Existence of solution for stochastic differential equations driven by G-Lévy process with discontinuous coefficients

Abstract

Abstract The existence theory for the vector-valued stochastic differential equations driven by G-Brownian motion and pure jump G-Lévy process (G-SDEs) of the type d Y t = f ( t , Y t ) d t + g j , k ( t , Y t ) d 〈 B j , B k 〉 t + σ i ( t , Y t ) d B t i + ∫ R 0 d K ( t , Y t , z ) L ( d t , d z ) $dY_{t}=f(t,Y_{t})\, dt+g_{j,k}(t,Y_{t})\, d\langle B^{j},B^{k}\rangle _{t}+\sigma_{i}(t,Y_{t}) \, dB^{i}_{t}+\int _{R_{0}^{d}}K(t,Y_{t},z)L(dt,dz)$ , t ∈ [ 0 , T ] $t\in[0,T]$ , with first two and last discontinuous coefficients, is established. It is shown that the G-SDEs have more than one solution if the coefficients f, g, K are discontinuous functions. The upper and lower solution method is used.

Keywords

• stochastic differential equations
• G-Lévy process
• upper and lower solution
• discontinuous coefficients