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Lamperti drift-implicit CIR scheme converges on finite horizons in probability

Statement

Assume AC, $κ, θ, σ > 0$ , $2 κ θ \geq σ^{2}$ , and deterministic $X_{0} = x_{0} > 0$ . Let $X$ be the positive CIR solution from CIR paths stay positive and do not explode under the Feller condition. For $h = T / N$ , initialize $Z_{0} = \sqrt{x_{0}}$ and apply Positive Lamperti drift-implicit scheme for the CIR process to obtain $Z_{n}$ and $X_{n}^{h} = Z_{n}^{2}$ . Then for every $T > 0$ , $\max_{0 \leq n \leq N} ∣ X_{t_{n}} - X_{n}^{h} ∣ ⟶ 0 in probability as N \to \infty .$ This is convergence in probability on each finite horizon; no $L^{p}$ error rate is asserted.

Facts & Assumptions

Given: AC; positive CIR parameters and initial state; the exact positive solution and the transformed drift-implicit algorithm.

[F1]

The positive transformed solution $Z = \sqrt{X}$ satisfies $d Z = q (Z) d t + c d W$ , where $q (z) = a / z - κ z / 2$ , $a > 0$ , $c = σ / 2$ ; the algorithm returns the unique positive root of $z_{n + 1} = z_{n} + h q (z_{n + 1}) + c Δ W_{n}$ . CIR Lamperti equation and unique positive implicit root Positive Lamperti drift-implicit scheme for the CIR process

[F2]

The exact CIR solution is global, continuous, and strictly positive on each finite horizon. CIR paths stay positive and do not explode under the Feller condition

[F3]

Brownian increments over deterministic intervals have centered normal law with variance equal to interval length. $d$ -dimensional Brownian motion Standard normal and normal laws

[F4]

For a nonnegative random variable $U$ and $r > 0$ , $P (U > r) \leq E U / r$ . Markov's inequality for random variables

[A1]

AC is inherited through the Brownian and numerical inputs; no additional choice is used. The Axiom of Choice

Proof

Given: The hypotheses and suppliers in the Statement and Facts.

1.1F2given

The exact path $Z = \sqrt{X}$ is continuous and strictly positive on the compact interval $[0, T]$ by [F2]. Hence its pathwise minimum is positive and its maximum is finite. For every $η > 0$ choose $0 < ε < R$ so that the event $E_{ε, R} = {ε \leq Z_{t} \leq R for every t \leq T}$ has probability at least $1 - η$ ; these events increase to a probability-one event as $ε ↓ 0$ and $R ↑ \infty$ .

1.2F1givenalgebra

Write $c = σ / 2$ and $r_{n + 1} = \int_{t_{n}}^{t_{n + 1}} (q (Z_{s}) - q (Z_{t_{n + 1}})) d s .$ The exact SDE and the implicit recursion give $Z_{t_{n + 1}} = Z_{t_{n}} + h q (Z_{t_{n + 1}}) + c Δ W_{n} + r_{n + 1}, Z_{n + 1} = Z_{n} + h q (Z_{n + 1}) + c Δ W_{n} .$ Let $e_{n} = Z_{n} - Z_{t_{n}}$ . Since $q^{'} (z) = - a / z^{2} - κ / 2 \leq - κ / 2$ , the mean-value theorem gives $(u - v) (q (u) - q (v)) \leq - κ ∣ u - v ∣^{2} / 2$ for positive $u, v$ . Subtract the two updates and take the inner product with $e_{n + 1}$ : $(1 + h κ / 2) ∣ e_{n + 1} ∣^{2} \leq ∣ e_{n + 1} ∣ ∣ e_{n} - r_{n + 1} ∣ .$ If $e_{n + 1} = 0$ the next bound is immediate; otherwise divide by $(1 + h κ / 2) ∣ e_{n + 1} ∣$ to get $∣ e_{n + 1} ∣ \leq ∣ e_{n} ∣ + ∣ r_{n + 1} ∣$ . As $e_{0} = 0$ , $\max_{n \leq N} ∣ e_{n} ∣ \leq \sum_{n < N} ∣ r_{n + 1} ∣$ .

2.1F1F3step 1.2givenalgebra

On $E_{ε, R}$ , the drift $q$ is bounded by $B$ and is Lipschitz with a constant $L$ . For $s \in [t_{n}, t_{n + 1}]$ , the exact transformed equation gives $∣ Z_{s} - Z_{t_{n + 1}} ∣ \leq B h + c ∣ W_{s} - W_{t_{n + 1}} ∣ .$ A centered normal variable of variance $v > 0$ has absolute first moment $\frac{2}{\sqrt{2 π v}} \int_{0}^{\infty} x e^{- x^{2} / (2 v)} d x = \sqrt{2 v / π}$ by direct integration of its density; at $v = 0$ both sides are zero. Thus, using $1_{E_{ε, R}} \leq 1$ (without assuming independence of this path event from the increment) and [F3], $E [1_{E_{ε, R}} ∣ r_{n + 1} ∣] \leq L \int_{t_{n}}^{t_{n + 1}} (B h + c \sqrt{2 (t_{n + 1} - s) / π}) d s \leq C_{ε, R} (h^{2} + h^{3 / 2}) .$ Summing over $n < N$ and using $N h = T$ gives $E [1_{E_{ε, R}} \sum_{n < N} ∣ r_{n + 1} ∣] \leq C_{ε, R, T} \sqrt{h}$ .

3.1A1F2F4step 1.1step 1.2step 2.1givenalgebra∎

For any $ρ > 0$ , steps 1.2–2.1 and [F4] give $P (\max_{n \leq N} ∣ e_{n} ∣ > ρ) \leq P (E_{ε, R}^{c}) + ρ^{- 1} C_{ε, R, T} \sqrt{h} .$ First let $N \to \infty$ with $ε, R$ fixed, then let $η \to 0$ using step 1.1. Thus $\max_{n} ∣ Z_{n} - Z_{t_{n}} ∣ \to 0$ in probability. On $E_{ε, R} \cap {\max_{n} ∣ e_{n} ∣ \leq 1}$ , $∣ Z_{n} + Z_{t_{n}} ∣ \leq 2 R + 1$ , and hence $\max_{n} ∣ X_{n}^{h} - X_{t_{n}} ∣ \leq (2 R + 1) \max_{n} ∣ Z_{n} - Z_{t_{n}} ∣ .$ The probability of the complementary event is arbitrarily small by the event construction in step 1.1 and the convergence just proved. This gives the stated convergence in probability for $X_{n}^{h} = Z_{n}^{2}$ . No $L^{p}$ rate follows from this localization argument. AC is exactly [A1].

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Sources

Aurélien Alfonsi, Strong convergence of some drift implicit Euler scheme. Application to the CIR process, equations (2)–(3) and the positive root formula (standard reference, not scraped)
Steffen Dereich, Andreas Neuenkirch and Lukasz Szpruch, An Euler-type method for the strong approximation of the Cox–Ingersoll–Ross process, Theorem 1.1 (standard reference, not scraped)