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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'). \...
Leonardo S. Vieira's user avatar
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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 $$g(t) := \int_0^t W(u) du$$?
athos's user avatar
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7 votes
2 answers
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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 ...
agassi's user avatar
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7 votes
1 answer
10k views

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
user22715's user avatar
  • 165
7 votes
1 answer
6k views

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 $dX_t dX_t = \sigma^2 dW_tdW_t = \sigma^2 dt$ ...
Pandaaaaaaa's user avatar
10 votes
2 answers
1k views

Change of measure and Girsanov's Theorem: Do the following models admit arbitrage and are they complete?

Let $S_{t}$ denote the price of stock, $\beta_{t}$ denote the savings account. For each model below state with reason whether it admits arbitrage and whether it is complete. (a) $\beta_{t}=e^{t}, S_{t}...
randorando's user avatar
7 votes
1 answer
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Integral of Wiener process w.r.t. time

I have a doubt with regards to the calculation of the below integral- $\int_0^t W_sds$ where $W_s$ is the Wiener Process. This has been solved very ably in the following page. It turns out to be a ...
Amrit Prasad's user avatar
2 votes
2 answers
615 views

How to numerically simulate exponential stochastic integral

For example given an integral $$ \int^t_0 \exp(aW(t'))\,dt', t\in\mathbb R_+ $$ where $W(t')$ is a standard Wiener process. I've been very confused about stochastic integrals like $\int^t_0 W(t')\,...
null's user avatar
  • 123
3 votes
2 answers
991 views

Covariance between integral of brownian motion and brownian motion

Let $$ I = \int_0^1W_tdt, $$ where $W_t$ is a Brownian motion. From Integral of Brownian motion w.r.t. time we have that $$ \mathbb{E}[I]=0, $$ by Fubini's theorem. And that $$ \mathbb{V}\text{ar}[I] =...
Oliver's user avatar
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1 vote
2 answers
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Integral of the square of Brownian motion using definition of variance

Let $B = \{ B(t); t \ge 0\}$ and let $Z = \{ Z(t); t \ge 0 \}$ where $$Z(t) = \int_0^t B^2(s) ds.$$ How do we find $E[Z(t)]$ and $E[Z^2 (t)]$ in order to get the variance $Var [Z^2(t)] = E[Z^2 (t) ] -...
holala's user avatar
  • 121
4 votes
1 answer
489 views

Limits of integration when applying stochastic Fubini theorem to Brownian motion

I'm looking at the solution below from Quantuple, it's a nice, succinct solution but I'm confused about how the limits of the integrals in the second line come from. Could someone please elaborate on ...
Bazman's user avatar
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1 vote
1 answer
550 views

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
  • 548
1 vote
1 answer
659 views

Continuous Geometric Asian Options

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; ...
Anon's user avatar
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3 votes
0 answers
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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) + \...
David Addison's user avatar
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0 answers
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Proof that integral of Brownian motion wrt time is not a martingale

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 ...
Lars Ericson's user avatar

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