Question 33708: One Fixed Affine Transformation Preserves the Same-Filtration Martingale

Original question and givens

Use the declared equiprobable one-step martingale with constant initial=-5 and complete terminal values=[-6, -4, -5, -5]. The initial information is trivial and the terminal information reveals these outcomes. Fixed scale=0, shift=10. Give (initial,terminal,scale,shift,transformed_initial,transformed_terminal,martingale?,affine_transform_preserves_same_filtration_martingale).

• G001 — Use the declared equiprobable one-step martingale with constant initial=-5 and complete terminal values=[-6, -4, -5, -5]. The initial information is trivial and the terminal information reveals these outcomes. Fixed scale=0, shift=10. Give (initial,terminal,scale,shift,transformed_initial,transformed_terminal,martingale?,affine_transform_preserves_same_filtration_martingale).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-martingale

Step-by-step derivation

source roles = [qf-mod31-martdeep-affine-stability, [domain, initial, operation, scale, shift, terminal]]

derived = [martdeep-affine-stability, Exact_finite_martingale_construction]

Affine ORIGINAL terminal sum := sum(-6, -4, -5, -5)

Affine ORIGINAL terminal sum = -20

Affine ORIGINAL initial times outcome count := mul(-5, 4)

Affine ORIGINAL initial times outcome count = -20

source roles = [-5, [-6, -4, -5, -5], -20, -20]

derived = [Yes]

Affine ORIGINAL scale times initial := mul(0, -5)

Affine ORIGINAL scale times initial = 0

Affine ORIGINAL scaled initial plus fixed shift := sum(0, 10)

Affine ORIGINAL scaled initial plus fixed shift = 10

Affine ORIGINAL terminal product0 := mul(0, -6)

Affine ORIGINAL terminal product0 = 0

Affine ORIGINAL transformed terminal0 := sum(0, 10)

Affine ORIGINAL transformed terminal0 = 10

Affine ORIGINAL terminal product1 := mul(0, -4)

Affine ORIGINAL terminal product1 = 0

Affine ORIGINAL transformed terminal1 := sum(0, 10)

Affine ORIGINAL transformed terminal1 = 10

Affine ORIGINAL terminal product2 := mul(0, -5)

Affine ORIGINAL terminal product2 = 0

Affine ORIGINAL transformed terminal2 := sum(0, 10)

Affine ORIGINAL transformed terminal2 = 10

Affine ORIGINAL terminal product3 := mul(0, -5)

Affine ORIGINAL terminal product3 = 0

Affine ORIGINAL transformed terminal3 := sum(0, 10)

Affine ORIGINAL transformed terminal3 = 10

Affine ORIGINAL transformed sum := sum(10, 10, 10, 10)

Affine ORIGINAL transformed sum = 40

Affine ORIGINAL transformed initial times count := mul(10, 4)

Affine ORIGINAL transformed initial times count = 40

source roles = [0, 10, -5, [-6, -4, -5, -5]]

derived = [10, [10, 10, 10, 10], Yes]

Reverse checks

• S017 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S016 · premise_and_result_audit — Recompute Affine ORIGINAL transformed initial times count with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Affine ORIGINAL transformed sum with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Affine ORIGINAL transformed terminal3 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Affine ORIGINAL terminal product3 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Affine ORIGINAL transformed terminal2 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Affine ORIGINAL terminal product2 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Affine ORIGINAL transformed terminal1 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Affine ORIGINAL terminal product1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Affine ORIGINAL transformed terminal0 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Affine ORIGINAL terminal product0 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Affine ORIGINAL scaled initial plus fixed shift with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Affine ORIGINAL scale times initial with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S003 · premise_and_result_audit — Recompute Affine ORIGINAL initial times outcome count with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Affine ORIGINAL terminal sum with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.

Result

(-5, [-6, -4, -5, -5], 0, 10, 10, [10, 10, 10, 10], Yes, affine_transform_preserves_same_filtration_martingale)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.