By Ewald C.-O.

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**Additional resources for Mathematical finance: continuous time**

**Sample text**

A sequence X k of elementary processes, such that X k Proof. As before we define an increasing sequence of stopping times as σn := inf{t : |Xt | ≥ n}. We have limn→∞ σn = ∞ and hence =0 lim P( sup |Xsσn − Xs | ≥ ǫ) ≤ n→∞ 0≤s≤t lim P( sup |Xsσn − Xs | ≥ ǫ) n→∞ 0≤s≤σn =0 + ≤ lim P( sup |Xsσn − Xs | ≥ ǫ) n→∞ σn ≤s≤t lim P(σn ≤ t) = 0 n→∞ ucp / X and all X n are bounded processes. Hence we can Hence X n assume in the following that X is also bounded. Define the rcll process Z as Z = X+ . For ǫ > 0 define a sequence of stopping times ǫ τ0ǫ := 0, τn+1 := inf{t|t > τnǫ and |Zt − Zτnǫ | > ǫ} ∀ n.

2. n ) = ∅. A fundamental theorem of asset pricing in this context is some kind of theorem which asserts that some class of financial markets satisfies the fundamental law of asset pricing. Most of the results so far have been established in the context of semi-martingales ( see [Delbaen/Schachermayer] [Stricker] ). In the context presented here, this is still up to further research. Conjecture 1. Fundamental Theorem of Asset Pricing Let Mm,n = ((Xt , Ft )t∈I , Φ) be a financial market where Φ is an ample cone, then the corresponding market Mm,n = ((Xt , Ft )t∈I , Φa ) satisfies the fundamental law of asset pricing.

If he does this by investing in the financial market, then we speak of hedging. 1. Let Mm,n = ((Xt , Ft )t∈I , Φ) be a financial market and g : Ω → R be a contingent claim. A trading strategy ϕ ∈ Φ is called a 1. s. 2. s. Often we loosely speak of a hedge respectively super hedge, meaning a hedging strategy respectively super hedging strategy. If there exists 36 a hedging strategy for g then g is called Φ-attainable. If there exists a super hedging strategy for g then g is called Φ-super attainable.