# Tagged Questions

This tag is used for questions about stochastic integrals - especially for calculations . For questions related to more theoretic aspects of stochastic integrals such as its construction. Stochastic-analysis may be a more appropriate tag.

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### Given ${\rm d}X_t=v_t(X_t){\rm d}t+\text{white noise}$, obtain a SDE for $v$ in a Hilbert space $H\subseteq L^2$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $U,H\subseteq L^2(\Omega,\mathbb R^d)$ be separable $\mathbb R$-Hilbert spaces Let $t\mapsto X_t\in\mathbb R^d$ be the trajectory of a ...
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### Application of Feynman-Kac to find an expected value by solving PDEs

So I have a stochastic calculus question that really boils down to a PDE question. " Find $$\mathbb{E}_{W_0=x}e^{-k\int_0^T W_s^2ds},$$ where $W_t$ is standard Brownian Motion at time $t$, $k>0$ ...
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### Supremum of expected value over equivalent measures

I'm not sure how one can proof the following statement: We have a probability space $(\Omega, \mathbb{F}, \mathbb{P})$ and a $\mathbb{F}$-measurable random variable $X$. Furthermore we have a set of ...
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### Square Integrable Martingales and the Unit Process

Let $X_t$ be a continuous square-integrable martingale. Then it is true (I think, please correct me if I am wrong) that: $$\forall t \in [0,\infty), \quad \int_0^t 1_{[0,t]}(s) dX_s = X_t - X_0$$ So ...
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### Non standard stochastic integral

I don't know how to deal with the following stochastic integral: $\int_0^t \frac{1}{\sqrt{t-s+1}} d W(s)$ As you can see, the variable $t$ appears both as an endpoint of the interval of integration ...
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### When is a stochastic integral a martingale?

In what follows, let the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ as well as the chosen filtration $(\mathcal{F}_t)_{t \ge 0}$ be known, and let $f$ denote an arbitrary locally bounded ...
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### Moment bounds for solutions to SDEs

I've been looking at Kuo's book on stochastic integration, and I noticed that in Section 10.7, he proves some estimates for solutions to SDEs, written as $$dX_t = f(t,X_t)dt + \sigma(t,X_t)dB_t$$. ...
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### Are martingales progressively measurable? (Application to square integrable martingales)

This is an incredibly dumb question, but I'm not sure if I know the correct answer, and it doesn't seem to be stated anywhere on the internet, so here goes: Are martingales progressively ...
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### What is the rate of convergence of Brownian motions Increments?

Would like to know what the rate of convergence of brownian motion is? I know each brownian motion increment is distributed with N(0,t) so do i need to apply a CLT?
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### Computation of an Ito Integral [closed]

I have the following Ito Integral \begin{align}\int_{0}^{t}B_se^{-\sigma B_S}dB_s&& \end{align} Here, $\sigma \gt 0$. Can someone please show me how to compute this Ito integral? ...
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### Expected value of the exponential of a Geometric Brownian motion

I am trying to compute the following expectation: $$E[ \exp (A_T)],$$ where $A_T = - C \int_{0}^{T} \exp( 2 \alpha W_t - \alpha^2 t) dt$, with $C$ and $\alpha$ positive constants, $W_t$ a standard ...
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### Mean and Variance of SDE

How would I compute the mean and variance of the following SDE? $dX_t = \alpha X_t dt + \sigma dB_t$ I know $E[X_t]$ produces the mean and $E[(X_t)^2]$ produces the variance, but I'm not sure how to ...
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### JL Doob / KL Chung paper: Fields Optionality and Measurability

I was reading the following paper: https://www.jstor.org/stable/2373011?seq=1#page_scan_tab_contents I had a question about a proof that is part of the paper. Here are some of the definitions and ...
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### Solving an Itô Integral

Can someone please show me how to solve this Itô Integral? \begin{align}\int_{1}^{t}\frac{dB_s}{B_s^2 + B_s^4} && \end{align}
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### Can Local Martingales be characterized only using their FV process and BM?

Prove or Disprove: A process $(X_t)_{t \ge 0}$ is a (continuous) local martingale if and only if it can be represented in the form: $$\int_0^t \xi dB = \large B_{\int_0^t \xi_s^2 ds}$$ where the ...
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### Simulating a Stochastic Integral of OU process

The stochastic integral I want to simulate is $$\int_{0}^{1}J_c(s)dJ_c(s)$$ where $J_c(s) = \int_{0}^{s}e^{-c(s-r)}dB(r)$, is an OU process. I simulate the data using Matlab and the sample codes are ...
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### System of stochastic equations

I want to know if this system of SDE: $$dX_{t}=b(X_{t})dt+\sigma( X_{t}) dB_{t}$$ $$dY_{t}=b_{0}(Y_{t})dt+\sigma( Y_{t}) dB_{t}$$...
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### brownian noise and stochastic differential equations

Consider the SDE $$dx=3x(t)dt+dW(t)$$ Where we're dealing with Brownian noise. Now, dW comes from $$dW(t)=\int_0^{dt}ds\ \eta (s)$$ As far as I understood, $\eta$ is the noise distribution (...
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### System of Stochastic Differential Equations (SDEs) from Diffusion on Manifold

I am looking at a system of SDEs due to Brownian motion on a 3d Riemannian manifold (see e.g. Ito, 1962, The Brownian Motion and Tensor Fields on Riemannian manifolds). I have reduced the associated ...
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### Expectation value of stochastic process

For which $k>0$ process $X=(e^{kW_s^2})_{s \ge 0}$ belong to $\mathcal{L}^2_{\infty }(W)$ and for which belong to $\Lambda ^2_{\infty }(W)$. Set one localization sequence $(\tau_n)_{n \ge 0}$ for ...
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### Using of Ito formula with martingales

We have exam test - $\alpha,\beta \in \mathbb{R}$ and $N(t)=e^{\beta t}cos(\alpha W(t)).$ It is necessary to calculate $\mathbb{E}[cos(\alpha W(t))]$. I know that $\beta$ can be chosen so that $N$ ...
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### What's the variance of the following stochastic integral?

The stochastic integral is defined as $$u_t = \int_{t-1}^t e^{-\kappa(t-s)}\int_0^s e^{-c(s-r)} \, dW(r) \, ds.$$ where $W(t)$ is a standard Brownian motion, $\kappa$ and $c$ are both positive. I ...
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