Let (X_{t}, t geq 0), be defined as [X_{t}=left{B_{t} mid B_{t} geq 0 ight}, quad forall t>0]

Question:

Let $X_{t}, t \geq 0$, be defined as

\[X_{t}=\left\{B_{t} \mid B_{t} \geq 0\right\}, \quad \forall t>0\]

that is, the process has the paths of the Brownian motion conditioned by the current value being positive.

(a) Show that the pdf of $X_{t}$ is

\[f_{X_{t}}(x)=2 f_{B_{t}}(x), \forall x \geq 0\]

(b) Calculate $\mathbf{E}\left[X_{t}\right]$ and $V\left(X_{t}\right)$.

(c) Is $X_{t}$ a Gaussian process?

(d) Is $X_{t}$ stationary?

(e) Are $X_{t}$ and $\left|B_{t}\right|$ identically distributed?

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Quantitative Finance

ISBN: 9781118629956

1st Edition

Authors: Maria Cristina Mariani, Ionut Florescu

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Question Posted: Apr 20, 2024 05:57 AM

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Let (X_{t}, t geq 0), be defined as [X_{t}=left{B_{t} mid B_{t} geq 0 ight}, quad forall t>0]

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a The process Xt is the Brownian motion Bt conditioned to be posit...View the full answer

Quantitative Finance

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