By P. S. Milojevic
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Sample text
Assume first that fn is an elementary symmetric function. ,jn W (Aj1 ) · · · 1Ajl (t) · · · W (Ajn ) = nIn−1 fn (·, t) . For a general fn ∈ L2 (An ), the result follows easily by an obvious approximation argument. We recall that for F ∈ D1,2 , the derivative DF belongs to L2 (Ω; H). L2 (Ω × A); thus DF is a In the setting of this section L2 (Ω; H) function of two variables, ω ∈ Ω and t ∈ A. As usual, we shall not write the dependence on ω. We note DF (t) = Dt F . Finally we study the divergence operator .
For example, bonds belong to the first type and stocks to the second one. The price dynamics for safe investments is given by dA(t) = ρ(t)A(t) dt, where ρ(t) is the interest rate at time t. s. s. We assume that the coefficients in the above equations are adapted stochastic processes satisfying the appropriate conditions ensuring existence and uniqueness of solution. © 2005, First edition, EPFL Press Representation of Wiener Functionals 55 A portfolio consists of a random number of assets of each type — safe and risky — and this number varies with time.
Let u be an H-valued random variable of the form n Fj hj , u= j=1 with Fj ∈ Sb . We notice that the duality relation between D and δ holds for F ∈ D1,1 ∩ L∞ (Ω) and u of the kind described above. Moreover, u is total in L1 (Ω; H), that is, if v ∈ L1 (Ω; H) satisfies E v, u H = 0 for every u in the class, then v = 0. Therefore, E ϕ (F ) DF, u H = E D(Ψ (F ) , u = E Ψ (F )δ(u) © 2005, First edition, EPFL Press ≤ H ϕ ∞E δ(u) . 40 Local property of the operators Taking limits as tends to zero, we obtain E 1(F =0) DF, u H = 0.
