Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider an Ito stochastic differential equation with delay, driven by Brownian motion, whose solution, by an appropriate reformulation, defines a Markov process $X$ with values in a space of continuous functions $\mathbf{C}$, with generator $\mathcal{L}$.  then consider a backward stochastic differential equation depending on $X$, with unknown processes $(Y,Z)$, and  study properties of the resulting system, in particular  identify the process $Z$ as a deterministic functional of $X$.  next prove that the forward-backward system provides a suitable solution to a class of parabolic partial differential equations on the space $\mathbf{C}$ driven by $\mathcal{L}$, and  apply this result to prove a characterization of the fair price and the hedging strategy for a financial market with memory effects.  also include applications to optimal stochastic control of differential equation with delay: in particular  characterize optimal controls as feedback laws in terms of the process $X$.
['Marco Fuhrman', 'Federica Masiero', 'Gianmario Tessitore']
I am writing a research paper on a idea Stochastic Equations with Delay: Optimal Control via BSDEs and Regular Solutions of Hamilton-Jacobi-Bellman Equations
[{"from": "human", "value": "I am writing a research paper on a idea Stochastic Equations with Delay: Optimal Control via BSDEs and Regular Solutions of Hamilton-Jacobi-Bellman Equations"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider an Ito stochastic differential equation with delay, driven by Brownian motion, whose solution, by an appropriate reformulation, defines a Markov process $X$ with values in a space of continuous functions $\\mathbf{C}$, with generator $\\mathcal{L}$.  then consider a backward stochastic differential equation depending on $X$, with unknown processes $(Y,Z)$, and  study properties of the resulting system, in particular  identify the process $Z$ as a deterministic functional of $X$.  next prove that the forward-backward system provides a suitable solution to a class of parabolic partial differential equations on the space $\\mathbf{C}$ driven by $\\mathcal{L}$, and  apply this result to prove a characterization of the fair price and the hedging strategy for a financial market with memory effects.  also include applications to optimal stochastic control of differential equation with delay: in particular  characterize optimal controls as feedback laws in terms of the process $X$."}]