By Shaked M., Singpurwalla N. D.

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**Additional info for A Bayesian approach for quantile and response probability estimation with applications to reliability**

**Example text**

16. 8) satisfies 2 2 2 (k) 2 E((x(k) n ) + (yn ) ) = 1 + σ tn − σ tn h2 + tn · O(hk+1 ). 26) (k) Proof. 25). For details see [15]. 13), it is found that as h → 0 and k > 1, the leading error of the second moment of the numerical solution arising from the P (EC)k scheme tends to that from the midpoint rule. 18) as k = 4l + 1 and k = 4l + 2 (l ≥ 0). 17. 8) with x0 = 1, y0 = 0 will switch signs infinitely many times as n → ∞, almost surely, for k = 4l + 1 and k = 4l + 2 (l ≥ 0). Proof. See [15]. 18).

2. STOCHASTIC SYMPLECTIC INTEGRATION or equivalently, 37 2 ˜ n+1 = Xn (1 − σ h) . 12) is unapplicable because the denominator 1 − σ∆Wn can vanish for any small h, due to the unboundedness of ∆Wn for any small h. Milstein √ et. al. proposed the method √ of truncating ∆Wn to make it bounded. Model ∆Wn by hζh instead of the original hξ with ξ ∼ N (0, 1). 12) with order larger than or equal to 21 . The proof of this fact is based on the following two theorems. 11. 15) for arbitrary t0 ≤ t ≤ t0 + T − h and some K > 0, and 1 p2 ≥ , 2 1 p 1 ≥ p2 + .

3) imply ∂H ∂q ∂H ∂p ∂ q˙ ∂L ∂L ∂ q˙ ∂L − − =− , ∂q ∂q ∂ q˙ ∂q ∂q ∂ q˙ ∂L ∂ q˙ = q˙T + pT − = q˙T . 2). 8) t0 among all curves q(t) that connect two given points q(t0 ) = q0 and q(t1 ) = q1 . 2) has the name, the Euler-lagrange equation for the functional S. 2 (Hamilton’s Principle). 8). Proof. 2). 6) preserves the natural symplectic structure of the phase space of a mechanical problem, which is the characteristic property of a Hamiltonian system. To explain it, we first introduce the definition of a symplectic mapping.

### A Bayesian approach for quantile and response probability estimation with applications to reliability by Shaked M., Singpurwalla N. D.

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