2023年11月21日 · Brownian motion is the seemingly random motion of particles within a liquid or gas that emerges from constant collisions and redirection from impacting the atoms or molecules within the fluid. All ...

For one thing, it would have to have either a random starting point, or a jump immediately after time 0, which are typically things we don't want to have in our definition of Brownian motion. (However, you might like to look up the stationary Ornstein-Uhlenbeck process, which is a nontrivial continuous process with random starting point whose ...

2018年2月14日 · The following is the definition of a Wiener process that I am following: I am confused regarding the multivariate Brownian motion which is defined as follows: My question is, does $\mathbf{W}_t$ follow the same conditions 1-4 for a univariate Wiener process?

2020年7月26日 · I am a bit confused about how the geometric brownian motion process is commonly defined. On this reference it seems to imply that the $\mu$ and $\sigma$ are the mean and the standard deviation of the normal distribution where the logarithm of the ratios of consecutive points are drawn from:

2016年2月4日 · Definition 2: In Stochastic Differential Equations with Applications to Physics and Engineering, Modeling, Simulation, and Optimization of Integrated Circuits and Generalized Functions - Vol 4: Applications of Harmonic Analysis they take a real-valued Brownian motion $(B_t)_{t\ge 0}$ on $(\Omega,\mathcal A,\operatorname P)$ and define $$\langle …

2020年3月29日 · I am having trouble proving two definition of brownian motions are equivalent. Let $(\Omega, F, (F_t), P)$ be a filtered probability space satisying the usual conditions. Let $(X_t)$ be a continuos adapted process valued in $\mathbb{R}$. Then …

2020年11月26日 · Currently I'm learning about Brownian motion. In the lecture slides the following definition is given ...

2015年1月14日 · In 5.3.1, after the Theorem 5.3.1 (Martingale representation, one dimension), Shreve explains: "The assumption that the filtration in Theorem 5.3.1 is the one generated by the Brownian motion is more restrictive than the assumption of Girsanov's Theorem, Theorem 5.2.3, in which the filtration can be larger than the one generated by Brownian ...

2015年10月10日 · Brownian motion isn't a stationary process in this sense. What is true is that it has stationary ...

2022年2月13日 · I have two small questions about the definition of Brownian motion. First question: Kuo presents it as a measurable function defined on the product space $[0,\infty)\times \Omega=T\times \Omega$ , where $\Omega$ is a measuable space $(\Omega, \mathbf{F},P)$ , with several characteristics.