Mean-field equilibrium price formation under single-default risk

Abstract: We study equilibrium price formation in an incomplete financial market with a large population of agents, where stock prices are subject to a single-default event. Agents are assumed to be heterogeneous in their risk aversion and terminal liabilities, and maximize exponential utility of terminal net wealth. We first characterize each agent's optimal strategy by a quadratic-growth backward stochastic differential equation (BSDE) driven by Brownian motions and a compensated default martingale. We then formulate the market-clearing condition in terms of aggregate optimal demand and derive a mean-field quadratic-growth BSDE for the equilibrium risk premium. The resulting characterization quantifies how default intensity, jump size, and agent heterogeneity jointly shape the default-risk component of equilibrium security risk premia. Under a Markovian factor model, we establish short-time solvability of the mean-field BSDE through a fixed-point argument based on estimates for a coupled semilinear PDE system. Finally, we show that the risk premium characterized by the mean-field BSDE asymptotically clears the market as the population size tends to infinity.

• The paper develops a quadratic-growth BSDE framework with a compensated default martingale to characterize optimal investment by heterogeneous agents facing a single default event.
• The paper decomposes the equilibrium risk premium into Brownian hedging and default-risk components, with the latter given by $-\widehat\gamma\lambda\psi^*\beta$ and reflecting default intensity, jump size, and aggregate risk tolerance.
• The paper proves short-time Markovian equilibrium existence through a contraction argument and establishes an $O(1/N)$ mean-square market-clearing error for finite populations, while leaving global horizons and multiple defaults open.

This paper develops a mean-field equilibrium model of asset pricing in an incomplete market exposed to a single default event, extending the purely diffusive framework of Fujii and Sekine to a defaultable setting (2607.17502). Agents with heterogeneous risk aversion and terminal liabilities maximize exponential utility of terminal net wealth; the equilibrium risk premium is constructed so that aggregate optimal demand clears the market in the large-population limit. The central mathematical object is a mean-field quadratic-growth backward stochastic differential equation (BSDE) driven by Brownian motions and a compensated default martingale, coupled with a scalar fixed-point problem for the aggregate default-exposure factor.

Market structure and individual optimization

The model is built on a product probability space combining a common component, generated by a Brownian motion $W^0$ and the indicator $N$ of a totally inaccessible default time $\tau$ with bounded compensator density $\lambda$, and idiosyncratic components carrying agents' private Brownian motions and characteristics. There are $n \le d_0$ risky stocks with dynamics

$$S_t = S_0 + \int_0^t \mathrm{diag}(S_{s-})\bigl(\mu_s\,ds + \sigma_s\,dW^0_s + \beta_s\,dN_s\bigr),$$

where $\sigma\sigma^\top$ is uniformly non-degenerate, $\beta^i_t > -1$ (so prices remain positive), and $$|\beta_t|^2 \ge \underline\beta^2 \mathbf 1_{\{\lambda_t>0\}}$$, which guarantees that the jump exposure direction $$\eta_t := \sigma_t^\top(\sigma_t\sigma_t^\top)^{-1}\beta_t$$ is uniformly non-degenerate on $N$0. Each agent maximizes $N$1 over self-financing strategies, with $N$2 bounded between $N$3 and $N$4 and $N$5 a bounded terminal liability.

Applying the martingale optimality principle of Hu–Imkeller–Müller in the jump setting, the author derives a quadratic-growth BSDE for the value process $N$6, with integrands $N$7 against $N$8, the idiosyncratic Brownian motion, and the compensated default martingale $N$9. The key structural difference from the diffusive case is that the first-order condition for the optimal demand $\tau$0 no longer admits an explicit closed-form solution: it reads

$\tau$1

where $\tau$2 is determined implicitly by the scalar equation $\tau$3 for a predictable input $\tau$4. The inverse map $\tau$5 is well-defined, $\tau$6-Lipschitz, and satisfies uniform a priori bounds on $\tau$7 and on the exponential term, which underpin the entire analysis.

Economically, the last term in the optimal demand is the default-induced demand component: its sign is governed by the jump size $\tau$8, and its magnitude is scaled by $\tau$9, which the paper identifies as the potential jump ratio of conditional expected utility at default. A security that loses value at default commands reduced pre-default demand, and conversely for securities paying off at default.

Well-posedness of the individual BSDE is established in $\lambda$0 via a driver linearization, a change-of-measure argument, and a monotone approximation through truncated admissible sets in the spirit of Morlais. A crucial technical lemma shows that the Doléans-Dade exponential of a single-default stochastic integral is a true martingale of class-$\lambda$1; the author notes explicitly that this uniqueness argument relies on the single-jump identity and does not extend directly to multiple-jump (e.g., Lévy-driven) models. A verification theorem then confirms that the candidate $\lambda$2 from the first-order condition is the unique optimal strategy.

The mean-field BSDE and decomposition of the risk premium

With $\lambda$3 conditionally i.i.d. agents, the market-clearing condition is $\lambda$4. Passing to the mean-field limit via De Finetti's theorem replaces the average by the $\lambda$5-conditional mean $\lambda$6 for a representative agent. Solving the aggregate clearing condition for the risk premium yields the candidate

$\lambda$7

where the consistency condition $\lambda$8 must hold simultaneously with the implicit equation for $\lambda$9. This is the structural novelty relative to the diffusive framework: the equilibrium construction couples a mean-field BSDE with an additional scalar fixed-point problem rather than reducing to a single self-contained equation.

The resulting equilibrium expected return decomposes into two economically interpretable components. The Brownian hedging component $n \le d_0$0 is proportional to the covariance between each stock's diffusive shock and the population-average common-noise exposure, exactly as in the purely diffusive model. The default-risk component $n \le d_0$1 is new: it is proportional to the default intensity and the jump-size vector, scaled by the endogenous factor $n \le d_0$2, which aggregates across the population how unfavorable the default state is, weighted by risk tolerance. Since $n \le d_0$3, the component has the sign opposite to $n \le d_0$4: securities that lose value at default carry a positive pre-default risk premium compensating holders, while securities paying off at default carry a negative component reflecting their insurance value. The component vanishes after default since $n \le d_0$5.

The scalar fixed-point problem

The consistency condition for $n \le d_0$6 is analyzed on the ball $n \le d_0$7, where $n \le d_0$8 is an explicit constant depending on the model parameters and the bound $n \le d_0$9 on candidate integrands. The induced map $$S_t = S_0 + \int_0^t \mathrm{diag}(S_{s-})\bigl(\mu_s\,ds + \sigma_s\,dW^0_s + \beta_s\,dN_s\bigr),$$0 is shown to map $$S_t = S_0 + \int_0^t \mathrm{diag}(S_{s-})\bigl(\mu_s\,ds + \sigma_s\,dW^0_s + \beta_s\,dN_s\bigr),$$1 into itself and to be a contraction with modulus $$S_t = S_0 + \int_0^t \mathrm{diag}(S_{s-})\bigl(\mu_s\,ds + \sigma_s\,dW^0_s + \beta_s\,dN_s\bigr),$$2, where

$$S_t = S_0 + \int_0^t \mathrm{diag}(S_{s-})\bigl(\mu_s\,ds + \sigma_s\,dW^0_s + \beta_s\,dN_s\bigr),$$3

is explicit. Banach's theorem therefore yields a unique fixed point $$S_t = S_0 + \int_0^t \mathrm{diag}(S_{s-})\bigl(\mu_s\,ds + \sigma_s\,dW^0_s + \beta_s\,dN_s\bigr),$$4, which is strictly positive and bounded. A companion stability lemma provides an $$S_t = S_0 + \int_0^t \mathrm{diag}(S_{s-})\bigl(\mu_s\,ds + \sigma_s\,dW^0_s + \beta_s\,dN_s\bigr),$$5-type Lipschitz estimate of $$S_t = S_0 + \int_0^t \mathrm{diag}(S_{s-})\bigl(\mu_s\,ds + \sigma_s\,dW^0_s + \beta_s\,dN_s\bigr),$$6 in terms of perturbations of the input triple $$S_t = S_0 + \int_0^t \mathrm{diag}(S_{s-})\bigl(\mu_s\,ds + \sigma_s\,dW^0_s + \beta_s\,dN_s\bigr),$$7, with constants depending only on $$S_t = S_0 + \int_0^t \mathrm{diag}(S_{s-})\bigl(\mu_s\,ds + \sigma_s\,dW^0_s + \beta_s\,dN_s\bigr),$$8 and the model parameters. This contraction structure is the mechanism by which the default term is folded into a well-defined BSDE driver.

Markovian factor model and short-time existence

Under a Markovian assumption in which the default intensity, market coefficients, and liabilities are functions of a common factor $$S_t = S_0 + \int_0^t \mathrm{diag}(S_{s-})\bigl(\mu_s\,ds + \sigma_s\,dW^0_s + \beta_s\,dN_s\bigr),$$9 (with regime switch at default), an idiosyncratic factor $\sigma\sigma^\top$0, and the default indicator, the author reduces the mean-field BSDE to a coupled system of two semilinear parabolic PDEs in the two regimes:

$\sigma\sigma^\top$1

with terminal data $\sigma\sigma^\top$2. The coupling term $\sigma\sigma^\top$3 encodes the value jump at default. Using Gaussian bounds on the fundamental solution, the paper proves uniform bounds on $\sigma\sigma^\top$4 and its gradient of the form

$\sigma\sigma^\top$5

with $\sigma\sigma^\top$6 independent of $\sigma\sigma^\top$7 and $\sigma\sigma^\top$8. These estimates show that, for $\sigma\sigma^\top$9 above an explicit threshold $\beta^i_t > -1$0 and $\beta^i_t > -1$1, the solution map $\beta^i_t > -1$2 of the BSDE with frozen input maps the closed Markovian ball $\beta^i_t > -1$3 into itself and is a contraction with modulus $\beta^i_t > -1$4 as $\beta^i_t > -1$5, where $\beta^i_t > -1$6. The Banach fixed-point theorem then yields the first main result: existence (and uniqueness within the Markovian ball) of a solution to the mean-field BSDE for sufficiently short horizons. The author is explicit that uniqueness is claimed only within $\beta^i_t > -1$7, not in the larger space $\beta^i_t > -1$8, and that the short-horizon restriction is inherent to the contraction argument.

Asymptotic market clearing

The second main result justifies the mean-field construction from the finite-agent market. Given a bounded solution of the mean-field BSDE with bounded $\beta^i_t > -1$9 and $$|\beta_t|^2 \ge \underline\beta^2 \mathbf 1_{\{\lambda_t>0\}}$$0, the induced risk premium $$|\beta_t|^2 \ge \underline\beta^2 \mathbf 1_{\{\lambda_t>0\}}$$1 is shown to be essentially bounded and to satisfy the mean-field clearing condition exactly: $$|\beta_t|^2 \ge \underline\beta^2 \mathbf 1_{\{\lambda_t>0\}}$$2, $$|\beta_t|^2 \ge \underline\beta^2 \mathbf 1_{\{\lambda_t>0\}}$$3-a.e. The proof exploits pathwise uniqueness of the individual BSDE and the Yamada–Watanabe theorem of Kurtz to obtain a measurable map from the common market data and each agent's idiosyncratic input to the agent's solution, establishing that $$|\beta_t|^2 \ge \underline\beta^2 \mathbf 1_{\{\lambda_t>0\}}$$4 are $$|\beta_t|^2 \ge \underline\beta^2 \mathbf 1_{\{\lambda_t>0\}}$$5-conditionally i.i.d. Decomposing the optimal demand into $$|\beta_t|^2 \ge \underline\beta^2 \mathbf 1_{\{\lambda_t>0\}}$$6, where both families are conditionally i.i.d. with vanishing conditional means and square-integrable, a conditional variance calculation gives the quantitative bound

$$|\beta_t|^2 \ge \underline\beta^2 \mathbf 1_{\{\lambda_t>0\}}$$7

with $$|\beta_t|^2 \ge \underline\beta^2 \mathbf 1_{\{\lambda_t>0\}}$$8 independent of $$|\beta_t|^2 \ge \underline\beta^2 \mathbf 1_{\{\lambda_t>0\}}$$9. The paper also notes that the almost-sure limit $$\eta_t := \sigma_t^\top(\sigma_t\sigma_t^\top)^{-1}\beta_t$$0, $$\eta_t := \sigma_t^\top(\sigma_t\sigma_t^\top)^{-1}\beta_t$$1-a.e., follows via a dyadic subsequence argument with Doob's maximal inequality. This $$\eta_t := \sigma_t^\top(\sigma_t\sigma_t^\top)^{-1}\beta_t$$2 rate in $$\eta_t := \sigma_t^\top(\sigma_t\sigma_t^\top)^{-1}\beta_t$$3 confirms that the mean-field risk premium asymptotically clears the finite-agent market.

Limitations and open questions

The paper concedes several restrictions at the points where they bind. First, existence of the mean-field equilibrium (Theorem 4.1) is local in the horizon; extension to arbitrary maturities would require stronger a priori estimates or a continuation argument and is left open. Second, the uniqueness argument for the individual BSDE depends essentially on the single-jump structure of the default compensator; multiple defaults or general Lévy random measures would require new techniques. Third, the equilibrium risk premium is constructed under an essential boundedness assumption on $$\eta_t := \sigma_t^\top(\sigma_t\sigma_t^\top)^{-1}\beta_t$$4 and $$\eta_t := \sigma_t^\top(\sigma_t\sigma_t^\top)^{-1}\beta_t$$5, whereas the no-default framework of Fujii–Sekine treats the risk premium as an $$\eta_t := \sigma_t^\top(\sigma_t\sigma_t^\top)^{-1}\beta_t$$6 process; extending the stability and solvability theory to that level of generality in the defaultable setting remains unresolved. Finally, the author suggests combining the equilibrium construction with filtering of default intensity in a partially observable market as a natural but unaddressed direction.

Conclusion

The paper extends mean-field equilibrium price formation with exponential utilities to markets with single-default risk. Its contributions are a quadratic-growth BSDE with a compensated default martingale characterizing individual optimality, a quantitative decomposition of the equilibrium risk premium into Brownian hedging and default-risk components, a contraction-based proof of short-time existence of the mean-field equilibrium under a Markovian factor model, and an $$\eta_t := \sigma_t^\top(\sigma_t\sigma_t^\top)^{-1}\beta_t$$7 bound establishing asymptotic market clearing. The default-risk component $$\eta_t := \sigma_t^\top(\sigma_t\sigma_t^\top)^{-1}\beta_t$$8 makes precise how intensity, jump size, and the cross-sectional distribution of risk aversion and liabilities are aggregated by market clearing into the price of default risk. The main open problems are global-in-time existence, multiple defaults, and unbounded risk premia.