Struct DefaultTimeMoments 

struct DefaultTimeMoments<R: Scalar> {
    shift: R,
    lift: R,
    intensity_mean: R,
    stein: R,
}

The rate–credit moments for default at s with the gap read at t ≥ s, each Cov(x_s, ·) carried to t along the factor’s mean reversion. With c₁ = Cov(x_s, ∫₀ˢy), c₂ = Cov(x_s, y_s) and B = B_a(s, t):

E[D_t·λ_s·S_s·g(x_t)] = P(0,t)·S(s)·lift·((h(s) + intensity_mean)·E[g] + stein·E[g′]), the expectations with the factor’s mean moved by −shift from its forward-measure mean, where

• shift = Cov(x_t, ∫₀ˢy) = e^{−a(t−s)}·c₁;
• lift = exp(Cov(∫ₛᵗx, ∫₀ˢy)) = exp(B·c₁), the discount past s against the survival to it;
• intensity_mean = Cov(x_s, ∫₀ˢy) − Cov(y_s, ∫ₛᵗx) = c₁ − B·c₂, what the measure adds to λ_s over the market hazard;
• stein = Cov(x_t, y_s) = e^{−a(t−s)}·c₂.

At s = t it is the survival-forward measure at t and Stein’s lemma.

Fields

shift: R

lift: R

intensity_mean: R

stein: R

Implementations

impl<R: Scalar> DefaultTimeMoments<R>

fn at( params: &HullWhiteParameters<R>, intensity: &GaussianIntensity<R>, s: f64, t: f64, ) -> Self