Does the derivative dWt/dt exist, where Wt is a Brownian Motion?
1. Yes, it exists, and it's a fundamental concept in stochastic calculus.
2. No, it doesn't exist, and I'm curious to know why!
3. I really don't know – let's explore this together!
Cast your vote and share your thoughts in the comments below! Let's unravel the mysteries of stochastic calculus together.
Answer 2. No, it doesn't exist, and I'm curious to know why!
The derivative dWt/dt does not exist for Brownian motion Wt. This is because Brownian motion has very erratic and irregular paths, making them nowhere differentiable. The paths are so rough
that the usual derivative does not exist at any point. Instead, in stochastic calculus, we use the concept of stochastic integrals (like the Ito integral) to handle such processes.
Brownian motion serves as the basic building block for more complex models in finance, like the Geometric Brownian Motion, used in the Black-Scholes option pricing model. The
non-differentiability of Wt is a fundamental concept in understanding why standard calculus doesn't work in the stochastic world and why we need a new set of tools.
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