Salut, voici une application de la formule d'Itô que je pige pas...Ca concerne le modèle de taux de HJM losqu'on spécifie des fonctionnelles particulières pour la volatilité.A l'origne $\frac{dB(t,T)}{B(t,T)} = r_t dt + < \sigma (t,T) , dW_t >$ sous la probabilité risque-neutre.Par définition: $ \gamma(t,T) = \frac{\partial}{\partial T} \sigma (t,T) $On spécifie pour $\gamma$ la dynamique suivante $ \gamma (t,T) = \gamma (t,t) e^{-\int_t^T k(x) dx} $($\gamma(t,T)$ est stochastique) Soit $\phi (t) = \int_0^t {\gamma (s,t)}^2 ds$Comment montrer que ceci équivaut à:(Je ne vois pas comment utiliser Itô)$d\phi (t) =( \gamma (t,t) ^2 - 2k(t) \phi (t) )ds$puis:$ \beta(t) = \int_0^t \gamma (s,t) \sigma (s,t) ds + \int_0^t \gamma (s,t) dW_t $Comment montrer que ceci équivaut à:(Je ne vois pas comment utiliser Itô)$ d\beta (t) = (\phi (t) - k(t) \beta (t)) dt + \gamma (t,t) dW_t $

Если все мы говорим на нашем собственном языке не будет никакого заказа.

Ok Ok I wonder why: $\beta(t) = \int_0^t \gamma (s,t) \sigma (s,t) ds + \int_0^t \gamma (s,t) dW_t $<=>$ d\beta (t) = (\phi (t) - k(t) \beta (t)) dt + \gamma (t,t) dW_t $ where $ \gamma(t,T) = \frac{\partial}{\partial T} \sigma (t,T) $$\phi (t) = \int_0^t {\gamma (s,t)}^2 ds$

Ok Ok I wonder why: $\beta(t) = \int_0^t \gamma (s,t) \sigma (s,t) ds + \int_0^t \gamma (s,t) dW_t $<=>$ d\beta (t) = (\phi (t) - k(t) \beta (t)) dt + \gamma (t,t) dW_t $ where $ \gamma(t,T) = \frac{\partial}{\partial T} \sigma (t,T) $$\phi (t) = \int_0^t {\gamma (s,t)}^2 ds$

this is the last time I'm drinking gin

nmaughan, are you Ukrainian (or a Russian from Moscow)? I am just wondering 'cos your language sounds funny

And nobody for my question !?

Neither, I'm a Londoner of Italian and Scottish blood who lives in NYC. That is why I speak poor Russian.

what's k(t) in your eqn?

nmaughan, no you Russian is very good!Just I got confused by your using "zakaza" instead of "poryadka". But then realised that these two Russian words are translated into English by the word "order".This led me to believe you were either a Ukrainian (who sometimes use in Russian words from the Ukrainian language, which meaning in Russian is different), or a Moscovite with their sometimes absolutely illogical slang

t->k(t) is a real-valued function, you can take k(t) = cte if you want ...

today not feeling like reading in french...

Ok see your answer later ...