A stochastic approach to path-dependent nonlinear Kolmogorov equations via BSDEs with time-delayed generators and applications to finance

摘要

We prove the existence of a viscosity solution of the following pathdependent nonlinear Kolmogorov equation: [ egin{cases}partial_{t}u(t,phi)+mathcal{L}u(t,phi)+f(t,phi,u(t,phi),partial_{x}u(t,phi)sigma(t,phi),(u(cdot,phi)) _{t})=0,;tin[0,T),;phiinmathbb{Lambda},, u(T,phi)=h(phi),;phiinmathbb{Lambda}, end{cases} ] where$mathbb{Lambda}=mathcal{C}([0,T];mathbb{R}^{d})$, $(u(cdot,phi))_{t}:=(u(t+heta,phi))_{hetain[-delta,0]}$ and [mathcal{L}u(t,phi):=langle b(t,phi),partial_{x}u(t,phi)angle+dfrac{1}{2}mathrm{Tr}ig[sigma(t,phi)sigma^{st}(t,phi)partial_{xx}^{2}u(t,phi)ig]. ] The result is obtained by a stochastic approach. Inparticular we prove a new type of nonlinear Feynman-Kac representation formulaassociated to a backward stochastic differential equation with time-delayedgenerator which is of non-Markovian type. Applications to the large investor problem and risk measures via$g$-expectations are also provided.