# Tag Info

Accepted

### Clarification on Deriving Ito's Lemma

Just a few notes How to make sense of $\text dW_t$ is the entire point of stochastic calculus. It's far beyond the scope of any answer here. You should read some introductory lecture notes/books on ...
• 16.1k
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### List: Behavioural characteristics of key Ito processes used in finance

I will provide some references such that you can see where the different processes are used. These papers typically motivate their models and show which effect the single paramaters have and what ...
Accepted

• 5,053
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### conditional expectation of stochastic integral

What a great question! I've had a go at it below, I'd say I'm about 75% sure of the result I've got to but I'd love feedback from others. I'm going to use the definition of the Ito integral, \begin{...
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• 5,053
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### Quantile normal and lognormal

Quantiles are preserved under monotonic transformations, hence the quantile for $Y$ is simply the exponential of the quantile of $X$, no need for corrections whatsoever (see here for instance). Put ...
• 14.7k
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### Stochastic differential equation of a Brownian Motion

In stochastic calculus, only stochastic integrals are defined. The differential form is just a notation. That is, $$dF=g(t)dW_t$$ is just another expression for the integral $$F=\int_0^t g(s) dW_s.$$ ...
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• 16.1k

### Application of Ito's Lemma in expected utility theory

The risky and riskless assets follow processes, $$\frac{dS_t}{S_t}= \mu \, dt + \sigma \, dB_t, \,\,\, \frac{dM_t}{M_t}= r \, dt$$ If the proportion invested in the risky asset at time $t$ is $p_t$, ...
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### Itos Lemma Derivation notation

The theory behind the actual reasoning is a bit complicated than the coverage in Hull's, but staying within the simple reasoning, the difference comes down to the following: The Brownian increments ...

No. Itō’s formula helps you derive the dynamics of $f (S_\cdot )$ given the SDE followed by $S$. Here this is not the case. You simply have: $$\mathrm{d} \left[\int{g(S_t)\mathrm{d}S_t}\right] = g(... • 2,680 6 votes Accepted ### Proving that a stochastic process is a martingale using Ito's Lemma$$ d Y \left(t\right) := d \left[\int_0^t{a \left(s\right)\mathrm{d}W_s}\right] = a \left(t\right) dW_t  Note that since $Y$ is a driftless process, it is a local martingale, and because $a$ is ...
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The dynamics \begin{align*} \frac{dS_t}{S_t} =\mu dt + \sigma dW_t. \end{align*} is under the real-world measure $\mathbb{P}$. Then, \begin{align*} d\ln S_t =\Big(\mu-\frac{1}{2}\sigma^2 \Big) dt + \...