Brownian Motion-III
TLDRIn this lecture on Brownian motion, the focus is on the quadratic variation of the process. The lecturer explains the concept of quadratic variation and its significance, using mathematical analysis to explore the properties of Brownian motion, particularly its non-differentiable nature. Through partitioning intervals and applying the mean value theorem, the lecturer demonstrates how to compute the first and second variations of a function. Key mathematical tools like Chebyshev’s lemma and Borel-Cantelli lemma are employed to prove that the quadratic variation of Brownian motion equals time T almost surely, highlighting its unique characteristics compared to smooth functions.
Takeaways
- 📈 Today we are discussing the quadratic variation of Brownian motion.
- 🧑🔬 W stands for Wiener, after Norbert Wiener, who formalized the study of Brownian motion.
- 📝 To analyze Brownian motion, we break the interval [0, T] into n partitions and take the limit as n approaches infinity.
- 📊 The first variation of a continuously differentiable function over [0, T] is 0.
- 📉 For Brownian motion, the quadratic variation accumulates, unlike smooth functions.
- 🧮 The quadratic variation of Brownian motion is T, almost surely.
- 🔍 The first variation measures the total up and down movements of the function.
- 🧩 The quadratic variation is a sum of squared differences over the partitions.
- 📚 To prove quadratic variation, we use the mean value theorem, Riemann sums, and integration.
- 🔢 Chebyshev's lemma and Borel-Cantelli lemma are crucial in proving convergence in probability and almost surely.
Q & A
What is the primary topic of today's lecture?
-Today's lecture focuses on the quadratic variation of Brownian motion.
Who is Norbert Wiener and what is his contribution to Brownian motion?
-Norbert Wiener was a mathematician who formally studied Brownian motion in detail and made numerous applications of it.
How does one break the interval for studying the quadratic variation?
-The interval 0 to T is broken into n partitions, with each partition having a step length of T/n.
What happens to the interval size as n tends to infinity?
-As n tends to infinity, the maximum size of any interval within the partition goes to 0.
What is the first variation of a continuously differentiable function?
-The first variation of a continuously differentiable function over the interval 0 to T is given by the integral from 0 to T of the absolute value of the derivative of the function.
How is the second (quadratic) variation of a function defined?
-The second variation is defined as the limit of the sum of the squares of the function's increments over the partitions, as the partition size goes to 0.
What is the key difference between first and second variation?
-The first variation counts the total up and down crossings of a function, while the second variation sums the squares of these variations, resembling the mean square error.
Why does a continuously differentiable function not accumulate any quadratic variation?
-A continuously differentiable function does not accumulate quadratic variation because the integral of the square of its derivative over any interval is finite, resulting in the limit of the quadratic variation going to 0.
How is quadratic variation different for a Brownian motion?
-Brownian motion accumulates quadratic variation, which is distinct from the behavior of continuously differentiable functions.
What is the significance of proving that the quadratic variation of Brownian motion is equal to T?
-Proving that the quadratic variation of Brownian motion is equal to T almost surely is significant as it highlights the unique nature of Brownian paths and is useful in applications like Ito calculus in finance.
What mathematical tools are used to prove the quadratic variation property of Brownian motion?
-Chebyshev's lemma and the Borel-Cantelli lemma are used to prove the quadratic variation property of Brownian motion.
What does it mean for a partitioned sample quadratic variation to be a random variable?
-Since Brownian motion is a stochastic process, the sample quadratic variation for a given partition is a random variable dependent on the specific realization (omega) of the process.
How does the expectation and variance of the quadratic variation help in proving the main theorem?
-By showing that the expected value of the quadratic variation converges to T and that its variance goes to 0 as n tends to infinity, one can use probabilistic tools to prove convergence almost surely.
What does 'almost surely' mean in the context of Brownian motion?
-'Almost surely' means that the event (here, the quadratic variation converging to T) happens with probability 1.
Why is proving convergence to T almost surely more complex than proving convergence in probability?
-Proving convergence almost surely requires more sophisticated techniques and a stronger notion of convergence compared to convergence in probability, which is generally easier to establish.
Outlines
📚 Introduction to Quadratic Variation of Brownian Motion
The script begins with an introduction to the concept of quadratic variation in the context of Brownian motion, also known as Brownian motion-III. It discusses the difficulty in defining a step length for Brownian motion, as opposed to symmetric random walks, and introduces the idea of taking a limit as the number of steps, n, approaches infinity. The concept of first and second variation of a continuously differentiable function is also briefly mentioned, with an emphasis on the continuous but non-differentiable nature of Brownian motion paths.
📐 Calculating First Variation and Its Implications
This paragraph delves into the calculation of the first variation of a function, particularly in the context of a continuously differentiable function over a time interval. It explains the partitioning of the interval and the use of the mean value theorem to simplify the calculation of the first variation. The paragraph also touches on the properties of Brownian motion, highlighting that unlike differentiable functions, Brownian motion does accumulate quadratic variation.
📉 Exploring Quadratic Variation with Mean Value Theorem
The script continues with an exploration of quadratic variation, symbolized and defined in terms of a limit as the partition size approaches zero. It uses the mean value theorem to express the quadratic variation in terms of the square of the function's derivative, leading to an integral expression. The paragraph concludes with the observation that for a continuously differentiable function, the quadratic variation is zero, contrasting with the behavior of Brownian motion.
🔍 The Unique Behavior of Brownian Motion's Quadratic Variation
This section contrasts the behavior of Brownian motion with that of smooth functions, emphasizing that Brownian motion does accumulate quadratic variation. It introduces the theorem that the quadratic variation of Brownian motion is equal to the time interval almost surely, and mentions the importance of understanding this property for applications in finance and Ito calculus.
🎲 Proof Techniques for Brownian Motion's Quadratic Variation
The script outlines the proof techniques used to establish that the quadratic variation of Brownian motion converges to the time interval almost surely. It discusses the use of expected value and variance to show that the sample quadratic variation converges to the time interval on average and has zero variance. The paragraph also hints at the application of Chebyshev's lemma and the Borel-Cantelli lemma in the proof.
📘 Convergence in Probability and Almost Surely
The final paragraph focuses on the convergence of the quadratic variation to the time interval, both in probability and almost surely. It explains the use of Chebyshev's lemma to show convergence in probability and suggests that further techniques are required to prove almost sure convergence, which will be detailed in supplementary notes rather than in the lecture.
Mindmap
Keywords
💡Brownian Motion
💡Quadratic Variation
💡Wiener Process
💡Partitioning
💡First Variation
💡Second Variation
💡Continuously Differentiable Function
💡Mean Value Theorem
💡Chebyshev's Lemma
💡Borel-Cantelli Lemma
💡Ito Calculus
Highlights
Introduction to quadratic variation of a Brownian motion and its importance in understanding stochastic processes.
Explanation of the Brownian motion as a continuous but non-differentiable function, emphasizing its zigzagging nature.
Partitioning the interval into n partitions and forming a sum to compute the quadratic variation.
Clarification that the limit of the sum as n tends to infinity is T, despite it not being apparent initially.
Discussion on first and second variation of a continuously differentiable function and their respective calculations.
Definition and calculation of first variation using mean value theorem and Riemann sum.
Introduction of quadratic variation and its computation using a similar partitioning approach.
Distinction between the behavior of continuously differentiable functions and Brownian motion regarding quadratic variation.
Proof that Brownian motion accumulates quadratic variation while a smooth function does not.
Theorem stating that the quadratic variation of Brownian motion Wt is equal to T almost surely for t ≥ 0.
Explanation that quadratic variation in Brownian motion is a random variable, leading to convergence in the almost surely sense.
Use of Chebyshev’s lemma and Borel-Cantelli lemma to prove convergence of quadratic variation to T in probability.
Detailed step-by-step proof of quadratic variation convergence using expected value and variance of Q pi n.
Discussion on the practical applications of quadratic variation in finance, especially in the context of Ito calculus.
Final remarks on the complexity of proving convergence to T almost surely and the plan to provide further details in class notes.
Transcripts
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