Questions tagged [integral]

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27 views

How to derive Parameter Derivative within an FFT integral

I have the following function (Carr-Madan) of which I am trying to take the derivative wrt $\theta$: $c(k)=\int_0^\infty \frac{e^{-iuk}}{\alpha^2 + \alpha - u^2 + i(2\alpha+1)u} e^{\phi_T(u-(\alpha+1)...
6
votes
1answer
669 views

Is there a closed-form solution for the following integral?

The integral under consideration is as follows: $$ F=\int_{a}^{1} \exp\Big\{c\Phi^{-1}(x+b) + d\Big\}\; \mathrm dx, $$ where $0<a, b<1$, and $c>0, d\in\mathbb{R}$ are constants, and the ...
5
votes
2answers
142 views

Expectation of integral where one of limits of integration is a random variable

Is it correct to write \begin{equation} E_t \int_0^{X_T} f(z) dz = \int_0^\infty \left(\int_0^x f(z) dz \right) p(x)dx \,\,? \end{equation} Here $X_T$ is a positive random variable with density $p(x)...
1
vote
1answer
36 views

Regression of stochastic integral on Wiener process

This question is a follow-up from the following: conditional expectation of stochastic integral so I won't repeat myself regarding assumptions and notation. Using Brownian bridge approach, we know ...
1
vote
2answers
140 views

Show that $\int_0^T(T-t)dW_t=\int_0^TW_t \ dt$

I want to show that $$\int_0^T(T-t)dW_t=\int_0^TW_t \ dt \tag1$$ By Ito's lemma we can write $$d((T-t)W_t)=(T-t)dW_t-W_t \ dt.\tag{2}$$ Now If I integrate this expression and use that $W_0=0$ I should ...
6
votes
1answer
88 views

Esscher Premium: Integral Transform Proof

I have some difficulty understanding the following proof and I hope someone can help me with that. Claim: I want to show that $E_\alpha(S)=\frac{d}{dr} \log M_S(r)|_{r=\alpha} $, where $M_S(r)=E(\exp(...
4
votes
1answer
331 views

Ornstein–Uhlenbeck process – integration by parts

While deriving the solution for the stochastic differential equation that models the Ornstein–Uhlenbeck process, Paul Wilmott (Paul Wilmott on Quantitative Finance, chapter 4, page 87) performs the ...
9
votes
2answers
345 views

conditional expectation of stochastic integral

let $M_t$ be the following stochastic integral $$ M_t = \int_0^t \sigma_s dW_s $$ where $\sigma_t$ is a sufficiently regular deterministic function and $W_t$ is a standard Wiener process (that is $...