Questions tagged [stochastic-integral]

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

I am not quite sure how to solve this integral to be able to do numerical calculations with it. $\lambda$ is a constant, $u$ is time, and $Z_u$ is a wiener process. Can anyone provide some direction ...
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1 vote
1 answer
266 views

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

From the post Integral of Brownian motion w.r.t. time we have an argument for $$\int_0^t W_sds \sim N\left(0,\frac{1}{3}t^3\right).$$ However, how does this generalise for the interval $[t;T]$? I.e. ...
  • 359
0 votes
0 answers
65 views

Distribution of Stochastic Integral Example

I am looking for help on justifying how the integral $$\int_{0}^{t} (t-s) \, dW_{s}$$ is normally distributed. I realize that the general fact that Ito Integrals with deterministic integrands are ...
  • 1
2 votes
1 answer
204 views

Why this stochastic integral is calculated with Riemann integral

This picture is from Neftci's textbook, 'An Introduction to the Mathematics of Financial Derivatives, Third Edition' What makes me uncomfortable is equation [10.61] In above picture. In this equation,$...
6 votes
1 answer
234 views

Parametric Stochastic Integral

I need help. Defining the parametric stochastic integral $$ F_t = \int_t^T\xi(t,s)g(s)ds $$ $\\\\$ with $\xi$ a generic stochastic process such that $d\xi(t,s) = \mu(t,s)dt + \sigma(t,s)dW_t$, I'm ...
  • 61
1 vote
1 answer
208 views

What does it mean to "compute" an Itô integral?

I'm reading Shreve's Stochastic Calculus for Finance II. On page 191, Exercise 4.6, we are given the problem Exercise 4.6. Let $S(t)=S(0)\exp\Big \{\sigma W(t)+(\alpha-\frac{1}{2}\sigma^2)t\Big\}$ be ...
  • 437
1 vote
1 answer
58 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 ...