Table 2 Comparison of Lundberg exponent: exponential case. \((\theta _{1}, \theta_{1}^{R})=(0.2, 0.4)\) and \((\theta_{2}, \theta_{2}^{R})= (0.25, 0.5)\), values of Lundberg exponent and upper bound of ruin probability with different dependence parameters for Hu’s model and our model

\((\alpha_{11},\alpha_{12})\)

(0,1)

(0.2,0,8)

(0.4,0.6)

(0.6,0.4)

(0.8,0.2)

(1,0)

\((\alpha_{21},\alpha_{22})\)

(1,0)

(0.8,0.2)

(0.6,0.4)

(0.4,0.6)

(0.2,0.8)

(0,1)

\((\lambda_{1},\lambda_{2})\)

(4,2)

(3.6,2.4)

(3.2,2.8)

(2.8,3.2)

(2.4,3.6)

(2,4)

ρ

0

0.111111

0.160714

0.160714

0.111111

0

\(M^{*}_{1}\)

1.374693

1.426397

1.382661

1.292643

1.22716

1.286787

\(M^{*}_{2} \)

1.65657

1.679524

1.741289

1.766061

1.70904

1.55064

\(R_{H}^{*} \)

0.244762

0.201829

0.188391

0.191707

0.211816

0.261482

\(e^{-10R_{H}^{*}}\)

0.0865

0.1329

0.1520

0.1470

0.1203

0.0732

\(R^{*} \)

0.273025

0.228736

0.203113

0.202304

0.231029

0.271354

\(e^{-10R^{*}}\)

0.0652

0.1015

0.1312

0.1323

0.0992

0.0663