[Paper Review] Stochastic equations with delay: optimal control via BSDEs and regular solutions of Hamilton-Jacobi-Bellman equations

Marco Fuhrman, Federica Masiero+1 more|arXiv (Cornell University)|Jun 11, 2008

Stochastic processes and financial applications3 references4 citations

TL;DR

This paper develops a backward stochastic differential equation (BSDE) approach to solve optimal control and pricing problems for stochastic differential equations with delay, by reformulating the state process as a Markov process in a space of continuous functions. The key contribution is a characterization of the value function and hedging strategies via a mild solution to a Hamilton-Jacobi-Bellman equation on the path space, generalizing the Black-Scholes framework to memory-dependent systems.

ABSTRACT

We 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 $\mathcal L$. We then consider a backward stochastic differential equation depending on $X$, with unknown processes $(Y,Z)$, and we study properties of the resulting system, in particular we identify the process $Z$ as a deterministic functional of $X$. We 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 we apply this result to prove a characterization of the fair price and the hedging strategy for a financial market with memory effects. We also include applications to optimal stochastic control of differential equation with delay: in particular we characterize optimal controls as feedback laws in terms the process $X$.

Motivation & Objective

• To address optimal stochastic control problems for stochastic differential equations with delay (SDDEs) by leveraging the Markov property of the delayed state process.
• To characterize the value function of the control problem as a solution to a Hamilton-Jecobi-Bellman (HJB) equation on the space of continuous paths.
• To extend the BSDE method to infinite-dimensional state spaces, particularly path spaces, to solve parabolic PDEs driven by the generator of the delayed process.
• To apply the framework to financial markets with memory effects, providing a characterization of fair option prices and hedging strategies.

Proposed method

• Reformulate the SDDE as a Markov process $X_t$ with values in the space $\mathbf{C} = C([-r,0]; \mathbb{R}^n)$, enabling the use of diffusion processes on path space.
• Construct a forward-backward stochastic differential system (FBSDE) coupling the forward SDDE with a backward equation for $(Y,Z)$, where $Y_t = v(t,X_t)$ and $Z_t = \nabla_0 v(t,X_t)\sigma(t,X_t)$.
• Use the theory of BSDEs to show that the solution $v(t,x)$ satisfies a semilinear parabolic PDE on $\mathbf{C}$, specifically $\partial_t v + \mathcal{L}v = \psi(t,x,v,\nabla_0 v \sigma)$, with terminal condition $v(T,x) = \phi(x)$.
• Define the Hamiltonian $\psi(t,x,z) = \inf_u \{ g(u) + z h(t,x,u) \}$ to link the control problem to the HJB equation.
• Apply the framework to a financial model with memory by introducing a risk-neutral measure via Girsanov's theorem, transforming the dynamics to a martingale under the new measure.
• Prove that the fair price and hedging strategy are uniquely determined by $V_0 = v(0,s)$ and $\pi_t = \nabla_0 v(t,S_{t+\cdot})$, where $v$ solves the HJB equation $\partial_t v + \mathcal{L}v = r v$, $v(T,x) = \phi(x)$.

Experimental results

Research questions

• RQ1 How can optimal control problems for stochastic differential equations with delay be characterized using backward stochastic differential equations (BSDEs)?
• RQ2 What is the connection between the solution of a forward-backward system on path space and the solution of a Hamilton-Jacobi-Bellman (HJB) equation in infinite-dimensional settings?
• RQ3 How can the fair price and hedging strategy for a contingent claim be derived in a financial market with memory effects using the BSDE and HJB framework?
• RQ4 What conditions ensure the existence and uniqueness of a mild solution to the HJB equation on the space of continuous paths?
• RQ5 Can the classical Black-Scholes equation be generalized to include path-dependent (memory) effects through this framework?

Key findings

• The value function $v(t,x)$ of the optimal control problem coincides with the solution of the HJB equation $\partial_t v + \mathcal{L}v = \psi(t,x,v,\nabla_0 v \sigma)$, with terminal condition $v(T,x) = \phi(x)$, where $\psi$ is defined via the infimum over control actions.
• The process $Z_t$ in the BSDE is identified as $Z_t = \nabla_0 v(t,X_t)\sigma(t,X_t)$, showing that the control can be expressed as a feedback law in terms of the current path $X_t$.
• The fair price of a contingent claim $\phi(S_{T+\cdot})$ is given by $V_0 = v(0,s)$, where $v$ solves the HJB equation $\partial_t v + \mathcal{L}v = r v$, $v(T,x) = \phi(x)$, under the risk-neutral measure.
• The hedging strategy is characterized as $\pi_t = \nabla_0 v(t,S_{t+\cdot})$, which is a functional of the current path of the underlying asset.
• The solution $v$ is shown to be a mild solution of the HJB equation, ensuring well-posedness under appropriate regularity and boundedness conditions on the coefficients.
• The framework generalizes the classical Black-Scholes model to include memory effects by embedding the path-dependent dynamics into the infinite-dimensional state space $\mathbf{C}$.