We know that the forward Libor rate $L(t, T, T + \tau)$, in the absence of arbitrage, is a martingale under the measure $T + \tau$, i.e. $Q^{T+\tau}$. In this context:
$$ \tag{1}\label{1} L(t, T, T + \tau) = \mathbb{E}_t^{T + \tau} \left[ L(T, T, T + \tau \right], $$
with $t \leq T$. This means that the expectation under this specific measure of the spot Libor rate $L(T, T, T + \tau) = \hat{L}(T, T + \tau)$ can be computed analytically.
Now, briefly, what I would like to know if this property can be extended to the forward Libor rate:
$$ \tag{2}\label{2} L(t, T, T + \tau) = \mathbb{E}_t^{T + \tau} \left[ L(s, T, T + \tau \right], $$
with $t \leq s \leq T$. If you can answer this, there is no need to keep reading.
The result in \eqref{1} is very useful for many computations. For example, it is used for a vanilla swap valuation, showing that its price depends only on the term structure of interest rates observed on the valuation date.
Let's analyze a simpler case. Imagine the following stream of payment:
⋅------x//////////x------>
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t Tx = T Tp = T + τ
where Tx represents a fixing date and Tp a payment date. The value of this contract at time $t \leq T_x$ is given by:
$$ \begin{align} V(t) &= \mathbb{E}_t^Q \left[ D(t, T + \tau) \cdot \tau \cdot L(T, T, T + \tau) \right]\\ V(t) &= P(t, T + \tau) \cdot \tau \cdot \mathbb{E}_t^{T + \tau} \left[ L(T, T, T + \tau) \right] \end{align} $$
where $D(t, T)$ represents the discount factor and $P(t, T)$ the discount bond or zero coupon bond. The previous equation yields to (using the property defined in \eqref{1}):
$$ V(t) = P(t, T + \tau) \cdot \tau \cdot L(t, T, T + \tau) $$
So far, so good. Now I would like to compute the price of a generalized stream of payment, given by:
⋅------x----+//////////+----x------>
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t Tx Tb Te Tp
where Tx represents a fixing date, Tb a beginning of accrual date, Te a end of accrual date and Tp a payment date.
The value of this contract at time $t \leq T_x$ is given by:
$$ \begin{align} V(t) &= \mathbb{E}_t^Q \left[ D(t, T_p) \cdot \left(T_e - T_b \right) \cdot L(T_x, T_b, T_e) \right]\\ V(t) &= P(t, T_p) \cdot \left(T_e - T_b \right) \cdot \mathbb{E}_t^{T_p} \left[ L(T_x, T_b, T_e) \right] \end{align} $$
This last expectation seems to be not analytically tractable, right? What I would like to know is which restrictions I have to impose in order to solve it analytically. For example, is it sufficient to match only the end date and the payment date, i.e. $T_e = T_p$, which yields to:
$$ \begin{align} V(t) &= P(t, T_p) \cdot \left(T_p - T_b \right) \cdot \mathbb{E}_t^{T_p} \left[ L(T_x, T_b, T_p) \right] \end{align} $$
This expectation could be solved if \eqref{2} is true. Maybe \eqref{2} can be demonstrated using the definition of the Libor forward rate:
$$ L(t, T, T + \tau) = \frac{1}{\tau} \cdot \left( \frac{P(t, T)}{P(t, T + \tau)} - 1 \right) $$
Any ideas or this is not possible and both $T_x = T_b$ and $T_e = T_p$ should match in order to get an analytically tractable expectation?
Finally, just for completeness, I would like to point out that when $T_e \neq T_p$, the need of a convexity adjustment arises. This question is all about asking if a convexity adjustment or anything else is needed when $T_x \neq T_b$.
