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How to Prove a Process Is Adapted to a Filtration

Original question

Can you show me how to prove that a stochastic process is adapted to a given filtration?

Show that XtX_t is measurable with respect to Ft\mathcal F_t for each tt. If Ft\mathcal F_t is generated by observations up to tt, write XtX_t as a measurable function of those observations. For a counterexample, identify an event determined by XtX_t whose truth still depends on future information.

Show that every input is available at the right time

Let Fn=σ(R1,…,Rn)\mathcal F_n=\sigma(R_1,\ldots,R_n) for observed returns RjR_j. If Xn=fn(R1,…,Rn)X_n=f_n(R_1,\ldots,R_n) with Borel-measurable fnf_n, then for every Borel set AA,

{Xn∈A}={(R1,…,Rn)∈fn−1(A)}∈Fn.\{X_n\in A\} =\{(R_1,\ldots,R_n)\in f_n^{-1}(A)\}\in\mathcal F_n.
Sums, products, maxima and threshold indicators of currently measurable variables are measurable. For processes defined recursively, an induction step often works. If Xn−1X_{n-1} is measurable at time n−1n-1, it is measurable at time nn by the filtration inclusion. Applying a measurable update involving the new return RnR_n then yields an Fn\mathcal F_n-measurable XnX_n.

Let the returns be independent and equally likely to be +1+1 or −1-1. Start with V0=0V_0=0, set H0=0H_0=0, and choose Hn−1=1{Rn−1=1}H_{n-1}=\mathbf1_{\{R_{n-1}=1\}} for n≥2n\ge2. The value evolves by Vn=Vn−1+Hn−1RnV_n=V_{n-1}+H_{n-1}R_n. Prove that VV is adapted and explain why Hn−1H_{n-1} is a valid position for period nn.

By construction, Hn−1H_{n-1} is measurable with respect to Fn−1\mathcal F_{n-1}. At nn the position and the new return are known, and the induction argument proves VnV_n is Fn\mathcal F_n-measurable. The position is available one step before the move. If you also want a martingale proof, finite-time integrability follows from bounded increments and E[Hn−1Rn∣Fn−1]=Hn−1E[Rn]=0E[H_{n-1}R_n\mid\mathcal F_{n-1}]=H_{n-1}E[R_n]=0.
For comparison, consider the process An=1{Rn+1=1}A_n=\mathbf1_{\{R_{n+1}=1\}}. Its value is determined only after the next observation. More formally, if AnA_n were Fn\mathcal F_n-measurable, then E[An∣Fn]=AnE[A_n\mid\mathcal F_n]=A_n. Independence gives E[An∣Fn]=1/2E[A_n\mid\mathcal F_n]=1/2, whereas AnA_n takes only values zero and one. This proves the failure of adaptedness for this filtration.
Adaptedness is separate from integrability and path regularity. For example, a measurable transformation of observed returns is still adapted even if its values have infinite absolute expectation. That distinction matters when moving from an information check to a martingale proof.

For the induction, write down the measurability of both terms in Vn=Vn−1+Hn−1RnV_n=V_{n-1}+H_{n-1}R_n. The first term is Fn−1\mathcal F_{n-1}-measurable and therefore also Fn\mathcal F_n-measurable. The position is chosen with Fn−1\mathcal F_{n-1} information, and the new return is Fn\mathcal F_n-measurable. Their product is measurable at nn. This proves the portfolio value can be recorded at nn. A position used to earn the return from n−1n-1 to nn has the stronger requirement of being determined before that return.

References and next steps