10. Write probability density function of $W(t)$. Using this, compute $E[\exp(i\lambda W(t))]$ for any $\lambda \in \mathbb{R}$.
Added by Michael S.
Close
Step 1
The probability density function (pdf) of a one-dimensional Brownian motion is given by the Gaussian distribution with mean 0 and variance t. Therefore, the pdf of W(t) is: f(W(t)) = (1 / sqrt(2πt)) * exp(-(W(t))^2 / (2t)) Show more…
Show all steps
Your feedback will help us improve your experience
Vipin Kumawat and 88 other Physics 103 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Let Wt denote standard Brownian motion. Calculate the following probabilities: (a) P(W2 < 0 | W1 > 0) (b) P(W1 x W2 < 0) (c) P(W1 < 0 ∩ W2 < 0)
Sri K.
Let Xt be the geometric Brownian motion. dXt = rXt dt + aXt dBt Find E[XT|Ft] for t < T by a) using the Markov property and b) writing Xt = xertMt, where Mt = exp(aBt - (a^2)t/2) is a martingale.
Adi S.
Let α be a positive constant and assume X: Ω → R is a random variable with density f(x) = α * exp(-αx), x ≥ 0, f(x) = 0, x < 0. (1) 1) Compute X's Laplace transform L(t) = E[exp(-tX)] for t ≥ 0. 2) Compute X's expectation E[X], and compute X's variance V[X]. 3) Let Y be an independent random variable of X with the same density as X, i.e., Y has the density (1). What is the distribution of the random variable Z = min{X, Y}?
Jacob F.
Recommended Textbooks
University Physics with Modern Physics
Physics: Principles with Applications
Fundamentals of Physics
Watch the video solution with this free unlock.
EMAIL
PASSWORD