Stochastic Finance by Hans Föllmer Alexander Schied

Author:Hans Föllmer,Alexander Schied
Language: eng
Format: epub
Publisher: De Gruyter
Published: 2016-02-29T16:00:00+00:00

Proof. By dividing both sides of (5.3) by Sit is seen that condition (b) is a reformulation of Definition 5.4. Moreover, (b) holds if and only if

for t = 1, . . . , T − 1, and this identity is equivalent to (c).

Remark 5.8. The numéraire component of a self-financing trading strategy satisfies

Since

the entire process ξ0 is determined by the initial investment V0 and the d-dimensional process ξ. Consequently, if a constant V0 and an arbitrary d-dimensional predictable process ξ are given, then we can use (5.7) and (5.6) as the definition of a predictable process ξ0, and this construction yields a self-financing trading strategy := (ξ0, ξ). In dealing with self-financing strategies , it is thus sufficient to focus on the initial investment V0 and the d-dimensional processes X and ξ.

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Remark 5.9. Different economic agents investing into the same market may choose different numéraires. For example, consider the following simple market model in which prices are quoted in euros (€) as the domestic currency. Let S0 be a locally riskless €-bond with the predictable spot rate process r0, i.e.,

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