Questions tagged [stochastic-integral]
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11 questions
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How can one calculate third and fourth moments of a jump-diffusion process with time-varying parameters?
Suppose that $x_t$ is a random process that satisfies the mean-reversion jump-diffusion process governed by the stochastic differential equation
$$dx_t=\alpha(t)(\beta(t)-x_t)\,dt+\sigma(t)\,dW_t+J_t\,...
- 101
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Weak stationarity of continuous ARMA process from Brockwell
I am currently working on Brockwell "Levy-driven CARMA processes" (2001) and I am stuck in the introduction.
So we have a continuous AR process (CAR(p))
\begin{align*}
X_t=e^{At}X_0+\...
- 135
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Construction of Ito Integral
I am self-learning basic stochastic calculus. In my book, the author first defines the Ito integral for simple step adapted processes and then extends it to a larger class $\mathcal{L}_{c}^{2}(T)$ of ...
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Expected value and variance of the short rate under the Vasicek model
Would be grateful for any assistance.
Below are the expected value and variance of the integral of the short rate under the Vasicek model (https://www.researchgate.net/publication/41448002):
$E\left[ \...
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Path integral approach to price call option on zero coupon bonds
I am given the following identities:
$$
Z[J,t_1,t_2]=\int D W e^{\int_{t_1}^{t_2}dtJ(t)W(t)}e^{S}=e^{\frac{1}{2}\int_{t_1}^{t_2}dtJ(t)^2}
$$
$$
\int_t^Tdx\alpha(t,x)=\frac{1}{2}\left[\int_t^Tdx\sigma(...
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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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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. ...
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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,$...
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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 ...
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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 ...
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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 ...
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