stochastic Riccati equation

A backward stochastic Riccati differential equation (BSRDE) is a nonlinear BSDE with a quadratic generator that arises naturally in linear-quadratic stochastic control problems. In the transaction cost equilibrium literature, the BSRDE determines the optimal tracking speed at which agents trade towards their frictionless targets.

In Herdegen, Muhle-Karbe, Possamai (2020), the BSRDE takes the form (equation 3.5):

c_t = integral_t^T (gamma/lambda * sigma_s^2 - c_s^2) ds - integral_t^T Z_s^c dW_s

where c_t >= 0 is the tracking speed, gamma is risk aversion, lambda is the transaction cost parameter, and sigma_t is the (possibly stochastic) volatility process. The unique solution (c, Z) lies in S^infinity x H^2_BMO, satisfying 0 ⇐ c_t ⇐ (gamma/lambda) * ||sigma||^2_{H^2_BMO}.

For constant volatility sigma, the BSRDE reduces to a scalar Riccati ODE with solution c = delta * tanh(delta(T-t)) where delta = sqrt(gamma sigma^2 / (2 lambda)), recovering the explicit formulas of Herdegen, Muhle-Karbe (2018).

The BSRDE is the key technical object in the Picard iteration scheme for proving equilibrium existence with endogenous volatility: given a candidate volatility sigma, the BSRDE determines c, which determines the optimal strategy, which determines the equilibrium volatility — and the iteration contracts in the H^2_BMO norm.

Key Details

• Generator: quadratic in c: f(c) = gamma/lambda * sigma^2 - c
• Terminal condition: c_T = 0 (trading stops at terminal time)
• Bounds: 0 ⇐ c_t ⇐ gamma/lambda * ||sigma||^2_{H^2_BMO}
• For constant sigma: c_t = delta * tanh(delta(T-t)), delta = sqrt(gamma * sigma^2 / (2*lambda))
• Stability (Lemma 7.1): E^alpha[sup |c^sigma - c^{sigma-tilde}|^2] ⇐ g_c * ||sigma - sigma-tilde||^2_{H^2_BMO}
• Role: determines the discount kernel for future target values and the tracking speed of optimal portfolios

Textbook References

Stochastic Controls - Hamiltonian Systems and HJB Equations (Yong & Zhou, 1999)

The stochastic Riccati equation in its general matrix form, arising from the stochastic LQ optimal control problem, is:

$\dot{P}+PA+A^{T}P+C^{T}PC+Q-(B^{T}P+S+D^{T}PC{)}^T(R+D^{T}PD{)}^{-1}(B^{T}P+S+D^{T}PC)=0$, $P(T)=G$, with $R(t)+D(t{)}^{T}P(t)D(t)\ge 0$.

This equation is called “stochastic” because it arises from a stochastic LQ problem (and becomes a genuine BSDE when coefficients are random), even though the equation itself is a deterministic ODE when coefficients are deterministic. The term $D^{T}PD$ in the denominator, absent from the classical deterministic Riccati equation, is the distinguishing feature.

• Equation (6.6) (p. 314): The stochastic Riccati equation derived via the maximum principle / Hamiltonian system approach
• Theorem 6.1 (p. 315): Solvability of (6.6) implies solvability of Problem (SLQ) with state feedback control $\bar{u}=-\Psi x-\Theta$; also derived independently via HJB (Section 6, pp. 317—318) and completion of squares (p. 317)
• Proposition 7.1 (p. 319): Uniqueness of solutions under (L2) continuity assumption on $D$ and $R$
• Theorem 7.2 (p. 320): Global existence in the standard case ($R\gg 0$, $Q-S^T{R}^{-1}S\ge 0$, $G\ge 0$) via monotone iterative scheme with exponential convergence (Proposition 7.4, p. 323: $|P_i(t)-P(t)|\le K{c}^{i-2}/(i-2)!$)
• Theorem 7.5 (p. 325): For $C=0$, $S=0$, $Q,G\ge 0$: solvability equivalent to existence of a fixed point $\hat{R}=R+D^T\Gamma (\hat{R})D$, where $\Gamma$ is the solution operator of the deterministic Riccati equation
• Theorem 7.7 (p. 328): Necessary and sufficient condition: $\exists \hat{R}\ge 0$ such that $R+D^T\Gamma (\hat{R})D\ge \hat{R}$; $R$ can be indefinite but not “too negative”
• Theorem 7.9 (p. 330): Complete characterisation of the maximal solvability interval $\Theta$ for the one-dimensional constant-coefficient case, with explicit formulas for all sub-cases depending on the signs of $\alpha =(2A+C^2)-(B+C{)}^2$, $\beta$, $\gamma =-[R(B+C)-S{]}^2$, and the discriminant $\Delta ={\beta }^2-4\alpha \gamma$
• Example 7.8 (p. 327): For $dx=udt+udW$, $J=E\{-\frac{r}{2}\int u^{2}dt+\frac{1}{2}x(1{)}^2\}$ with $0<r<1$: solvability on $[0,1]$ requires $(1-r{)}^2\ge r$, i.e., $r\le 3-2\sqrt{2}$

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