New answers tagged brownian-motion
1
The examples provided by Sin in their article Complications with Stochastic Volatility Models might help to answer your questions.
I'm transcribing the abstract below:
We show a class of stochastic volatility price models for which the most natural candidates for martingale measures are only strictly local martingale measures, contrary to what it is usually ...
3
$$ X_t = B_t 1_{t<0.5} + (x+ B_t) 1_{t\geq 0.5} = B_t + x1_{t\geq 0.5}$$
$$ [X, X]_t = [B, B]_t + x^2 1_{t\geq 0.5} = t+ x^2 1_{t\geq 0.5}$$
(the author probably intended to use $0.5$ as jump size too)
4
As stated here, for $f =
f(t, x) ∈ C^{1,2}(\mathbb{R}^2)$ a deterministic function and Ito process $$X_t = W_t^2,$$ the stochastic process
$$Y_t = f(t,X_t)$$
is an Ito process and we have
$$df (t,X_t) = \partial_tf(t,X_t)\,dt + \partial_xf(t,X_t)\,dX_t +
\frac{1}{2} \partial_{xx}^2f(t,X_t)(dX_t)^2. $$
Since
$$ dX_t = 2W_t dW_t + dt $$ and
$$ (dX_t)^2 = ...
4
Answering the title question:
Let $f(t,W_t)=W_t^2-t$, then it is easier to derive the dynamics using the "general formula" for Itô's lemma (reference, see eq. 10):
$$df(t,W_t)=\frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial W_t} dW_t + \frac{1}{2}\frac{\partial^2f}{\partial W_t^2} dW_t^2$$
where,
$$\frac{\partial f}{\partial t} = -1, \...
2
One can use the Euler-Maruyama discretization scheme for CIR, 'fixed' for $v$ positivity, to get:
$$ v(t+\epsilon) -v(t)\approx \kappa (\bar{v} -v(t)^+)\epsilon + \omega \sqrt{v(t)^+} (W_v(t+\epsilon) - W_v(t)). $$
So, one approximation of the Brownian increment, when $v(t)$ and $v(t+\epsilon)$ are given, is:
$$ W_v(t+\epsilon) - W_v(t) \approx \frac{v(t+\...
1
Edit: This is probably incorrect.
The Quadratic Exponential scheme is the best one I have seen as it converges in distribution and is pretty fast, so nice choice there!
When $\eta$ is constant you can simplify the integral
$$
\int_t^{t+\varepsilon}\eta dW(u)=\eta\int_t^{t+\varepsilon}dW(u)=\eta\left(W(t+\varepsilon)-W(t)\right)
$$
In the QE scheme you either ...
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