Conditional Brownian motion

In summary, Conditional Brownian motion is a type of stochastic process that takes into account the previous positions of particles to model their random movement in a fluid medium. It differs from regular Brownian motion in that it is not completely random and has many applications in scientific research, such as modeling diffusion, chemical reactions, and other types of random motion. It is mathematically described using a stochastic differential equation and can be used to model various systems, including the movement of molecules, the spread of diseases, and stock prices.
  • #1
IniquiTrance
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I computed the distribution of [itex]B_s[/itex] given [itex]B_t[/itex], where [itex]0\leq s <t[/itex] and [itex]\left\{B_t\right\}_{t\geq 0}[/itex] is a standard brownian motion. It's normal obviously..

My question is, how do I phrase what I've done exactly? Is it that I computed the distribution of [itex]B_s[/itex] over [itex]\sigma(B_t)[/itex]?
 
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  • #2
Hey IniquiTrance.

If you partition the distributions so that they don't overlap then you can use the properties of a Wiener (or Brownian motion) process and that should be enough in terms of the justification used.
 

1. What is Conditional Brownian motion?

Conditional Brownian motion is a type of stochastic process that models the random movement of particles or molecules in a fluid medium. It is a continuous-time process where the position of the particle at any given time is dependent on its previous positions.

2. How is Conditional Brownian motion different from regular Brownian motion?

Regular Brownian motion is completely random and does not depend on any previous positions. In contrast, Conditional Brownian motion takes into account the past positions of the particle and adjusts its future movement accordingly.

3. What are the applications of Conditional Brownian motion in scientific research?

Conditional Brownian motion has many applications in various fields of science, including physics, chemistry, biology, and finance. It is used to model diffusion, chemical reactions, and other types of random motion in systems.

4. How is Conditional Brownian motion mathematically described?

Mathematically, Conditional Brownian motion is described using a stochastic differential equation, where the position of the particle at a given time is determined by the previous positions and a random component. It is often represented using the Wiener process, a type of continuous-time random walk.

5. What are some real-life examples of systems that can be modeled using Conditional Brownian motion?

Examples of systems that can be modeled using Conditional Brownian motion include the movement of molecules in a liquid, the spread of diseases in a population, and the stock prices in financial markets. It is also used in physics to study the motion of particles in a fluid or gas medium.

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