euclid.jap.1261670699

# In particular 21 holds for deterministic functions r

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Unformatted text preview: n An inequality for variances of the discounted rewards and T E 0 rt d t = E = 0 = 0 = ∞ 0 ∞ 1211 rt 1{T ≥ t } dt E[rt 1{T ≥ t }] dt ∞ ∞ 0 E[rt ] E[1{T ≥ t }] dt E rt P{T ≥ t } dt ∞ =E 0 e−αt rt dt. In particular, (2.1) holds for deterministic functions r and R , and, therefore, E[J1 | F∞ ] = E[J2 | F∞ ] P -a.s., (2.2) if either E[|J1 | | F∞ ] < ∞ or E[|J2 | | F∞ ] < ∞ P-a.s. However, the second moments can be different. Indeed, we have the following statement. Theorem 2.1. If either E[|J1 | | F∞ ] < ∞ or E[|J2 | | F∞ ] < ∞ P-a.s., then var (J1 ) ≤ var (J2 ), and the equality holds if and only if var (J2 | F∞ ) = 0 P-a.s. Proof. By the total variance formula (see [6, p. 83] or [3, p. 454]), for i = 1, 2, var (Ji ) = E[var (Ji | F∞ )] + var (E[Ji | F∞ ]). Therefore, because of (2.2), var (E[J1 | F∞ ]) = var (E[J2 | F∞ ]). In addition, E[var (J1 | F∞ )] = 0 and E[var (J2 | F∞ )] ≥ 0. Hence, var (J2 ) − var (J1 ) = E[var (J2 | F∞ )] ≥ 0, i.e. var (J1 ) ≤ var (J2 ). Example 2.1. Cons...
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## This note was uploaded on 02/02/2014 for the course AMS 507 taught by Professor Feinberg,e during the Fall '08 term at SUNY Stony Brook.

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