Probabilistic representation for solution of some coupled system of quasilinear parabolic PDEs

• 1、School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023

Abstract：In this paper, we obtain a probabilistic representation for thesolution of the following coupled system of quasilinear parabolicPDEs:egin{equation*}left{egin{tabular}{ll}$partial_t u^0+ b u_x^0+rac{1}{2}sigma^2 u_{xx}^0+(Deltau-delta u_x^0)gamma_t+f(t, x, u^0, u_x^0 sigma, Delta u)=0,$\$partial_t u^1+ b u_x^1+rac{1}{2}sigma^2 u_{xx}^1+f(t, x, u^1,u_x^1sigma, Delta u)=0,$\$u^0(T, x)=arphi(0, x)in mathbb{R},$\$u^1(T,x)=arphi(1, x)in mathbb{R},$end{tabular} ight.end{equation*}where $Delta u(t, x)=u^1(t, x+delta(t, x))-u^0(t, x)$ and $b$,$sigma$, $delta$ are $mathbb{R}$-valued functions defined on $[0,T] imes mathbb{R}$, by introducing a new kind of backwardstochastic differential equation, called BSDE with random defaulttime.

keywords： probability theory backward stochastic differential equation random default time coupled system of quasilinear parabolic PDEs probabilisticrepresentation.

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