Brownian Motion-II
TLDRThe lecture delves into the concept of Brownian motion, a continuous stochastic process that mirrors the behavior of a symmetric random walk. It discusses the properties of independence and normal distribution of increments, highlighting the distinction between Brownian motion and stock prices, which are modeled using geometric Brownian motion to ensure non-negativity. The instructor introduces filtration associated with Brownian motion, proving its Martingale property, and explains the importance of exponential Martingales in finance. The summary also touches on the computation of joint probabilities and conditional density functions for understanding the transitions in Brownian motion.
Takeaways
- 🚶 Brownian motion is a continuous stochastic process that exhibits properties similar to a symmetric random walk but in a continuous time setting.
- 🌪 The increments of Brownian motion are independent, meaning the change in the process from one time point to another does not depend on previous values.
- 📉 Brownian motion has the property that its increments are normally distributed with a mean of 0 and a variance determined by the time interval.
- 💡 The concept of Brownian motion is often used as a model for random fluctuations seen in various phenomena, such as the movement of particles in a fluid.
- 📉 Stock prices do not strictly follow Brownian motion due to the fact that stock prices cannot be negative, unlike the values of a Brownian motion.
- 📈 Geometric Brownian motion is a modification used to model stock prices, ensuring non-negativity through the use of exponential functions.
- 🔍 Filtration in the context of Brownian motion is a collection of sigma-algebras that represent the information available up to a certain time, with Brownian motion being adapted to this filtration.
- 🎲 Brownian motion is a Martingale, a sequence of random variables for which the expected value of the next step in the sequence is equal to the current value, given the information available.
- ✂️ Exponential Martingale is an important concept in finance, particularly for calculating risk-neutral probabilities, and is derived from Brownian motion with an exponential function.
- 📚 The joint probability of a Brownian motion at given time points can be computed using the transition probability densities, which describe the likelihood of moving from one state to another over time.
- 🧩 Understanding the integrability of random variables in the context of Brownian motion is crucial for correctly applying mathematical results and ensuring the finiteness of expectations.
Q & A
What is the relationship between symmetric random walk and Brownian motion?
-Brownian motion is a continuous stochastic process that exhibits properties similar to a symmetric random walk. It is considered the continuous analog of a symmetric random walk, behaving in a zigzag pattern, like the motion of a pollen grain in water.
What is the definition of a stochastic process Wt that is considered a Brownian motion?
-A stochastic process Wt, where W stands for Brownian motion and t is time, is defined as a Brownian motion if for any given time points, the increments Wt1 - Wt0 are independent, and W0 is always 0, making it an identically zero function.
Why can't stock prices be modeled directly using Brownian motion?
-Stock prices cannot be modeled directly using Brownian motion because Brownian motion can take negative values, whereas stock prices cannot be negative. Once a stock price reaches zero, it signifies the stock is out of the market.
Outlines
🚶♂️ Introduction to Brownian Motion and Symmetric Random Walk
This paragraph introduces the concept of Brownian motion as a continuous stochastic process that resembles the zigzagging pattern of a symmetric random walk. It explains that Brownian motion can be observed in phenomena such as the erratic movement of a pollen grain in water or a molecule in a gas chamber. The paragraph emphasizes that Brownian motion is a continuous analog to the symmetric random walk, with increments that are independent and identically distributed, following a normal distribution. The increments represent the zigzagging changes in the function value over time intervals. The concept of a stochastic process Wt with W0 always being zero is also introduced, setting the foundation for further discussion on Brownian motion.
📊 Properties of Brownian Motion and Its Distinction from Stock Prices
The second paragraph delves into the properties of Brownian motion, highlighting that its increments are independent random variables with a mean of zero and variance dependent on the time interval. It mentions that these increments can be proven to follow a normal distribution, although this proof is not provided due to time constraints. The paragraph also discusses the misconception that stock prices follow Brownian motion, clarifying that while they may appear similar, stock prices cannot take negative values unlike Brownian motion. It introduces the concept of geometric Brownian motion as a model for stock prices, which ensures non-negativity by using the exponential function. Additionally, the paragraph touches on the historical work of Bachelier, who linked financial markets to partial differential equations through Brownian motion.
📚 Filtration and Martingale Properties of Brownian Motion
This paragraph introduces the concept of filtration in the context of Brownian motion, defining it as a collection of Sigma-algebras that evolve over time and must contain information about the process up to a given time. It explains the conditions that must be met for a process to be considered adapted to the filtration and the independence of increments beyond a certain time point. The paragraph then discusses the Martingale property of Brownian motion, demonstrating through mathematical manipulation that the expected value of future states given the current state is equal to the current state itself. This property is crucial for understanding the probabilistic behavior of Brownian motion and its applications in finance.
📈 Ex
Mindmap
Keywords
💡Symmetric Random Walk
💡Brownian Motion
💡Stochastic Process
💡Increments
💡Normal Distribution
💡Martingale
💡Filtration
💡Exponential Martingale
💡Volatility
💡Geometric Brownian Motion
💡Transition Probability Density
Highlights
Introduction to Brownian motion as a continuous stochastic process exhibiting properties of a symmetric random walk.
Brownian motion's analogy to the motion of a pollen grain in water, illustrating its zigzagging nature.
The definition of Brownian motion in terms of independent increments and their normal distribution.
Clarification that W0 is always 0, emphasizing the stochastic process's starting point.
The distinction between Brownian motion and stock prices, noting that stock prices cannot be negative.
Historical context of Bachelier's use of Brownian motion to model stock market pricing in 1900.
Explanation of geometric Brownian motion as a model for stock prices that ensures non-negativity.
Introduction of filtration associated with Brownian motion and its properties.
Proof that Brownian motion is a Martingale, a key concept in probability theory.
Definition and importance of the exponential Martingale in finance, particularly for risk-neutral valuation.
Technical discussion on the topic.
Transcripts
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