Software Systems Scientific Computing

## Maple 11 Quick Reference Card

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E. measurable under Ft−1 ). 2 T and (st0 , st1 , . . , std ) form The probability space ( , F, P), the ﬁltration {Ft }t=1 what we call the market model . 3 d or also vt (φ) = The portfolio value in t is deﬁned by {sti φti }i=1 d i=1 φti sti . 4 We say that the market is viable (arbitrage-free) if, given φ1t = (φt0 , φt1 , . . , φtd ) and φ2t = (φt0 , φt1 , . . , φtd ), we have v1t (φ) = v2t (φ) for all t. 5 We deﬁne a numeraire as an increasing stochastic process st0 = (st0 ; t = 0, 1, . .

Another important law in probability theory is the so-called Borel LLN. Assume that Sn is the partial sum of the process. 3). 3) involves convergence n X i (s). e. e. e. 1/2 ); the larger the number of repetitions, the better the approximation tends to be. 4) the conditional expectation of X given the information on the random variable Y . It is obvious that, in this case, σ (Y ) does not carry more information than Y . 1 Simply, we can say that E(X ) = E(E(X |σ (Y ))) = X . This result will be clear later after discussing the rules for conditional expectations.

2 Consider the stochastic process f (t, X (t)), where X (t) satisﬁes t t A1 (s)ds + X (t) = X (0) + 0 A2 (s)dB(s) 0 Find t f (t, X (t))dX (t) 0 9 10 Lower case indicates derivatives. Therefore the geometric Brownian motion considered at the beginning is an Ito process. 1 THE RIEMANN–STIELJES INTEGRAL Deﬁne the following partition 0 = t0 < t1 < . . 1) 0 where f (t) is a deterministic function, say, for example, f (t) = σ . 2) 52 The Mathematics of Derivatives Securities The mean value theorem justiﬁes, in this case, the convergence of the partial sums above to the integral, regardless of the position of f (tk∗ ) within the interval tk ≤ tk∗ ≤ tk+1 .