# 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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### limit of Stochastic Integration

Suppose that $X_t:=\int_{0}^{t}b_sds+\int_{0}^{t}\sigma_tdB_t$ where $b,\sigma \in L^{\infty}(F)$. For $\pi:0=t_0<t_1<...<t_n=T$, denote $$S_L(\pi):=\sum_{i=0}^{n-1}X_{t_i}B_{t_i,t_{i+1}}$$. ...
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### Expectation of an Ito Process at a specific time

Suppose we have the following dynamics: $$dX_t = a_t X_t dt + b_t X_t dW_t; \;\;\; X_0 = x.$$ Would the expectation of $X_t$ at $t=T$ be $$\mathbf{E} [X_T] = x + \int_0^T a_t X_t dt$$? I know that it ...
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### SDE for a positive cadlag, bounded and non-increasing process

Giving a process $(D_{t})_{0\leq t\leq T}$ with the following properties: adapted, cadlag, non-increasing, $D_{0}<=1$ and $D_{T}=0$. Can I write the SDE associated to that process in one of the ...
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### Can we state the distribution of this Brownian motion as follows?

Consider the following Brownian motion: $W(e^{2t})$ Where $t\in[0,\infty)$. What can we then say about how $W(t)$ is distributed? Could we say: $W(e^{2t})\sim \sqrt{e^{2t}}N(0,1)$?
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### Summary of $\frac{dM(t)}{M(t)}$ using Ito's formula in dimension $d$

EDIT: This question has been solved! Consider a $d$-dimensional Brownian motion $W$ and the stochastic differential equation(s) \begin{align} dX&=b(X)dt+\rho(X)dW\\ X_0&=x, \end{align} ...
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### Convolution of gaussian process

Let $\text{GP}(\mu(x),k(x,x^{\prime})$ be a gaussian process. Here, $\mu$ is the mean function. Typically, $\mu$ equals to $0$. $k$ is the kernel function. Can you define a convolution of the gaussian ...
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### Construction of Ito integral in oksendal's book

In Oksendal's book of Stochastic Differential Equations, I have a problem assimilating the proof of the second step in the process of constructing Ito integral, this is the statement of the second ...
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### Martingale & Ito Integral Help

I have seen a few textbooks that say $\int_{0}^{t} W_{s}ds$ is NOT a martingale. I believe they are correct, but I'm confused what is wrong with the logic below that seems to show it is a martingale. ...
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### Stochastic Product Rule Example

When given a function I wanted to know the how the application relates to: $Y_t = X_tdY_t + Y_tdX_t + (dX_t)(dY_t)$ (I am okay with using the 2 variable 2 equation function form). For example consider:...
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### How to show the sum of the squares process is deterministic

Can I please get some feedback on my work for the following problem? Is there a more simple approach to arrive to the solutions for part a? Thank you for your time and consideration! Let $X$ be a ...
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### Integrating wrt a stochastic process

I am confronted with the following expectation $$E_t \left[\int_t^Tg(S_s)dS_s \right]$$ Where $S_t$ is a stochastic process. How would we go about computing this quantity? If we can't do so in the ...
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### Writing down the proof of Proposition 1.1(ii)(1.12) in Ikeda and Watanabe

I am studying chapter 2 of the book "Stochastic Differential Equations and Diffusion Processes by Nobuyuki Ikeda and Shinzo Watanabe." I am struggling to write down proof for the following ...
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### quadratic variation and local martingale

Consider a predictable process $H_t$ and a continuous local martingale $M_t$ for $t \in [0,T]$ I want to show that $$X_t:=\left(\int_0^t H_s dM_s\right)^2 - \int_0^t H^2_s d \langle M_s\rangle$$ is ...
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### Cross Variation of Ito Integral with $L^2$ bounded continuous martingale

I am trying to prove the identity: $$< I(K), N_{\cdot}>_t = \int_0^t K_s \ d < B,N_{\cdot}>_s$$ where $I(K)_t$ is by definition $\int_0^t K_s \ d B_s$, and we take $K$ as a ...
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### integrability of random variables w.r.t. Brownian motion

Consider random variables $Z_t>0 \ \forall t \in [0,T]$, where $E[Z_T]=1$ and $Z_t=E[Z_T|F_t]$, where $F_t$ is the Brownian filtration and $H$, which is integrable with respect to Brownian motion ...
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Let $g$ and $F$ be functions defined on $[0,1]$. The function $F$ is distribution function. There are estimators $\hat{g}_n$ and $\hat{F}_n$ such that $$\int_0^1 |\hat{g}_n(t) - g(t)| dt, \text{ and }... 1answer 44 views ### Is this stochastic integral well-defined? Let x_t = \int_{s=0}^t dW_s where W_t is the standard Weiner process. Now I define$$ f(x_t) = \frac{1}{(2-x_t)^4} $$Using Ito's Lemma,$$ df(x_t) = \frac{4}{(2-x_t)^5}dW_t - \frac{10}{(2-x_t)^6}...
Consider the stochastic exponential: $F[M] = e^{M(t)-\frac{1}{2}\langle M\rangle(t)}$ for an local martingale $M$. Define: $$M:= \log(L(0)) + \int_0^* \frac{1}{L} dL$$ where $L$ is a strictly ...