オルンシュタイン＝ウーレンベック過程
cov  ( r s , r t ) = E [ ( r s − E [ r s ] ) ( r t − E [ r t ] ) ] = E [ ∫ 0 s σ e θ ( u − s ) d W u ∫ 0 t σ e θ ( v − t ) d W v ] = σ 2 e − θ ( s + t ) E [ ∫ 0 s e θ u d W u ∫ 0 t e θ v d W v ] = σ 2 2 θ e − θ ( s + t ) ( e 2 θ ( s ∧ t ) − 1 ) {\displaystyle {\begin{aligned}\operatorname {cov} (r_{s},r_{t})&=\mathrm {E} [(r_{s}-\mathrm {E} [r_{s}])(r_{t}-\mathrm {E} [r_{t}])]\\&=\mathrm {E} \left[\int _{0}^{s}\sigma e^{\theta (u-s)}\,dW_{u}\int _{0}^{t}\sigma e^{\theta (v-t)}\,dW_{v}\right]\\&=\sigma ^{2}e^{-\theta (s+t)}\mathrm {E} \left[\int _{0}^{s}e^{\theta u}\,dW_{u}\int _{0}^{t}e^{\theta v}\,dW_{v}\right]\\&={\frac {\sigma ^{2}}{2\theta }}\,e^{-\theta (s+t)}(e^{2\theta (s\wedge t)}-1)\end{aligned}}}