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If $a,b,c \in R$ such that $a < b < c < d$ , show that $(x-a)(x-c) + 2(x-b)(x-d) = 0$ has real and distinct roots .
I'm not getting good manipulation idea to show discriminant positive , also wherever I searched i found only same not much intuitive solution . I m seeking for good intuitive sol...
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Assuming $X_t$ is a standard Brownian motion and $t$ is the time variable, I have learned to derive Ito's lemma for a function $F(X_t, t)$ using the following results:
For any finite $\delta t > 0$ and a standard Brownian motion $X_t$, we can argue that:
\begin{align*}
\tag{1}
\delta X_t := X(\de...
In mathematics, Itô's lemma or Itô's formula (also called the Itô–Doeblin formula, especially in the French literature) is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serves as the stochastic calculus counterpart of the chain rule. It can be heuristically derived by forming the Taylor series expansion of the function up to its second derivatives and retaining terms up to first order in the time increment and second order in the Wiener process increment. The lemma is widely employed in mathematical finance, and its best known...
Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and stochastic differential equations.
The central concept is the Itô stochastic integral, a stochastic generalization of the Riemann–Stieltjes integral in analysis. The integrands and the integrators are now stochastic processes:
where H is a locally square-integrable process adapted to the filtration generated by X (Revuz & Yor 1999, Chapter IV), which is a Brownian motion or, more generally, a se...
