Stationary distribution of the stochastic theta method for nonlinear stochastic differential equations

Abstract

The existence and uniqueness of the stationary distribution of the numerical solution generated by the stochastic theta method are studied. When the parameter theta takes different values, the requirements on the drift and diffusion coefficients are different. The convergence of the numerical stationary distribution to the true counterpart is investigated. Several numerical experiments are presented to demonstrate the theoretical results.

Appendix

Proof of Lemma 3.4

Denote

$$ \begin{array}{@{}rcl@{}} \theta^{*}=1+\frac{K_{1}}{4K_{2}}, ~ \lambda=\frac{2K_{2}+K_{1}}{2K_{2}}\wedge(2\theta-1). \end{array} $$

When 𝜃 ∈ [1/2,𝜃^∗], using Lemma 3.2 with λ₁ = (1 − 𝜃)h and λ₂ = (1 − 𝜃 + λ)h we have

$$ \begin{array}{@{}rcl@{}} &&\left|(X_{k+1}^{x}-X_{k+1}^{y})-(1-\theta+\lambda)h(f(X_{k+1}^{x})-f(X_{k+1}^{y}))\right|^{2} \\ &\leq &\left|(X_{k+1}^{x}-X_{k+1}^{y})-\theta h(f(X_{k+1}^{x})-f(X_{k+1}^{y}))\right|^{2}\\ &&+2(2\theta-1-\lambda)h\langle X_{k+1}^{x}-X_{k+1}^{y},f(X_{k+1}^{x})-f(X_{k+1}^{y})\rangle\\ &\leq &\left|({X_{k}^{x}}-{X_{k}^{y}})-(1-\theta)h(f({X_{k}^{x}})-f({X_{k}^{y}}))\right|^{2}\\ &&+4(1-\theta)h\langle {X_{k}^{x}}-{X_{k}^{y}},f({X_{k}^{x}})-f({X_{k}^{y}})\rangle+|g({X_{k}^{x}})-g({X_{k}^{y}})|^{2}h\\ &&+2(2\theta-1-\lambda)h\langle X_{k+1}^{x}-X_{k+1}^{y},f(X_{k+1}^{x})-f(X_{k+1}^{y})\rangle+M_{k}\\ &\leq&\left|\frac{1-K_{2}(1-\theta)h}{1-K_{2}(1-\theta+\lambda)h}\right|^{2}\left|({X_{k}^{x}}-{X_{k}^{y}})-(1-\theta+\lambda)h(f({X_{k}^{x}})-f({X_{k}^{y}}))\right|^{2}\\ &&+4(1-\theta)h\langle {X_{k}^{x}}-{X_{k}^{y}},f({X_{k}^{x}})-f({X_{k}^{y}})\rangle+|g({X_{k}^{x}})-g({X_{k}^{y}})|^{2}h\\ &&+2(2\theta-1-\lambda)h\langle X_{k+1}^{x}-X_{k+1}^{y},f(X_{k+1}^{x})-f(X_{k+1}^{y})\rangle+M_{k}, \end{array} $$

where

$$ \begin{array}{@{}rcl@{}} M_{k}&=&2\langle({X_{k}^{x}}-{X_{k}^{y}})+(1-\theta)h(f({X_{k}^{x}})-f({X_{k}^{y}})),g({X_{k}^{x}})-g({X_{k}^{y}})\rangle{\Delta} B_{k}\\ &&+|g({X_{k}^{x}})-g({X_{k}^{y}})|^{2}(|{\Delta} B_{k}|^{2}-h). \end{array} $$

It is clear that $\mathbb {E}M_{k}=0$. Denote

$$W_{k}=\mathbb{E}\left|({X_{k}^{x}}-{X_{k}^{y}})-(1-\theta)h(f({X_{k}^{x}})-f({X_{k}^{y}}))\right|^{2},$$

and

$$L_{h}=\frac{(1-K_{2}(1-\theta)h)^{2}}{(1-K_{2}(1-\theta+\lambda)h)^{2}}.$$

By Conditions 2.5 and 2.5, we have

$$ \begin{array}{@{}rcl@{}} W_{k+1}&\leq& L_{h} W_{k}+[4(1-\theta)K_{2}h+K_{1} h]\mathbb{E}|{X_{k}^{x}}-{X_{k}^{y}}|^{2}\\ &&+2K_{2} h(2\theta-1-\lambda)\mathbb{E}|X_{k+1}^{x}-X_{k+1}^{y}|^{2}\\ &\leq&L_{h}^{k+1}[W_{0}-2(2\theta{\kern1.7pt}-{\kern1.7pt}1-\lambda)K_{2} h\mathbb{E}|x-y|^{2}]\!+\varphi_{\lambda}(\theta)h\sum\limits_{i=0}^{k}L_{h}^{k-i}\mathbb{E}|{X_{i}^{x}}-{X_{i}^{y}}|^{2}, \end{array} $$

where

$$\varphi_{\lambda}(\theta)=4(1-\theta)K_{2}+K_{1}+2(2\theta-1-\lambda)K_{2}L_{h}.$$

For 𝜃 ∈ [1/2,𝜃^∗], we have $\varphi _{\lambda }^{h}<0$. Since $\mathbb {E}|{X_{k}^{x}}-{X_{k}^{y}}|^{2}\leq W_{k}$ and L_h ∈ (0, 1), we obtain

$$ \mathbb{E}|{X_{k}^{x}}-{X_{k}^{y}}|^{2}\leq c_{3}, $$

where

$$ c_{3} := {L_{h}^{k}}[W_{0}-2(2\theta-1-\lambda)K_{2}h|x-y|^{2}]. $$

Therefore, the assertion holds for 𝜃 ∈ [1/2,𝜃^∗].

For 𝜃 ∈ (𝜃^∗, 1], choosing λ^′ < λ sufficiently small such that $\psi _{\lambda ^{\prime }}(\theta )<0$ for any h > 0, the assertion can be proved using the similar arguments above. □

Proof of Lemma 3.9

From (2.2), by applying Condition 2.5, we have

$$ \begin{array}{@{}rcl@{}} |X_{k+1}^{x}{\kern1.7pt}-{\kern1.7pt}X_{k+1}^{y}|^{2}&{\kern1.7pt}={\kern1.7pt}&\langle X_{k+1}^{x}-X_{k+1}^{y},{X_{k}^{x}}-{X_{k}^{y}}+(1-\theta)[f({X_{k}^{x}})-f({X_{k}^{y}})]h\\ &&\!+[g({X_{k}^{x}}){\kern1.7pt}-{\kern1.7pt}g({X_{k}^{y}})]{\Delta} B_{k}\rangle{\kern1.7pt}+{\kern1.7pt}\langle X_{k+1}^{x}{\kern1.7pt}-{\kern1.7pt}X_{k+1}^{y},f(X_{k+1}^{x}){\kern1.7pt}-{\kern1.7pt}f(X_{k+1}^{y})\rangle\theta h\\ &\!\leq\!&\frac{1}{2}|X_{k+1}^{x}-X_{k+1}^{y}|^{2}+\frac{1}{2}|{X_{k}^{x}}-{X_{k}^{y}}+(1-\theta)[f({X_{k}^{x}})-f({X_{k}^{y}})]h\\ &&+[g({X_{k}^{x}})-g({X_{k}^{y}})]{\Delta} B_{k}|^{2}+\theta hK_{2}|X_{k+1}^{x}-X_{k+1}^{y}|^{2} \\ &\!\leq\! & \left( \frac{1}{2}-K_{2}\theta h \right)^{-1} \left( \frac{1}{2}|{X_{k}^{x}}{\kern1.7pt}-{\kern1.7pt}{X_{k}^{y}}|^{2}+\frac{1}{2}(1{\kern1.7pt}-{\kern1.7pt}\theta)^{2}|f({X_{k}^{x}}){\kern1.7pt}-{\kern1.7pt}f({X_{k}^{y}})|^{2}h^{2}\right.\\ &&\!+\frac{1}{2}|g({X_{k}^{x}})-g({X_{k}^{y}})|^{2} |{\Delta} B_{k}|^{2}\\ &&\!\left.+(1-\theta)h\langle {X_{k}^{x}}-{X_{k}^{y}},f({X_{k}^{x}})-f({X_{k}^{y}})\rangle +Q_{1} \vphantom{\frac{1}{2}}\right), \end{array} $$

where

$$ Q_{1}=\langle ({X_{k}^{x}}-{X_{k}^{y}}) + (1-\theta)h f({X_{k}^{x}})-f({X_{k}^{y}}) ,g({X_{k}^{x}})-g({X_{k}^{y}})\rangle{\Delta} B_{k}. $$

Since $\mathbb {E}Q_{1}=0$, taking expectations on both sides and using Conditions 2.4, 2.5, and 3.6 yield

$$ \begin{array}{@{}rcl@{}} \mathbb{E} |X_{k+1}^{x}-X_{k+1}^{y}|^{2}\leq&\frac{1+(1-\theta)^{2}h^{2}K_{1}+hK_{1}+2K_{2}(1-\theta)h}{1-2K_{2}\theta h}|{X_{k}^{x}}-{X_{k}^{y}}|^{2}. \end{array} $$

By iterations, we have

$$ \mathbb{E}|X_{k+1}^{x}-X_{k+1}^{y}|^{2}\leq \bar{C}_{3}^{k+1}\mathbb{E}|x-y|^{2}, $$

where

$$ \bar{C}_{3}=\frac{1+(1-\theta)^{2}h^{2}K_{1}+hK_{1}+2K_{2}(1-\theta)h}{1-2K_{2}\theta h}. $$

Since h < −(2K₂ + K₁)/((1 − 𝜃)²K₁) and 1 − 2K₂𝜃h > 1, we have $\bar {C}_{3} \in (0,1)$. Let $C_{3} = \bar {C}_{3}^{k+1}$, the proof is completed. □

Proof of Lemma 3.10

By the elementary inequality and Condition 2.6, we derive from (2.2) that

$$ \begin{array}{@{}rcl@{}} |X_{k+1}|^{2}&=&\langle X_{k+1}, X_{k}+(1-\theta) f(X_{k})h+g(X_{k}){\Delta} B_{k} \rangle+\langle X_{k+1}, \theta f(X_{k+1})h \rangle\\ &\leq& \frac{1}{2}|X_{k+1}|^{2} \!+ \frac{1}{2}|X_{k}+(1{\kern1.7pt}-{\kern1.7pt}\theta) f(X_{k})h {\kern1.7pt}+{\kern1.7pt}g(X_{k}){\Delta} B_{k}|^{2} \!+ \theta h(\mu|X_{k+1}|^{2}\!+a). \end{array} $$

Applying Conditions 2.6, 2.7, and 3.7, we get

$$ \begin{array}{@{}rcl@{}} |X_{k+1}|^{2}&\leq & \frac{1}{2}\left( \frac{1}{2}-\theta h \mu\right)^{-1}[|X_{k}|^{2}+(1-\theta)^{2} h^{2} |f(X_{k})|^{2}+|g(X_{k})|^{2}|{\Delta} B_{k}|^{2}\\ &&+ 2 h (1-\theta)\langle X_{k},f(X_{k})\rangle+2 \langle X_{k},g(X_{k})\rangle {\Delta} B_{k}\\ &&+2 (1-\theta) h \langle f(X_{k}),g(X_{k})\rangle {\Delta} B_{k}]+\left( \frac{1}{2}-\theta h \mu\right)^{-1}\theta h a\\ &\leq&D_{1}|X_{k}|^{2}+D_{2}|g(X_{k})|^{2}|{\Delta} B_{k}|^{2}+D_{3}, \end{array} $$

where

$$ D_{1}= \frac{1+\frac{1}{2}\kappa h^{2}(1-\theta)^{2}+h\mu(1-\theta)+\frac{1}{2}\kappa h(1-\theta)}{\frac{1}{2}-h\mu\theta}, $$

$$ D_{2}= \frac{1+\frac{1}{2} h (1-\theta)}{\frac{1}{2}-h \mu \theta}, $$

and

$$ D_{3}= \frac{\frac{1}{2}h(1-\theta)c+\frac{1}{2}ch^{2}(1-\theta)^{2}+ah(1-\theta)+ah\theta}{\frac{1}{2}-h\mu\theta}. $$

Summarizing both sides yields

$$ \begin{array}{@{}rcl@{}} \sum\limits_{i=1}^{k+1}|X_{i}|^{2}=D_{1}\sum\limits_{i=0}^{k}|X_{i}|^{2}+D_{2}\sum\limits_{i=0}^{k}|g(X_{i})|^{2}|{\Delta} B_{i}|^{2}+(k+1)D_{3}. \end{array} $$

Now, we have

$$ \begin{array}{@{}rcl@{}} |X_{k+1}|^{2}=(D_{1}-1)\sum\limits_{i=0}^{k}|X_{i}|^{2}+|X_{0}|^{2}+D_{2}\sum\limits_{i=0}^{k}|g(X_{i})|^{2}|{\Delta} B_{i}|^{2}+(k+1)D_{3}. \end{array} $$

Taking the supreme and expectation on both sides gives

$$ \begin{array}{@{}rcl@{}} \mathbb{E} \left( \sup\limits_{0 \leq k \leq n} |X_{k}|^{2} \right) &\leq&(D_{1}-1)\sum\limits_{i=0}^{k}\mathbb{E} \left( \sup\limits_{0 \leq k \leq n} |X_{i}|^{2} \right)\\ &&+D_{2}\mathbb{E} \left( \sup\limits_{0 \leq k \leq n}(\sum\limits_{i=0}^{k}|g(X_{i})|^{2}|{\Delta} B_{i}|^{2})\right)+(n+1)D_{3}+|X_{0}|^{2}. \end{array} $$

Now, we obtain

$$ \begin{array}{@{}rcl@{}} \mathbb{E} \left( \sup\limits_{0 \leq k \leq n} |X_{k}|^{2} \right) &\leq&(D_{1}-1+D_{2}h^{2}\sigma)\sum\limits_{j=0}^{n}\mathbb{E} \left( \sup\limits_{0 \leq i \leq j} |X_{i}|^{2} \right)\\ &&+D_{2}h^{2}nb+(n+1)D_{3}+|X_{0}|^{2}, \end{array} $$

where $\mathbb {E} |{\Delta } B_{k}|^{2} = h$ is used. By Lemma 2.10, we have

$$ \mathbb{E} \left( \sup\limits_{0 \leq k \leq n} |X_{k}|^{2} \right) \leq (D_{2}h^{2}kb+(k+1)D_{3}+|X_{0}|^{2})\exp ((k+1)(D_{1}-1+D_{2}h^{2}\sigma)). $$

The proof is completed. □