In equilibrium models with quadratic transaction costs, the individually optimal portfolio and trading rate for each agent are characterised by a coupled system of forward-backward stochastic differential equations (FBSDEs). The forward component describes the evolution of the portfolio position; the backward component describes the optimal trading rate (or equivalently, the marginal cost of trading).
For agent n with quadratic transaction costs, the individual optimality FBSDE is (Herdegen et al. 2018, Lemma 4.1; Shelley 2023, Lemma 2.2.2):
d(phi_t^{lambda,n}) = phi-dot_t^{lambda,n} dt, phi_0 = 0 d(phi-dot_t^{lambda,n}) = Z_t^n dW_t^n + (gamma_n sigma^2)/(2 lambda) * (phi_t^{lambda,n} - phi_t^n) dt + delta * phi-dot_t^{lambda,n} dt
with terminal condition phi-dot_T = 0 (for T < infinity) or transversality condition (for T = infinity).
Market clearing (sum phi^n = 0 or = s) couples these into a system of N-1 FBSDEs. For quadratic costs, this system is linear, enabling explicit solution via:
- Matrix exponentials for T = infinity (Theorem A.2 in Herdegen et al.)
- Hyperbolic matrix functions for T < infinity (Theorem A.4)
For general convex costs (Gonon et al. 2020), the FBSDEs become nonlinear (equations 6.7-6.8), and the coupled system is fully nonlinear when volatility is endogenous. These are solved numerically via deep learning (Section 7).
Key Details
- Linear for quadratic costs: enables matrix power series solutions
- Nonlinear for general costs: wellposedness is an open problem for non-quadratic costs
- Coupling mechanism: market clearing condition sum phi^n = s links all agents’ FBSDEs
- Key matrices: B (coupling in backward component), A (cross-agent effects), Theta^varphi, Theta^zeta, Theta^epsilon encode risk aversions and cost structure
- Tracking speed: for quadratic costs, agents track frictionless targets at speed sqrt(Delta) - delta/2, where Delta = (gamma_1 + gamma_2)/(2) * Lambda^{-1} Sigma + delta^2/4