mathman said:There are discrete processes with the Markov property. Brownian motion is continuous.
woundedtiger4 said:Hi all,
I know that Brownian process can be shown as Markov process but is the converse possible? I mean can we show that a markov process is a brownian process?
Thanks in advance.
mathman said:Brownian motion is continuous.
Borek said:I believe in the scale large enough Brownian motions are indistinguishable from (continuous) diffusion, but on microscopic scale it is just a random walk.
Edit: but then perhaps I am thinking about a real process, and you are thinking about a mathematical model.
You misunderstood what mathman wrote. Brownian motion is a simple example of a Markov process. He picked one example of a Markov process that is not a Wiener process.woundedtiger4 said:Sorry, I didn't understand.
The book I am reading shows Brownian as Markov.
lavinia said:Not sure what you mean by a Brownian process but if you mean a Weiner process then there are many Markov processes that are not Weiner processes. For instance,in finance, geometric Brownian motions are commonly use to model securities prices.
lavinia said:In general, Brownian motion in mathematics is not necessarily continuous. Sample paths are only continuous almost surely. There is a version of it where the paths are continuous.
As far as real processes are concerned, you do not know whether they are continuous or not since you never have anything except discrete samples of them. For instance, even though stock prices are recorded in discrete jumps, the underlying process may be continuous or may be a continuous time process.
woundedtiger4 said:what are real processes?
chiro said:I think Borek is talking a process that actually exists in nature as opposed to one that is a mathematical curiosity or representation on paper (that either doesn't exist or at hasn't yet been observed).
woundedtiger4 said:I believe that in your reply you said that it is not necessary for a Markov process to be a Brownian (with Gaussian distribution & 0 mean) & then you have given the example of geometric Brownian motions, am I correct?
lavinia said:Correct. More generally there are Markov processes that are called Ito diffusions that are not Brownian motions. Infinitesimally they look like a Brownian motion multiplied by a function plus a drift term.
A Markov process is a stochastic process that models the evolution of a system where the future state depends only on the current state and not on any previous states. It is a memoryless process and is widely used in various fields, including physics, biology, finance, and engineering.
A Brownian process is a continuous-time stochastic process that models the random motion of particles in a fluid or gas due to collisions with other particles. It is characterized by its properties of being continuous, non-differentiable, and having independent increments.
Markov process and Brownian process are related in that both are stochastic processes used to model the behavior of a system over time. However, while all Brownian processes are Markov processes, not all Markov processes are Brownian processes. Brownian process is a specific type of Markov process that satisfies additional properties, such as being continuous and having independent increments.
The key differences between Markov process and Brownian process are that Markov process is memoryless and its future state depends only on the current state, while Brownian process has a continuous and non-differentiable path and its future state also depends on its previous states.
Markov process and Brownian process are widely used in various fields of scientific research, such as physics, biology, finance, and engineering. Markov process is used to model the behavior of complex systems, such as the spread of diseases or the movement of particles in a gas. Brownian process is used to model random phenomena, such as the stock market or the diffusion of molecules in a liquid. Both processes are valuable tools for understanding and predicting the behavior of complex systems in science and engineering.