# Questions tagged [stochastic-integrals]

This tag is used for questions about stochastic integrals – especially for calculations. For questions related to more theoretical aspects of stochastic integrals such as their constructions, (stochastic-analysis) may be more appropriate.

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### Using the definition of a stochastic integral directly show that $\int_0^t W_s^2 dW_s=\frac 13 W_t^3 - \int_0^t W_s ds$

Using the definition of a stochastic integral directly show that $$\int_0^t W_s^2 dW_s=\frac 13 W_t^3 - \int_0^t W_s ds$$ There are many solutions to this task on the forum, but they use the the ...
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### Ito-Formula for a Poisson-Process [closed]

I am new to Stochastic Theory and trying to understand (Prop 20.13) of this Article https://personal.ntu.edu.sg/nprivault/MA5182/stochastic-calculus-jump-processes.pdf (The Ito-Formula for a Poisson-...
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1 vote
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### Linear SDE with locally Lipschitz coefficients and without linear growth

I am seeking clarification on the existence and uniqueness of a strong solution to a stochastic differential equation (SDE) in the context of a Brownian motion. Let $\mathrm{W}$ be a $q$-dimensional ...
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### Proof that the stochastic exponential is a local martingale

I'm struggling to understand the proof as to why the stochastic exponential is a continuous non-negative local martingale. My notes say the following: where 3.6 is $Z_t = 1 + \int^t_0 Z_s dX_s$. I ...
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### Convergence of approximating integral sum in case of stochastic integrals

Is it true, that if the integrand in the $$\int_{0}^{T}X\left(t\right)dY\left(t\right)$$ integral is deterministic (so it is the same “function” for each $\omega\in\Omega$) and continuous, then it ...
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### Stochastic integration by parts of geometric brownian motion

My objective is to calculate the integral of a geometric brownian motion $Y=e^{\alpha t+\beta W_t}$, i.e. $$\int_{0}^T e^{\alpha s+\beta W_s}ds$$ and to characterize the moments of resulting random ...
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1 vote
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### Expectation of the Ito integral of a progressively measurable process

I consider $\phi\in H_{2}^{2}$ the set of progressively measurable process such that $$\lVert \phi\rVert_{H_{2}^{2}} = \mathbb{E}\left(\int_{0}^{\infty}\phi_{s}^{2}ds\right) <\infty$$ I would ...
1 vote
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### solving ODE contain matrix

I am currently researching the specific image generation problem in this paper 'Score_Based Generative modeling through stochastic differential equation' At the end of page 14, the authors are using ...
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### Are the integral and differential definitions of Ito process equivalent? [closed]

I think an ito process $X_t$ can be defined as $$X_t := X_0 + \int_0^t\sigma_s dB_s + \int_0^t\mu_s ds.$$ (Is this an Ito drift-difussion process?) (Why use the subscript $s$? Eg. why is it $\sigma_s$ ...
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1 vote
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### Expectation Value of the Product of a Time integral and a Ito Integral

Consider a stochastic process $X_t$ \begin{equation} dX_t = a(X_t)dt + \sigma dW_t \end{equation} where $W_t$ is a Wiener Process. I know the expectation value of the product of two stochastic ...
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### How to compute the second derivative of a function of a stochastic process?

Background Suppose we have a state $\mathbf{x}(t) \in \mathbb{R}^{d \times 1}$ evolving according to \begin{equation} \mathrm{d}\mathbf{x}(t) = \mathbf{F}(\mathbf{x}(t)) \mathrm{d}t + \mathbf{G}(...
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### Differentiating Stochastic Integrals (Ito Integrals)

If I wanted to differentiate a stochastic integral, is this logic correct? X_t = \int_0^t B^2_s dB_s\\ dX_t = d \int_0^t B^2_s dB_s \\ dX_t = B^2_tdB_t - B^2_0dB_0 \\ dX_t = B^2_tdB_t \quad (B_0 = ...
1 vote
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### How is Leibniz integration rule applied to stochastic integrals? [duplicate]

I am working through the mathematics behind the Hull-White short rate model and am currently stuck on how to take the partial derivative of and evaluate a stochastic integral when looking at how bond ...
1 vote
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### Doubt in representation of multiple stochastic integral, Kallenberg Th. 11.25

This is a question about Kallenberg's textbook, Foundations of Modern Probability, theorem 11.25. Let $I_n$ denote the Weiner-Ito stochastic integral on $L^2([0,1]^n,\lambda^n)$, with $\eta$ an ...
1 vote
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### Deriving stochastic process trajectory integral variance

Suppose the process $x_t=u_t+v_t\int\limits_{0}^tg_t\,dW_t$ is given. Here $W_t$ is the standard Wiener process, $u,v,g$ are some deterministic funcions, and $g$ is such that the integral is well ...
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I have two variables $X_1$ and $X_2$ $\in [0,1]$, where $X_2$ (strictly) first order stochastically dominate (FOSD) $X_1$, i.e., $F_{X_2}(x) \leq F_{X_1}(x)$ for all $x$. Then I have, $Y$ and $Z$, ...
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 ...