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Self financing portfolio

A self-financing portfolio is characterised by that all trades are financed by selling or buying assets in the portfolio. No money is withdrawn or inserted after the intial forming of the portfolio. Let {S(t)}, t≥0 be an N-dimensional price process.

1. A portfolio {h(t)}, t≥0 is an N-dim adapted process.
2. The corresponding value process {V^h(t)}, t≥0 is given by
   V^h(t)=∑ⁿ_i=1 hⁱ(t)Sⁱ(t)
3. A portfolio is self-financing if
   V^h(t+Δ)-V^h(t)=∑ⁿ_i=1 hⁱ(t)(Sⁱ(t+Δ)-Sⁱ(t)) (discrete time)
   dV^h(t)=∑ⁿ_i=1 hⁱ(t)dSⁱ(t) (continuous time)