Question 33705: One Fixed Affine Transformation Preserves the Same-Filtration Martingale

Original question and givens

Use the declared equiprobable one-step martingale with constant initial=-8 and complete terminal values=[-9, -7, -8, -8]. The initial information is trivial and the terminal information reveals these outcomes. Fixed scale=2, shift=7. 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=-8 and complete terminal values=[-9, -7, -8, -8]. The initial information is trivial and the terminal information reveals these outcomes. Fixed scale=2, shift=7. 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(-9, -7, -8, -8)

Affine ORIGINAL terminal sum = -32

Affine ORIGINAL initial times outcome count := mul(-8, 4)

Affine ORIGINAL initial times outcome count = -32

source roles = [-8, [-9, -7, -8, -8], -32, -32]

derived = [Yes]

Affine ORIGINAL scale times initial := mul(2, -8)

Affine ORIGINAL scale times initial = -16

Affine ORIGINAL scaled initial plus fixed shift := sum(-16, 7)

Affine ORIGINAL scaled initial plus fixed shift = -9

Affine ORIGINAL terminal product0 := mul(2, -9)

Affine ORIGINAL terminal product0 = -18

Affine ORIGINAL transformed terminal0 := sum(-18, 7)

Affine ORIGINAL transformed terminal0 = -11

Affine ORIGINAL terminal product1 := mul(2, -7)

Affine ORIGINAL terminal product1 = -14

Affine ORIGINAL transformed terminal1 := sum(-14, 7)

Affine ORIGINAL transformed terminal1 = -7

Affine ORIGINAL terminal product2 := mul(2, -8)

Affine ORIGINAL terminal product2 = -16

Affine ORIGINAL transformed terminal2 := sum(-16, 7)

Affine ORIGINAL transformed terminal2 = -9

Affine ORIGINAL terminal product3 := mul(2, -8)

Affine ORIGINAL terminal product3 = -16

Affine ORIGINAL transformed terminal3 := sum(-16, 7)

Affine ORIGINAL transformed terminal3 = -9

Affine ORIGINAL transformed sum := sum(-11, -7, -9, -9)

Affine ORIGINAL transformed sum = -36

Affine ORIGINAL transformed initial times count := mul(-9, 4)

Affine ORIGINAL transformed initial times count = -36

source roles = [2, 7, -8, [-9, -7, -8, -8]]

derived = [-9, [-11, -7, -9, -9], 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

(-8, [-9, -7, -8, -8], 2, 7, -9, [-11, -7, -9, -9], 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.