24 questions linked to/from Integral of Brownian motion w.r.t. time
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### Integral of Brownian Motion w.r.t Time: what is wrong with this solution? [duplicate]

My question is about a stochastic integral of brownian motion w.r.t time. Let $W(t)$ the Wiener process (or brownian motion). I want to calculate this: \begin{eqnarray} X(t)=\int_{0}^t dt' W(t'). \...
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### what is $\int t dW$ and $\int W dt$? [duplicate]

More explicitly, if $W(t)$ is Brownian motion, what would be $$f(t) := \int_0^t u dW(u)$$ and $$g(t) := \int_0^t W(u) du$$?
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### More questions about integral of Brownian Motion w.r.t time

A similar question have been posted earlier but one part has remained unanswered. Let us define: $$X_t = \int_0^t W_s ds,$$ where $W_t$ is a standard Brownian Motion. Is $X_t$ an Itô process or a ...
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### Correlation coeffitiont between two stochastic processes

I want to find correlation coeffitiont between $W_t$ and $\int_{0}^{t}W_s ds$. I think that these are uncorrelated. But Why? So thanks
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### Can I always use quadratic variation to calculate variance?

Suppose we have a Brownian Motion $BM(\mu,\sigma)$ defined as $X_t=X_0 + \mu ds + \sigma dW_t$ The quadratic variation of $X_t$ can be calculated as $dX_t dX_t = \sigma^2 dW_tdW_t = \sigma^2 dt$ ...
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Assume the risk-free bond $B_t$ and the stock $S_t$ follow the dynamics of the Black & Scholes model without dividends (with interest rate r, stock drift $\mu$ and volatility $\sigma$). Let $c(t; ... • 281 3 votes 0 answers 1k views ### Properties of Geometric Brownian Motion Integrated w.r.t. Time (i.e., distribution of a Yor Process) Let$S_t$be a process which follows a Geometric Brownian Motion:$\frac{dS_\tau}{S_\tau} = \mu \,d\tau + \sigma \,dW_\tau$By Ito's lemma, we have:$S_T = S_t e^{(\mu-{\sigma^2 \over 2})(T-t) + \...
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Let $X_t=\int_0^t W_s ds$ where $W_s$ is Brownian motion, so $E[W_s]=0$. Then $E[X_t]=\int_0^t E[W_s] ds=\int_0^t 0 ds=0$. So $E[X_t|{\cal F}_s]=0\neq X_s$, almost everywhere. So by previous ...