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9 votes

Calculate value of Integral of Wiener process $\int_{0}^t e^{\lambda u } dZ_u$

This is just a Wiener integral (stochastic integral with respect to a Brownian motion and deterministic integrand), hence a centered Gaussian random variable with variance $$ \int_0^t{e^{2\lambda u}\...
siou0107's user avatar
  • 2,680
5 votes
Accepted

Why this stochastic integral is calculated with Riemann integral

For any semi martingale $X$ (in particular for $X_t=W_t$ or for $X_t=t$) we have $$\tag{1} \int_0^t dX_s=X_t-X_0\,. $$ You are correct that the Ito integral uses the limit procedure $$\tag{2} \int_0^...
Kurt G.'s user avatar
  • 2,033
4 votes

Integral of brownian motion wrt. time over [t;T]

The last integral is correct as $$\int_t^T W_s ds = \int_t^T (T-s) dW_s \sim N\left(0, \int_t^T(T-s)^2ds\right) = N\left(0,\frac{1}{3}(T-t)^3\right).$$ Ref. Arbitrage Theory in Continuos Time (Björk, ...
Landscape's user avatar
  • 548
2 votes
Accepted

Calculate value of Integral of Wiener process $\int_{0}^t e^{\lambda u } dZ_u$

I'll try to use Ito's Lemma to come up with a solution. Ito's lemma states that: $$F(Z_t,t)=\int_0^t\left(\frac{\partial F}{\partial u}+\frac{\partial F}{\partial Z}a+0.5\frac{\partial^2 F}{\partial Z^...
Jan Stuller's user avatar
  • 6,178
2 votes
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What does it mean to "compute" an Itô integral?

Note that SDE (4) does have a "closed-form" representation. Let $X$ be $$X := S^p, $$ so (4) is a geometric Brownian motion SDE $$dX = (p\alpha + 2^{-1}p(p-1) \sigma^2) X dt + p \sigma X dW,...
ir7's user avatar
  • 5,043
1 vote

Parametric Stochastic Integral

I am having trouble to understand your notation $$ \int_t^Td\xi(t,s)g(s)\,ds\,. $$ What is the meaning of this when you switch from the differential form $dF_t$ to the integral form $$ F_t=F_0-\int_0^...
Kurt G.'s user avatar
  • 2,033
1 vote
Accepted

Regression of stochastic integral on Wiener process

By definition, $$ {\mathbb Cov}(M_t,W_T) = {\mathbb E}[M_t W_T] - {\mathbb E}[M_t] {\mathbb E}[W_T] = {\mathbb E}[M_t W_T] $$ since ${\mathbb E}[M_t] = {\mathbb E}[W_T] = 0 $. We now consider the ...
Gabriele Pompa's user avatar

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