\[\begin{align} f_{Y_{1:n}}(r_1, \ldots, r_n | \theta) &= \Bigg( \prod_{j \in C} F_X(r_{j-1}) f_{Z_j}(r_j) \Bigg) \Bigg( \prod_{j \in D} (1 - F_X(r_{j-1})) \Bigg) \\ &= \Bigg( \prod_{j \in C} F_X(r_{j-1}) \cdot \frac{1}{F_X(r_{j-1})} f_X(r_j) \chi_{r_j \leq y_{j-1}} \Bigg) \Bigg( \prod_{j \in D} (1 - F_X(r_{j})) \Bigg) \\ &= \Bigg( \prod_{j \in C} f_X(r_j) \Bigg) \Bigg( \prod_{j \in D} (1 - F_X(r_{j})) \Bigg) \end{align}\]where \(C\) denotes the set of time indices for the records which changed the running minimum and \(D\) denotes the set of time indices for the records which didn't change the running minimum. Note that we can determine the sets \(C\) and \(D\) by checking successively checking whether or not arecord changed since the previous record. Note also that we drop the factors \(\chi_{z \leq y_{j-1}}\) from the likelihood since they are redundant ; \(j \in C\) if and only if \(r_j \leq y_{j-1}\). Presuming that we can evaluate the CDF and PDF of \(X\) for any \(\theta\),  the evaluation of this form of the likelihood is straightforward. If we seek a model for a running maximum rather than a minimum, we can make a similar argument to find that the likelihood in this case becomes
\[\begin{align} f_{Y_{1:n}}(r_1, \ldots, r_n | \theta) &= \Bigg( \prod_{j \in C} (1 - F_X(r_{j-1})) \cdot \frac{1}{(1 - F_X(r_{j-1}))} f_X(r_j) \chi_{r_j \geq y_{j-1}} \Bigg) \Bigg( \prod_{j \in D} F_X(r_{j}) \Bigg) \\ &= \Bigg( \prod_{j \in C} f_X(r_j) \Bigg) \Bigg( \prod_{j \in D} F_X(r_{j}) \Bigg) \end{align}\]with the appropriate modifcations in notation for switching to the maximum.
Lemma 1: Marginal distribution of a historical record
Let \(X_1, \dots, X_n\) be a collection of iid random continuous variables with PDF \(f_X\) and CDF \(F_X\), and define \(Y_n := \max_{i\le n} X_i\). Then the marginal likelihood of \(Y_n\) is equal to:
\[\mathcal{L}_{Y_{n}}(o_n) = n [F_X(o_n)]^{n-1}f_X(o_n)\]
Proof: The CDF of \(Y_n\) is:\[\begin{align} F_{Y_n}(o_n) &= P(Y_n \le o_n) = P(\max\{X_1, \dots, X_n\} \le o_n)\\ &= \prod_{i\le n} P(X_i \le o_n) \\ &= \prod_{i\le n} F_X(o_n) \\ &= [F_X(o_n)]^n \end{align}\]
Differentiation of the CDF gives us the desired result. \(\square\)