I think an ito process $X_t$ can be defined as $$X_t := X_0 + \int_0^t\sigma_s dB_s + \int_0^t\mu_s ds.$$ (Is this an Ito drift-difussion process?) (Why use the subscript $s$? Eg. why is it $\sigma_s$ and not just $\sigma$?) And I've also seen: $$dX_t := \sigma_t dB_t + \mu_t dt$$ Are the 2 defin...

In the series expansion to obtain the form of higher order Milstein schemes you need to evaluate these two sister integrals (where W_t is the Wiener process): $$ \int_{t_i}^{t_{i+1}}dW_u\int_{t_i}^{u}ds\\ \int_{t_i}^{t_{i+1}}du\int_{t_i}^{u}dW_s, $$ after I evaluate the innermost integral I obtai...

In this article I read that two topologies can be compared. Suppose that $f : X \longrightarrow X $ be the identity map, $f (x) = x$ for any $x$. Consider on $X$ two topologies $\tau_1 , \tau_2 $. Suppose that $\tau_1 < \tau_2$ . Is $f : (X, \tau_1 ) \longrightarrow (X, \tau_2 ) $ always contin...

In this article I read that two topologies can be compared. Suppose that $f : X \longrightarrow X $ be the identity map, $f (x) = x$ for any $x$. Consider on $X$ two topologies $\tau_1 , \tau_2 $. Suppose that $\tau_1 < \tau_2$ . Is $f : (X, \tau_1 ) \longrightarrow (X, \tau_2 ) $ always contin...