Limit theorems for BSDE with local time applications to non-linear PDE

Given a d -dimensional Wiener process W , with its natural filtration F t , a F T -measurable random variable ξ in R , a bounded measure x on R , and an adapted process ( s , y , z ) M h ( s , y , z ), we consider the following BSDE: Y t = ξ + Z t T h ( s , Y s , Z s ) d s + Z R ( L T a ( Y ) m L t a ( Y )) x (d a ) m Z t T Z s d W s for 0 h t h T . Here L t a ( Y ) stands for the local time of Y at level a . For h =0, we establish the existence and the uniqueness of the processes ( Y , Z ), and if h is continuous with linear growth we establish the existence of a solution. We prove limit theorems for solutions of backward stochastic differential equations of the above form. Those limit theorems permit us to deduce that any solution of that equation is the limit, in a strong sense, of a sequence of semi-martingales, which are solutions of ordinary BSDEs of the form Y t = ξ + Z t T f ( Y s ) Z s 2 d s m Z t T Z s d W s . A comparison theorem for BSDEs involving measures is discussed. As an application we obtain, with the help of the connection between BSDE and PDE, some corresponding limit theorems for a class of singular non-linear PDEs and a new probabilistic proof of the comparison theorem for PDEs.

Keywords:

• Backward Stochastic Differential Equation
• Local Time
• Comparison Theorem
• Singular Partial Differential Equations