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Quantitative Finance · शब्दावली

Square-root process, Feller condition क्या है?

अन्य नाम: square-root process · Feller condition

Definition 4.7 Quantitative Methods · अध्याय 4 — Stochastic Differential Equations

The square-root process is dv=κ(vˉ−v) dt+ηv dWdv = \kappa(\bar v - v)\,dt + \eta\sqrt v\,dW with κ,vˉ,η>0\kappa, \bar v, \eta > 0 and v0>0v_0 > 0. The Feller condition is 2κvˉ≥η22\kappa\bar v \ge \eta^2; the ratio 2κvˉ/η22\kappa\bar v/\eta^2 is the Feller ratio.

Left: three Ornstein–Uhlenbeck paths from 0.5 with = 2 (half-life 0.35 years), x = 0, = 0.2, and their expected path 0.5e-2t (dashed). Right: three paths of the desk’s square-root process (= 2, v = 0.04, = 0.6), sampled exactly each day; with a Feller ratio of 0.44 they spend long spells near zero. Data: the chapter’s tutorial, seeded.
Figure 4.1. Left: three Ornstein–Uhlenbeck paths from 0.5 with κ=2\kappa = 2 (half-life 0.35 years), xˉ=0\bar x = 0, σ=0.2\sigma = 0.2, and their expected path 0.5e−2t0.5e^{-2t} (dashed). Right: three paths of the desk’s square-root process (κ=2\kappa = 2, vˉ=0.04\bar v = 0.04, η=0.6\eta = 0.6), sampled exactly each day; with a Feller ratio of 0.44 they spend long spells near zero. Data: the chapter’s tutorial, seeded.
Share of plain-Euler paths of the square-root process that produce a negative variance within one year, against the Feller ratio (vol-of-vol from 0.2 to 1.0). Refining the step does not remove the failures; exact and full-truncation schemes have none. Data: 20 000 and 10 000 seeded paths per point.
Figure 4.2. Share of plain-Euler paths of the square-root process that produce a negative variance within one year, against the Feller ratio (vol-of-vol η\eta from 0.2 to 1.0). Refining the step does not remove the failures; exact and full-truncation schemes have none. Data: 20 000 and 10 000 seeded paths per point.
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