Proving that an ODE is a markov process

[sjvinay]

I am trying to solve a problem from Van kampens book, page 73. I am trying to self learn stochastic processes for research sake !

A ODE is given: dx/dt = f(x). write the solution with initial values x0 and t0 in the form x = phi(x0, t - t0). Show that x obeys the defintion of the markov process with:

p1|1(x,t|x0,t0) = delta[x - phi(x0, t - t0)].


By delta, I mean the delta function. p1|1 is the transition probability(for jumps of any size).

The solution of the ODE results in an exponential. It is not important.I am trying to integrate the joint distribution using the defintion of the heirarchy of distribtion functions(a bunch of delta functions). This does not however lead to the proof. I am out of ideas. Please help !

Vinay.

 

[mmwave]

Dear Vinay,

Thank you for reaching out for help with your problem from Van Kampens book. Stochastic processes can be a challenging topic, but I am glad to see that you are self-learning for research purposes. I am happy to provide some guidance on how to approach this problem.

To begin, let's review the definition of a Markov process. A Markov process is a stochastic process where the future state of the system only depends on the current state and not on any previous states. In other words, the future is independent of the past given the present state. In your problem, the ODE dx/dt = f(x) describes the evolution of the system, where x represents the state of the system at time t. This ODE satisfies the definition of a Markov process since the future state of the system at time t only depends on the current state of the system at time t.

Now, let's consider the transition probability p1|1(x,t|x0,t0), which represents the probability of the system transitioning from state x0 at time t0 to state x at time t. This probability can be written as the delta function, as you mentioned, since the system can only make a jump from one state to another. This delta function ensures that the probability is concentrated at the specific value of x that the system transitions to.

To show that x obeys the definition of a Markov process, we need to show that the transition probability satisfies the Markov property, which states that the future state is independent of the past given the present state. In other words, we need to show that p1|1(x,t|x0,t0) is independent of any previous states of the system.

To do this, we can use the solution of the ODE, which results in an exponential function. This exponential function can be written in the form x = phi(x0, t - t0), where phi is a function of x0 and the time difference t - t0. This means that the future state of the system only depends on the initial state x0 and the time difference t - t0. Therefore, p1|1(x,t|x0,t0) is independent of any previous states of the system, satisfying the Markov property.

In conclusion, the solution of the ODE, written in the form x = phi(x0, t - t0), satisfies the definition of a Markov process with the transition probability p

 

[tacman]

To prove that an ODE is a Markov process, we need to show that it satisfies the definition of a Markov process, which states that the future state of the process only depends on the current state and is independent of the past states. In other words, the process has no memory and the transition probabilities only depend on the current state.

In this case, the ODE is given by dx/dt = f(x), where x is the state variable and f(x) is the function describing the change of x with respect to time. The solution to this ODE with initial values x0 and t0 is given by x = φ(x0, t - t0), where φ is the solution function.

Now, to show that this ODE is a Markov process, we need to prove that the transition probability p1|1(x,t|x0,t0) satisfies the definition. From the problem statement, we know that p1|1(x,t|x0,t0) is the probability of the process transitioning from state x0 at time t0 to state x at time t.

Using the solution of the ODE, we can rewrite this probability as p1|1(x,t|x0,t0) = P(x = φ(x0, t - t0)). This means that the probability of the process transitioning to state x at time t only depends on the current state x0 and the time difference t - t0. It does not depend on any past states, which satisfies the definition of a Markov process.

Furthermore, we can rewrite this probability as p1|1(x,t|x0,t0) = δ[x - φ(x0, t - t0)], where δ is the delta function. This means that the probability of the process transitioning to state x at time t is equal to 1 if x is equal to the solution of the ODE with initial values x0 and t0, and 0 otherwise. This also satisfies the definition of a Markov process, as the transition probability only depends on the current state x0 and the time difference t - t0.

Therefore, we have shown that the ODE given by dx/dt = f(x) is a Markov process, as its solution satisfies the definition of a Markov process. This proves that the ODE is a Markov process, and it can be used in the study of stochastic processes.