Questions tagged [stochastic-integral]

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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 ...
Quasar's user avatar
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Backward induction: equation including expected values of stochastic process

Given the following SDE: $$ d\psi_t = \rho dt + \mu \psi_t dX_t$$, where $G(t) = \rho t$, $\rho = \frac{1}{T}$ $\psi_0 =0$, $T=1$, $\mu > 0$ and $X_t$ is a standard Brownian Motion (assume we know ...
5ilver4rrow's user avatar
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integral of adapted process with respect to semimartingale is a martingale

Fix $T > 0$ a finite time horizon. Let $H$ be an adapted (or progressively measurable, if needed) continuous process and S be a continuous semi martingale, both on $[0,T]$. Under what conditions is ...
yrual's user avatar
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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 ( $E\left[ \...
user1171853's user avatar
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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(...
TheHunter's user avatar
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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 ...
NC520's user avatar
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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. ...
Landscape's user avatar
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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,$...
user13232877's user avatar
6 votes
1 answer

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 ...
Deros's user avatar
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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 ...
Gabriele Pompa's user avatar