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chiro said:Ito's lemma could be considered as more or less a 'substitution' where the brownian motion is modeled using a standard convention (i.e. B(t+h) - B(t) ~ N(B(t),h) if i recall correctly). If this is the case with your Weiner process then you can use the substitution.
omega_squared said:Hi guys,
It's been a while since high school, and now I'm faced with a problem I need to solve in a few days (attached). Would someone please help me through that? I would really appreciate support.
kai_sikorski said:Not really sure what you mean by this, but Ito's lemma is the stochastic calculus counterpart of the chain rule. Also B(t+h) - B(t) ~ N(B(t),h) doesn't really make sense. Think you meant N(0,h). There is no question about how the Weiner process is modeled though. Weiner process and Brownian motion are the same exact thing and increments over non overlapping time intervals being independent gaussian variables with variance given by the interval length is part of the definition.
chiro said:You have to remember that Ito's lemma is specific for Weiner processes and not for general distributions: you can't just use things like that for general distributions.
kai_sikorski said:Well Ito's lemma is often written out in it's simplest form which only applies to the Weiner process, but there is a more general formula for any X that is a semimartigale. The formula will include a quadratic covariation term [X,X]. For the Weiner process d[W,W]=dt.
kai_sikorski said:In your class has the distribution for the P already been derived, and now its a question of using this to get the distribution of Pρ? If not I must confess that I don't see a way to derive the distribution for Pρ without solving the SDE or the Fokker-Planck equation for P, and that doesn't seem consistent with the question saying it can be done in a straightforward manner.
SDE stands for stochastic differential equation, which is a type of equation used to model the evolution of a random process over time. Geometric brownian motion is a specific type of stochastic process that follows a log-normal distribution, and it is commonly used to model stock prices. SDEs are often used to describe the dynamics of geometric brownian motion.
Geometric brownian motion is commonly used to model stock prices because it has several desirable properties. It exhibits continuous paths, is Markovian (meaning that future movements do not depend on past movements), and allows for easy calculation of probabilities and expected values. These properties make it a useful tool for analyzing financial data and making predictions about future stock prices.
The solution to a geometric brownian motion SDE is a random variable that follows a log-normal distribution. To solve for this solution, you can use the Itô calculus, which is a mathematical tool used to solve SDEs. Alternatively, you can use the Euler-Maruyama method, which is a numerical method for approximating the solution of an SDE.
Volatility is a measure of the amount of uncertainty or risk associated with a particular stock or financial asset. In geometric brownian motion, volatility is a key parameter that determines the rate of change of the stock price. Higher volatility means that the stock price is more likely to experience large movements, while lower volatility means that the stock price is more likely to remain stable.
Geometric brownian motion is a key component in the Black-Scholes model, which is a mathematical formula used to price options. In this model, the stock price is assumed to follow geometric brownian motion, and this allows for the calculation of the fair value of an option based on the stock price, strike price, time to expiration, and other factors. Geometric brownian motion is also used in other option pricing models, such as the Cox-Ross-Rubinstein model.