Pricing of Zero Coupon bond under Risk-neutral pricing measure

The below topics are from Stochastic calculus for Finance II, Continuous time models by Steven E Shreve
(formula explained using LATEX code)

Pg 242 Topic 5.6.2: Futures contract

Risk-neutral pricing of a zero-coupon bond is given by the below formulae:

[math]B(t,T) \, = \,\frac{1}{D(t)}. \tilde E~[D(T)\mid F(t)], 0\,\leq \,t\,\leq\,T\,\leq\,\bar T[/math]
I understand here the interest rates are not constant, but is either deterministic or stochastic.

The interest rate path of D(t), it is either a subset or adaptable from the path of D(T).

If my understanding is correct, why cannot we nullify the common path for the period [0,t].

If we take D(t) inside the E-tilda then the following will be the steps:

[math]B(t,T) \, = \, \tilde E [ exp\{ -\int_{0}^{t} R(u)\,du+ \int_{0}^{T} R(u)\,du\}\mid F(t)][/math][math]B(t,T) \, = \, \tilde E [ exp\{ \int_{t}^{T} R(u)\,du\} \mid F(t)][/math]
Since it is now full expectation, or the period [t,T] is not dependent on Filtration (t), we can write the above as

[math]B(t,T) \, = \, \tilde E [ exp\{ \int_{t}^{T} R(u)\,du\}][/math]
If I turn to page Page 218 Topic 5.2.4 Pricing under the Risk-Neutral measure.

I compare Equation 5.2.30 and 5.2.31 In Equation 5.2.31, the D(t) is taken inside E-tilda

[math]V(T)\,=\, \tilde E [exp\{ − \int_{t}^{T} R(u)\,du \}.V(T) \mid F(t)][/math]
My doubt is, if it could be taken inside E-tilda in Equation 5.2.31 why I cannot consider D(t) in E-tilda in Topic 5.6.2 for Zero-coupon bond.

Kindly if anyone can help clarify this difference in approach.
 
If my memory serves me correctly, you can. The D(t) is left outside to get the result we are looking for. D(t) is F_t-measurable so you can bring it in. If you look at the 3rd line of V_k,j on page 242, you can see that it is allowed to be moved in.

I recommend Fima's book over Shreve's if you're wanting to learn stochastic calculus. Shreve's is pretty outdated and the commentary isn't as helpful as fima's. There's also a lot of unnecessary bloat like interest-rate modelling and ATS (other books explain it better and are more comprehensive).

But your questions are better suited for the quant stackexchange website.
 
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quant stackexchange website. has blocked me from posting any more questions.
thank you bro for taking time to read my posts.
I made an edit that might be helpful. Make a new stack account and follow their rules. If you have lots of questions, try to find similar questions and apply similar logic
 
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