Relative Arbitrage in Stochastic Portfolio Theory

• Relative arbitrage is the construction of self-financing strategies that guarantee outperformance of the market portfolio over a predetermined time interval under minimal model assumptions.
• It leverages nonlinear PDEs, geometric flows, and stochastic control techniques to precisely characterize minimal arbitrage horizons across different market models.
• Functionally generated portfolios and robust optimization frameworks provide practical methods to implement relative arbitrage in both continuous and discrete time contexts.

The relative arbitrage problem in stochastic portfolio theory concerns the explicit construction and structural characterization of self-financing strategies that outperform the market portfolio over deterministic time intervals, using only the observable path of market weights and subject to minimal model assumptions on volatility and diversity. This concept traverses continuous- and discrete-time formalisms, robust and “universal” (model-free) settings, and links the existence and optimality of such arbitrage to deep connections with nonlinear PDEs, geometric flows, stochastic control, and functional portfolio generation.

1. Definitions, Market Model, and Core Notions

Let $d\geq 2$ and consider $d$ stocks whose capitalization-weight processes $(\mu_1(t),\ldots,\mu_d(t))\in\Delta^d$ (the $d$-simplex). A self-financing trading strategy is identified by a predictable, $\mathbb{R}^d$-valued process $\theta(t)$, with relative wealth process

$V^\theta(t)=\theta(t)^\top\mu(t),$

satisfying

$$V^\theta(t)=V^\theta(0)+\int_0^t \theta(s)^\top d\mu(s).$$

The market portfolio corresponds to $\theta(t)\equiv(1,\ldots,1)$, so $V^\theta(t)=1$ at all times when normalized.

A strategy $\theta$ yields relative arbitrage over $[0,T]$ if

$$V^\theta(0)\geq 0,\quad V^\theta(T)\geq V^\theta(0)\ \mathrm{a.s.},\quad \mathbb{P}(V^\theta(T)>V^\theta(0))>0.$$

This defines a trading rule with no risk of underperformance and positive-probability strict gain relative to the dynamic market benchmark.

The minimal criterion for sufficient intrinsic volatility (SV) is

$$\operatorname{tr}[\mu,\mu](t)\geq t\quad \forall\,t\geq0,$$

where $\operatorname{tr}[\mu,\mu](t)$ is the trace of the quadratic covariation matrix of $\mu$ (Larsson et al., 2020).

2. Existence and Structural Results: Sharp Time Horizons

Minimal Arbitrage Horizon $T_*(d)$

The central question is: given $d$, what is the smallest $T_*(d)$ such that every sufficiently volatile $d$-stock market admits relative arbitrage on $[0,T]$ for all $T>T_*(d)$, and for $T<T_*(d)$ one can construct models without such arbitrage?

This problem is reduced via duality to a PDE/variational formulation: for each $d$, let

$u_d(y) = \sup\left\{\textrm{ess\,inf}\,\tau:\ \nu\ \textrm{is a continuous %%%%20%%%%-martingale},\ \nu(0)=y,\,\operatorname{tr}[\nu,\nu](t)=t \right\},$

where $\tau = \inf\{t\,:\,\nu(t)\notin\Delta^d\}$, and then

$T_*(d) = \sup_{y\in\Delta^d} u_d(y).$

Explicitly:

• For $d=2$ (two stocks): $T_*(2)=0$; arbitrage exists at any strictly positive horizon (Larsson et al., 2020).
• For $d=3$ (three stocks): $T_*(3)=\sqrt{3}/(2\pi)\approx 0.2757$, obtained by geometric analysis of mean-curvature flow on the triangle image of $\Delta^3$ (Larsson et al., 2020).
• For $d\geq4$: $T_*(d)$ is characterized as the maximal arrival time of a fully nonlinear geometric PDE (minimum curvature flow) on the simplex, and requires numerical or variational solution.

Geometric PDEs: Motion by (Mean/Minimum) Curvature

The sharp horizon for $d=3$ is determined via the mean curvature flow of the boundary triangle $K\subset\mathbb{R}^2$ (the image of $\Delta^3$), leading to the PDE

$$\frac{1}{|\nabla w|} = -\frac{1}{2}\operatorname{div} \left( \frac{\nabla w}{|\nabla w|} \right),\qquad w|_{\partial K}=0,$$

where $w(x)$ encodes the worst-case exit time from $K$ for a martingale process with quadratic variation rate 1. The extinction time for convex planar curves under this flow is initial area over $2\pi$, leading to $T_*(3)=\sqrt{3}/(2\pi)$ (Larsson et al., 2020).

In higher dimensions, the minimum curvature flow arises, where the arrival-time function $v$ solves

$$F(\nabla v(x), \nabla^2 v(x))=1,\ x\in K^\circ, \quad v|_{\partial K}=0,$$

with

$$F(p,M)=\inf\left\{ -\frac{1}{2}\operatorname{tr}(aM): a\in\operatorname{Sym}^{d-1}_+,\,\operatorname{tr}(a) =1,\, ap=0 \right\}.$$

This drives the boundary inward with velocity equal to the minimal principal curvature at each point (Larsson et al., 2020).

3. Portfolio Construction: Functionally Generated Portfolios

Classical stochastic portfolio theory shows that under diversity (no dominant stock, i.e., $\mu_i(t)\leq 1-\delta$ universality) and sufficient volatility, all portfolios that guarantee outperformance are functionally generated via strictly concave, positive functions $\Phi:\Delta^{(n)}\to (0,\infty)$: $$\pi_i(p) = p_i\left(1 + D_{e(i)-p}\log\Phi(p)\right).$$ Such portfolios have relative log-value decomposition: $$\log V(t) = \log\frac{\Phi(\mu(t))}{\Phi(\mu(0))} + \sum_{k=0}^{t-1} T(\mu(k+1)\mid \mu(k)),$$ with nonnegative pathwise L-divergence $T$ (Pal et al., 2014). Under diversity and sufficient volatility, this drift diverges, ensuring pseudo-arbitrage.

Structural theorems characterize functionally generated portfolios equivalently as those satisfying multiplicative cyclical monotonicity (MCM), and as solutions to natural optimal transport problems in the simplex. These portfolios align with the set of path-independent strategies that guarantee arbitrage in purely pathwise fashion (Pal et al., 2014).

4. Dynamic and Control-Theoretic PDE Formulations

The minimal time horizon problem is equivalently a stochastic optimal control problem over the class of martingale laws: $$T_* = \sup_{x\in K} \sup_\mathbb{P}\,\textrm{ess\,inf}\,\tau_K,$$ where $K$ is the image of $\Delta^d$ under suitable isometry, and $\tau_K$ is the exit time from $K$ under martingale dynamics with constrained quadratic variation structure (Lai et al., 19 Dec 2025, Larsson et al., 2020).

By dynamic programming, the value function $v(x)$ solves the viscosity PDE: $$F(\nabla v(x), \nabla^2 v(x))=1,\quad v|_{\partial K}=0,$$ with $F$ defined by the structure of the admissible quadratic variation matrices (via trace/eigenvalue or projection constraints). In special cases, this coincides with arrival-time PDEs for the (co-dimension) mean curvature flow; more generally, it embodies fully nonlinear degenerate elliptic equations (Larsson et al., 2020, Lai et al., 19 Dec 2025).

In model-uncertain or “robust” formulations, e.g., under Knightian uncertainty about model coefficients, the optimal arbitrage function $u(T,x)$ is the minimal supersolution of a degenerate HJB-type PDE: $$\partial_\tau u + F(z,u,Du,D^2u) \geq 0,\quad u(0,x)=1,$$ with $F$ encoding adversarial choices for drift and covariance in the model uncertainty set (Fernholz et al., 2012, Wang, 2015).

5. Necessary and Sufficient Conditions: Volatility, Diversity, and Short-Time Arbitrage

Sharp Sufficient Volatility

• In discrete time, diversity and pathwise sufficient volatility allow for relative arbitrage via functionally generated portfolios (Pal et al., 2014).
• In continuous time, under sufficiently strong lower eigenvalue bounds on the instantaneous covariance (strict uniform ellipticity), strong relative arbitrage exists at all time horizons (Fernholz, 2015, Fernholz et al., 2016, Lai et al., 19 Dec 2025).

Short-Term and Non-Existence Results

It is not sufficient for the total relative variation (e.g., quadratic variation of market weights or cumulative entropy drift) to merely grow linearly in time: explicit counterexamples show the possibility of markets with $\Gamma^G(t)\geq ct$ but no relative arbitrage on $[0,T]$ for any $T<T^*$, for appropriately constructed “worst-case” volatility structures (Fernholz et al., 2016, Larsson et al., 2020).

Short-term arbitrage is restored if, in addition to total volatility, the market weights revisit favorable configurations (time-homogeneous support/recurrence), uniform nondegeneracy holds (no collapse of volatility cone in any direction), or when dynamics are governed by volatility-stabilized-like structures (Fernholz, 2015, Larsson et al., 2020).

Analytic characterizations for $d=2,3$ are sharp and explicit, but for $d\geq4$, the critical horizon must be computed numerically for each domain and volatility constraint.

6. Extensions: Constraints, Optimization, and Multi-Agent Systems

Constraints and Generalized Benchmarks

Minimum horizon results extend under diversity constraints (e.g., restricting $\max_i\mu_i(t)\leq 1-\delta$), with the PDE solved in a smaller polytope domain (Larsson et al., 2020). The relative arbitrage problem generalizes to arbitrary (non-market) benchmarks, such as equal- or entropy-weighted portfolios, but non-existence/maximality theorems show no functionally generated portfolio consistently outperforms these under the same diversity and volatility (Wong, 2014).

Portfolio Optimization

The identification of functionally generated portfolios as optimizers subject to drift or divergence constraints yields not only explicit construction of arbitrage portfolios but also shape-constrained optimization problems for maximizing drift functional over the class of concave generating functions, with empirical or model-based transition measures (Wong, 2014).

Multi-Agent and Mean-Field Games

In the presence of multiple competitive investors, the relative arbitrage problem becomes a Nash equilibrium computation in a McKean–Vlasov system. The associated Cauchy PDE extends to additional variables (empirical measure of wealth) and requires compatibility (Fichera-drift) conditions for arbitrage existence. Mean-field limits and propagation of chaos results connect finite-population Nash equilibria to PDE-defined mean-field equilibria (Ichiba et al., 2020, Yang et al., 2023).

7. Numerical Methods and Practical Implementation

The high-dimensional geometric PDEs that characterize minimal arbitrage horizons in general dimensions can be approached via:

• Viscosity-solution-based numerical PDE solvers in polytopal domains
• Probabilistic Monte Carlo schemes based on time-changed Bessel processes and bridge interpolation to compute minimal nonnegative solutions of Cauchy problems in volatility-stabilized models (Yang et al., 6 Nov 2024)
• Empirical, gridwise or shape-constrained optimization for functionally generated portfolios (Wong, 2014)

Monte Carlo/Bessel bridge algorithms are especially tractable in volatility-stabilized markets, where explicit affine or quadratic generator functions lead to closed-form drift terms for the (backward) Cauchy PDE (Yang et al., 6 Nov 2024).

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Key result table for $T_*(d)$ under sufficient volatility (Larsson et al., 2020):

Number of Stocks ($d$)	Minimal Horizon $T_*(d)$	Characterization
2	$0$	Immediate arbitrage exists
3	$\sqrt{3}/(2\pi)\approx 0.2757$	Area-extinction time for mean-curvature flow
$\geq4$	$\sup_{x\in K}u_d(x)$	Maximal arrival-time for min-curvature flow PDE on $\Delta^d$; only numerical/variational solution possible

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The relative arbitrage problem thus centralizes the interplay between minimal time-horizon guarantees for universal outperformance, PDE and geometric-flow-based characterizations of market “volatility harvesting,” functional portfolio generation, and robust (model-uncertain) control, across both single- and multi-agent settings (Larsson et al., 2020, Lai et al., 19 Dec 2025, Pal et al., 2014, Fernholz et al., 2012, Wang, 2015, Wong, 2014, Ichiba et al., 2020, Yang et al., 2023, Yang et al., 6 Nov 2024, Fernholz et al., 2016, Fernholz, 2015, Mijatović et al., 2010).