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Hello I am studying about interest rate modeling

There is one good source about Vasicek (link: https://web.mst.edu/~bohner/fim-10/fim-chap4.pdf). However there is one equation that I try but unable to replicate which is:

$dP(t,T) = r(t)P(t,t)dt - \sigma B(t,T)P(t,T)dW(t)$

This equation on 2nd page (or page 18th according to document paging). It locates about 1/3 page top down. Anyone understand how we get this one? What border me is why there is $B(t,T)$ appear.

Besides, the side question is why in interest rate stochastics process it is always express under risk neutral $\mathbb{Q}$ why a traditional stock price S is often expressed in $\mathbb{P}$

Thank you so much

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  • $\begingroup$ Where is the stochastic PDE in this question? $\endgroup$
    – user658409
    Jul 22, 2019 at 18:19

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In the vasicek model, the short rate follows the following dynamic:

$dr_{t} = a(b-r_{t})dt + \sigma dW_{t}$

With :

$a$ constant positive, which represents the return force

$b$ constant positive, which represents the long-term rate

$\sigma$ constant positive, which represents the volatility

The parameters of the model then become $\vartheta$, $a$ and $\sigma$

Under vasicek, the short rate follows an Ornstein-Uhlenbeck process (O-U), and the corresponding solution is:

$$r_t = r_0 e^{-at} + b(1-e^{-at}) + \sigma e^{-at} \int_0^t e^{au} dW_u$$

This Ornstein-Uhlenbeck (O-U) process has been established, following a Gaussian law of parameters its expectation and variances, an Ornstein-Uhlenbeck process is in fact both Gaussian and Markovian.

Let be $X$ a random variable according to a normal distribution. Then $e^X$ follows a log-normal law and therefore:

$\mathbb{E}(e^X) = e^{\mathbb{E}(X) + \frac{1}{2} \mathbb{V}(X)}$

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