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Quantitative Finance · Glosarium

Apa itu Simple process, Itô integral?

Dikenal juga sebagai: simple process · Itô integral

Definition 3.1 Quantitative Methods · Bab 3 — Itô Calculus

A simple process is Ht=∑k=0n−1hk1(tk,tk+1](t)H_t = \sum_{k=0}^{n-1}h_k\mathbf 1_{(t_k, t_{k+1}]}(t) with 0=t0<⋯<tn=T0 = t_0 < \dots < t_n = T and each hkh_k bounded and Ftk\mathcal F_{t_k}-measurable. Its Itô integral is ∫0tHs dWs=∑khk(Wtk+1∧t−Wtk∧t)\int_0^t H_s\,dW_s = \sum_k h_k(W_{t_{k+1}\wedge t} - W_{t_k\wedge t}). For an adapted HH with E∫0THs2 ds<∞\E\int_0^T H_s^2\,ds < \infty, the Itô integral is the L2L^2 limit of the integrals of simple processes H(n)H^{(n)} with E∫0T(Hs(n)−Hs)2 ds→0\E\int_0^T(H^{(n)}_s - H_s)^2\,ds \to 0.

Contoh

Example 3.3 (The integral of WW against itself)

On a partition of [0,T][0, T], ∑kWtkΔk=12∑k(Wtk+12−Wtk2)−12∑kΔk2\sum_k W_{t_k}\Delta_k = \tfrac12\sum_k(W_{t_{k+1}}^2 - W_{t_k}^2) - \tfrac12\sum_k\Delta_k^2. The first sum telescopes to 12WT2\tfrac12 W_T^2 and the second tends to 12T\tfrac12 T by Theorem 2.12:

∫0TWt dWt=12(WT2−T).\int_0^T W_t\,dW_t = \tfrac12\bigl(W_T^2 - T\bigr).

The right-point sums tend to 12(WT2+T)\tfrac12(W_T^2 + T), and the midpoint sums to 12WT2\tfrac12W_T^2, the ordinary calculus answer (Figure 3.1).

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