By Arutyunov A.V., Jacimovic V.

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**Additional info for 2-Normal Processes in Controlled Dynamical Systems**

**Sample text**

So far, we have not insisted that a self-ﬁnancing strategy must at all times yield non-negative total wealth; that is, that Vt (θ) ≥ 0 for all t ∈ T. From now on, when we impose this additional restriction, we call such self-ﬁnancing strategies admissible; they deﬁne the class Θa . , have θti < 0 for some values of i and t), the overall value process must remain non-negative for each t. But the additional restriction has little impact on the mathematical modelling, as we show shortly. We use the class Θa to deﬁne our concept of ‘free lunch’.

Indeed, St = β T −t [EQ (ST − K)+ − EQ (K − ST )+ + K] = β T −t {ST ≥K} = β T −t Ω (ST − K)dQ − {ST

S. for u > t. 2. TRADING STRATEGIES 33 We amend θ to a new strategy φ by setting φu (ω) = 0 for all u ∈ T and ω ∈ Ω \ A, while on A we set φu (ω) = 0 if u ≤ t, and for u > t we deﬁne φ0u (ω) = θu0 (ω) − θ t · St i , φ (ω) = θui (ω) for i = 1, 2, . . , d. St0 (ω) u This strategy is obviously predictable. 5). We observe that φit = 0 on Ac for i ≥ 0 and that, on A, 0 ∆φ0t+1 = φ0t+1 = θt+1 − θ t · St i , ∆φit+1 = θt+1 for i = 1, 2, . . , d. St0 Hence (∆φt+1 ) · St = 1A (θt+1 · St − θt · St ) = 1A (θt · St − θt · St ) = 0 since θ is self-ﬁnancing.

### 2-Normal Processes in Controlled Dynamical Systems by Arutyunov A.V., Jacimovic V.

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