Locally Lipschitz BSDE driven by a continuous martingale: path-derivative approach

2016/06/13 by Nam, Kihun
#60H07 #60H10 #93E20 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1606.03836

Abstract

Using a new notion of path-derivative, we study well-posedness of backward stochastic differential equation driven by a continuous martingale M when f(s,γ,y,z) is locally Lipschitz in (y,z): Y_t=ξ(M_[0,T])+∫_t^Tf(s,M_[0,s],Y_s-,Z_sm_s)d\rm tr[M,M]_s-∫_t^TZ_sdM_s-N_T+N_t Here, M_[0,t] is the path of M from 0 to t and m is defined by [M,M]_t=∫₀^tm_sm_s^*d\rm tr[M,M]_s. When the BSDE is one-dimensional, we could show the existence and uniqueness of solution. On the contrary, when the BSDE is multidimensional, we show existence and uniqueness only when [M,M]_T is small enough: otherwise, we provide a counterexample that has blowing-up solution. Then, we investigate the applications to utility maximization problems.

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