Noise induced Stability of a Mean-Field model of Systemic Risk with uncertain robustness

TL;DR

This study develops a mean-field McKean–Vlasov diffusion model of systemic risk with uncertain robustness, where the barrier height of a symmetric bistable potential fluctuates and two noise channels (additive and multiplicative) interact. By allowing the stochastic integral interpretation to vary via a parameter nu, the authors show that phase behavior becomes richer and can exhibit noise-induced stability, unlike the classical Dawson–Shiino model. They derive the associated Fokker–Planck equation, a stationary density rho0, and a self-consistency function F(nu, sigma_a, sigma_m, a, theta)[mu], and analyze the phase boundary via F_mu_prime_0(0), including asymptotic regimes for sigma_m. The results reveal that multiplicative noise can either stabilize or destabilize the system depending on nu and the parameter regime, with explicit examples and contour plots illustrating 1 -> 3 -> 1 transitions. The work highlights how uncertainty in component robustness and stochastic integral choice reshape systemic-risk dynamics and open avenues for studying stability in more complex multi-well landscapes.

Abstract

We consider a model for systemic risk comprising of a system of diffusion processes, interacting through their empirical mean. Each process is subject to a confining double-well potential with some uncertainty in the coefficients, corresponding to fluctuations in height of the potential barrier seperating the two wells. This is equivalent to studying a single McKean-Vlasov SDE with explicit dependence on its moments and, novelly, independently varying additive and multiplicative noise. Such non-linear SDEs are known to possess two phases: stable (ordered) and unstable (disordered). When the potential is purely bistable, the phase changes from stable to unstable when noise intensity is increased past a critical threshold. With the recent advances, it will be shown that the behaviour here is far richer: indeed, depending on the interpretation of the stochastic integral, the system exhibits phase changes that cannot occur in any regime where there is no uncertainty in the potential. Strikingly, this allows for the phenomenon of noise induced stability; situations where more noise can reduce the risk of system failure.

Figures (6)

• Figure 1: Bifurcation diagrams of (left) the Dawson-Shiino model with a classic pitchfork shape, and (right) the model with uncertain robustness (parameters as inscribed), introduced in section \ref{['sec::model']}
• Figure 2: Panel of bifurcation diagrams, parameters inscribed. Left to right, top to bottom: $1\rightarrow3\rightarrow1$, $1\rightarrow$, $1\rightarrow3$ and $3\rightarrow 1$ for $(\nu,\,\sigma_a)$ as inscribed
• Figure 3: Bifurcation Diagrams for $a\leq0$. Top At $a=0$ a stable phase exists so long as $\nu=1$. Bottom Stable phase, and lack thereof, above and below the threshold. Note $\sqrt{10}\approxeq3.16$
• Figure 4: Left Gradient of the phase transition contour at $(\sigma_c,0)$ against $\theta$ for Itô, Stratonovich and Klimontovich noise. The roots at $\theta=1$ for $\nu=0.5$ has been recovered and those for $\nu=0$ and $\nu=1$ displayed. For $\theta$ above $\sim0.72$ for some range of $\nu$, noise induced stabilisation can be observed. It always occurs (regardless of $\nu$) for $\theta\gtrsim2.1$. Right the self-consistency function for MV-SDE (\ref{['funda']}) displaying a $1\rightarrow3\rightarrow1$ phase change.
• Figure 5: Panel of contour diagrams of $F^{'}(\nu,\cdot,\cdot){[}0{]}$ for increasing $\nu$, green positive, blue negative. The phase transition contour intersects the $\sigma_a$ axis at $\sigma_c$. Graph $\nu=\{0.75,1\}$ corresponds to column 4, $\nu=0.49$ and $\nu=0.35$ column 3, $\nu=0.2$ column 2 and $\nu=0$ column 1 of Table 1

Theorems & Definitions (9)

• Proposition 1: alecio Proposition 3.3
• Proposition 2: alecio
• proof
• Proposition 3: alecio Proposition 3.5
• Theorem 1: alecio Theorem 2.12
• Proposition 4: Asymptotic Properties of $F^{'}_\mu(\nu,\sigma_a,\sigma_m){[}\mu{]}$: $\sigma_m\downarrow 0$
• proof
• Proposition 5: Further Asymptotic Properties of $F^{'}_\mu(\nu,\sigma_a,\sigma_m){[}\mu{]}$: $\sigma_m\uparrow\infty$
• proof