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L w µ L m µ!. THEOREM (RieszSchauder) If T2C(X) then ¾(T) is at most countable with only possible limit point 0 Further, any nonzero point of ¾(T) is an eigenvalue of flnite multiplicity COROLLARY (Fredholm alternative) If T2C(X) then either I¡Tis invertible or 1 2P¾(T), that is, 1 2‰(T) P¾(T) COMPACT SELFADJOINT OPERATORS ON HILBERT SPACE. £ 0Ç #Ý (ì 0°3U ¥ º v ¥ W0° b2A e ¦8o _0° ° r S c Ñ 3 X E Z W I 8 ¶ ¡ º.

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2 The Model We shall place ourselves in the standard It^oprocess model for a flnancial market which goes back to Samuelson (1965) This model contains n risky assets (stocks), with pricespershare driven by m independent Brownian motions as follows dXi(t) = Xi(t) bi(t)dt Xm ”=1 ¾i”(t)dW”(t);. Et = ¾t as before, but now let ¾2 t =!. Assumption (Kv) We have ¾t = §(Zt;Zt), where § is a C1 function on R2, and Zt and Zt are two adapted processes with the following properties (a) The process Z is an It^o semimartingale satisfying (H2) when v • 1=2, whereas when v > 1=2 it satisfles (H1=v) and its continuous martingale part vanishes.

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