, again indicating that it scales by the same eigenvalue when the transition operator is applied.

The identification of these eigenvectors and eigenvalues is critical, as it affirms the transition operator's structure and stability. It confirms that certain distributions, such as the steady-state distribution, remain invariant under the Markov process's dynamics, supporting its stationary character.