0
votes
0answers
41 views

Finding the inverse of Laplace–Stieltjes transformation and Convolution related to Probability

I would like to ask you something I do not understand from my book. If I have G following exponential distribution with $G(t)=1-e^{-\lambda t},t\geq 0$ then ...
2
votes
0answers
59 views

Criteria for $L^1$ convergence looking at Laplace transforms

Let $(X_n)_{n \geq 0}$ be a sequence of integrable ($\mathbb{E} |X_n| < \infty$) random variables and denote by $l_n(t)$ the Laplace transforms of $X_n$. Similarly, let $X$ be a r.v. and $l(t)$ ...
5
votes
1answer
108 views

A Laplace transform question

Suppose I have a positive integrable random variable $X$ s.t. $$E[e^X]=+\infty$$ Now let's take a series with general term $p_n$, summing to one, and define $$Z=\sum_{n>0}p_ne^{X_n}$$ and $U=\ln Z$ ...
5
votes
2answers
325 views

Laplace transform of integrated geometric Brownian motion

Is there any closed form of the Laplace transform of an integrated geometric Brownian motion ? A geometric Brownian motion $X=(X_t)_{t \geq 0}$ satisifies $dX_t = \sigma X_t \, dW_t$ where ...
2
votes
1answer
117 views

Laplace Transform of a Truncated Random Variable

Let $B$ be a random variable, and $S=\mbox{min}(1,B)$. Can you help me see why the laplace stieltjes transform of $S$ is given by $$ E[e^{-\alpha S}]=1-\alpha\int_{0}^{1}e^{-\alpha y}P(B\geq y)dy$$
3
votes
1answer
92 views

Approximating the logarithm of a Laplace transform

Suppose $X$ is a random variable on $\mathbb R_+$ with finite mean, i.e. $\mathbb E X <+\infty$. Let $F_X(t)$ be its c.d.f. and $\mathcal{L}_X(\cdot)$ its Laplace transform, i.e. ...
0
votes
1answer
86 views

Computing the Restricted Laplace Transform of a Random Variable

Is there any way to calculate the restricted Laplace transform of the random variable $X$, i.e., $$ \int_{0}^{u}e^{-sx}dF(x)\ $$ $(u<\infty)$, based on its Laplace transform?
2
votes
1answer
124 views

Find the probability of certain measurement for a Laplace Operator on a state function

Let $H$ be the operator $ -\frac{d^{2}}{dx^{2}} $ and let its domain be $$\{f\in L^{2}(\mathbb{R},d\lambda)\text{ }:\int_{-\infty}^{\infty}|xF[f(x)]|^{2}dx<\infty\} $$ where $F$ is the Fourier ...