0
votes
0answers
25 views

Differential of stochastic process

How do I find the dynamics of $X_t=\int_0 ^t \sigma (s,t) dW_s$? It seems that the simple solution of $dX_t = \sigma(t,t)dW_t$ is not correct since I get $X_t = \int _0 ^t \sigma(s,s) dW_s$ if I ...
0
votes
0answers
43 views

Differentiating Exponential matrix Expression

To give the scalar version first: For the well known Ornstein-Uhlenbeck process: $dr(t)=\alpha(b-r(t))dt+\sigma dW(t)$ It is well known that the variance is: $\sigma_r^2=\sigma^2 \int_u^t\exp^{-2 ...
2
votes
0answers
59 views

Defining the scale function of a diffusion process

My question has to do with correctly calculating the scale function of a diffusion process, but ultimately might only have to do with calculus. I'll briefly set-up my calculations, so you can quickly ...
1
vote
1answer
34 views

Addition corresponds to convolution and subtraction?

We know that if two random variables have proper densities, than the density of the sum of them is given by the convolution. But what can we say about the difference of two random variables? $X-Y$ ...
0
votes
1answer
93 views

Simple integral with stochastic Brownian motion integrand

Consider $$\int_0^t \sin(B_s) ds$$ where $B_s$ is standard Brownian motion, I was wondering can I write $$\int_0^t \sin(B_s) ds = - ( \cos(B_t) - \cos(B_0)) = - \cos(B_t) ? $$ by using the ...
0
votes
1answer
49 views

Independence random variables

I found two theorems in my notes and they seem to be somewhat complementary which made me doubt that both of them are true: a) Let $X,Y: \Omega \rightarrow \mathbb{R}$ be a measurable function and ...
0
votes
1answer
88 views

Distribution of ceiling function and absolute value of random variable

Given a distribution function $f_X$, where $X$ is some random variable. I want to get the distribution functions of $|X|$ and $\lceil X \rceil$( the last one may only have an easy form if $X$ is ...
1
vote
1answer
35 views

Unbiased estimate $\lambda^2$

Given a Poisson distribution I want to figure out whether $d:(x_1,...,x_n) \mapsto x_1^2$ and $d':(x_1,...,x_n) \mapsto x_1x_2$ are unbiased estimations for $\lambda^2$ ? I mean it would sound ...
0
votes
1answer
98 views

Proof that a median minimizes 1-norm. [duplicate]

I was wondering whether there is an easy way to show the following: We have a data set $x_1,...,x_n$ and $m$ is a median if for at least half of the n data points we have that $x_i \le m$ and for ...
0
votes
0answers
41 views

How to prove d($\int_t^\infty$$e^{-ru}d\beta_u$)=-$e^{-rt}d\beta_t$?

I found it difficult to state clearly that: d($\int_t^\infty$$e^{-ru}d\beta_u$)=-$e^{-rt}d\beta_t$ , but intuitively it is correct, isn't it? I guess the Gaussian property of the integral may be used ...
0
votes
1answer
25 views

Meaning of my calculation card game

I have made a calculation and now I do not understand what I did there. It is about the following question: Imagine you have n cards of which there are 2 aces, what is the expectation value to get ...
1
vote
0answers
89 views

Expectation of $n$-dimensional Inverse Bessel Process

I think the main problem for me is to calculate the integral of $$\int_{0}^{\infty}\frac{e^{-\frac{r^2}{2t}}}{\sqrt{x^2+r^2}}r^{n-1}dr,n\geq2$$ For n=2, change of variable $y=\sqrt{x^2+r^2}$ would ...
1
vote
2answers
177 views

Average distance to perimeter of a polygon?

Trying to calculate heat transfer which is a function of distance of each molecule to the closest wall for various container shapes. For example, a rectangular prism versus a cylinder. So I think ...
0
votes
1answer
146 views

Second derivative of Brownian motion?

My question is, we give a meaning to the following expression: $$dX(t) = \mu(t,X(t))dt + \sigma(t,X(t))dW(t), \ \ X(0)=x.$$ where $W$ is a Wiener process. This equation can be thought as ...
2
votes
0answers
68 views

How to calculate the following expectation

I have a problem to find the expectation of the following expression, $$E\left[W_T e^{\int_0^T(W_s)ds}\right].$$ Here, $W_T$ is a Brownian motion. Any suggestions as to how to proceed with it? Many ...