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arxiv: 2605.00688 · v1 · submitted 2026-05-01 · 🧮 math.OC · math.PR· q-fin.MF

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Optimal Merton's Problem under Multivariate Affine Volterra Models with Jumps

Sigui Brice Dro , Emmanuel Gnabeyeu
Authors on Pith no claims yet

Pith reviewed 2026-05-09 18:47 UTC · model grok-4.3

classification 🧮 math.OC math.PRq-fin.MF
keywords Merton's problemportfolio optimizationVolterra-Heston modeljumpsRiccati BSDErough volatilityoptimal investmentstochastic control
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The pith

Merton's portfolio optimization in multivariate Volterra-Heston models with jumps reduces to solutions of time-dependent Riccati-Volterra equations and a Riccati BSDEJ.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper solves the classical Merton portfolio selection problem for investors with exponential, power, and logarithmic utility in multi-asset markets driven by a multivariate Volterra-Heston model that includes jumps from an independent Poisson random measure. The model is non-Markovian and fails to be a semimartingale, so standard stochastic control methods do not apply directly; instead the authors invoke the martingale optimality principle to construct a family of supermartingale processes whose dynamics are governed by a new Riccati backward stochastic differential equation with jumps. Optimal investment strategies then emerge in semi-closed form once the associated time-dependent multivariate Riccati-Volterra equations are solved, and the value function is recovered directly from the solution of the Riccati BSDEJ. Numerical experiments on a two-dimensional rough Heston specification illustrate how both path roughness and jump intensity shift the value function and the resulting optimal allocations.

Core claim

In a multivariate affine Volterra model with jumps, the optimal strategies for Merton's problems with exponential, power, and logarithmic utility are obtained in semi-closed form from the solutions to the corresponding time-dependent multivariate Riccati-Volterra equations, while the optimal value function is expressed in terms of the solution to the associated Riccati backward stochastic differential equation with jumps.

What carries the argument

The Riccati BSDEJ that defines the supermartingale family used in the martingale optimality principle to bypass the non-Markovian character of the Volterra dynamics.

If this is right

  • Optimal portfolios for the three standard utilities become numerically computable once the multivariate Riccati-Volterra system is solved for given parameters.
  • The value function explicitly encodes both the roughness of the volatility paths and the jump intensity through the BSDEJ solution.
  • The same construction supplies semi-closed strategies without requiring the state process to be Markovian.
  • Numerical illustrations confirm that higher roughness or stronger jumps systematically alter optimal weights and lower the attainable value.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same BSDEJ reduction could be tested on other non-Markovian specifications such as fractional Ornstein-Uhlenbeck volatility models.
  • High-dimensional asset extensions might exploit the affine structure of the Riccati-Volterra equations for efficient matrix solutions.
  • Calibration to empirical data that exhibit both rough volatility and jumps would quantify the economic value of the derived strategies.
  • The framework suggests a route to incorporate additional constraints such as no-short-sale rules by adjusting the admissible set inside the supermartingale construction.

Load-bearing premise

The Riccati BSDEJ admits a unique solution and the processes constructed from it are supermartingales under the given model assumptions.

What would settle it

A simulated path under the two-dimensional rough Heston model with jumps where the expected utility achieved by the candidate strategy differs from the value computed from the BSDEJ solution would disprove the claimed reduction.

Figures

Figures reproduced from arXiv: 2605.00688 by Emmanuel Gnabeyeu, Sigui Brice Dro.

Figure 1
Figure 1. Figure 1: Graph of 5 samples paths of the processes tk 7→ V 1 tk (left) and tk 7→ V 2 tk (right) over the time interval [0, 1], for the number of time steps n = 200. Figures 1 displays simulated paths of the instantaneous volatility V i tk for different values of the Hurst index Hi = αi − 1 2 ∈ (0, 1 2 ). Each upward spike corresponds to a jump arrival time τk of the Poisson measure N, at which the volatility is pus… view at source ↗
Figure 2
Figure 2. Figure 2: Graph of tk 7→ ψ 1 tk and tk 7→ ψ 2 tk over [0, 1] with the fractional Adams algorithm, γ = 0.2 and the number of time steps n = 200 both for Power (left) and Exponential (right) utilities functions. Figures 2 above also confirm the Remark following Proposition 3.2, that is the claim that ψ ≤ 0 for every ρi ∈ (0, 1) and every i ∈ {1, 2} in the exponential utility case. The lower the Hurst exponent H, the m… view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of the optimal portfolio strategy for different levels of the risk aversion parameter γ with T ∈ {1, 5, 10} for both the Power (left) and Exponential (right) utilities functions, r = 0.02 view at source ↗
read the original abstract

This paper is concerned with portfolio selection for an investor with exponential, power, and logarithmic utility in multi-asset financial markets allowing jumps. We investigate the classical Merton's portfolio optimization problem in a Volterra stochastic environment described by a multivariate Volterra--Heston model with jumps driven by an independent Poisson random measure. Owing to the non-Markovian and non-semimartingale nature of the model, classical stochastic control techniques are not directly applicable. Instead, the problem is tackled using the martingale optimality principle by constructing a family of supermartingale processes characterized via solutions to an original Riccati backward stochastic differential equation with jumps (Riccati BSDEJ).The resulting optimal strategies for Merton's problems are derived in semi-closed form depending on the solutions to time-dependent multivariate Riccati-Volterra equations, while the optimal value is expressed using the solution to this original Riccati BSDEJ. Numerical experiments on a two-dimensional rough Heston model illustrate the impact of both path roughness and jumps components on the value function and optimal strategies in the Merton problem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper solves Merton's portfolio optimization for exponential, power, and logarithmic utilities in multi-asset markets with jumps under a multivariate Volterra-Heston model. Owing to the non-Markovian, non-semimartingale dynamics, the authors apply the martingale optimality principle by constructing supermartingale processes via solutions to a novel Riccati backward stochastic differential equation with jumps (Riccati BSDEJ). Optimal strategies are derived in semi-closed form from solutions to associated time-dependent multivariate Riccati-Volterra equations, with the value function expressed via the Riccati BSDEJ solution. Numerical experiments on a two-dimensional rough Heston model illustrate effects of roughness and jumps.

Significance. If the well-posedness of the Riccati BSDEJ holds, the work extends classical Merton's results to affine Volterra models with jumps, providing semi-closed forms in a non-standard setting. It advances stochastic control methods for rough volatility models relevant to empirical finance features like volatility roughness and jumps, with numerical results offering practical insights into model impacts on strategies.

major comments (2)
  1. [Section on Riccati BSDEJ and martingale optimality principle] The central optimality claim rests on constructing supermartingales from solutions to the Riccati BSDEJ and verifying they satisfy the martingale optimality principle. However, no bespoke existence/uniqueness theorem is supplied for this BSDEJ under the multivariate affine Volterra dynamics with jumps (see the section defining the Riccati BSDEJ and applying the martingale principle). Standard BSDE results do not apply due to the non-Markovian/non-semimartingale nature, leaving the load-bearing step unverified.
  2. [Verification of supermartingale property] The paper states that the candidate value process is a supermartingale with equality attained under the candidate control, but provides no detailed verification of this property or the associated integrability conditions (see the verification step following the BSDEJ construction). This is required to rigorously apply the martingale optimality principle and confirm the derived strategies are indeed optimal.
minor comments (1)
  1. [Numerical experiments] The abstract references numerical experiments illustrating impacts of roughness and jumps, but the manuscript should expand on implementation details such as discretization schemes for the Volterra kernels and specific parameter values used in the two-dimensional rough Heston example to support reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their thorough review and valuable suggestions regarding the well-posedness of the Riccati BSDEJ and the supermartingale verification. We address each major comment in detail below, indicating the revisions we plan to make to enhance the rigor of the paper.

read point-by-point responses

  1. Referee: The central optimality claim rests on constructing supermartingales from solutions to the Riccati BSDEJ and verifying they satisfy the martingale optimality principle. However, no bespoke existence/uniqueness theorem is supplied for this BSDEJ under the multivariate affine Volterra dynamics with jumps (see the section defining the Riccati BSDEJ and applying the martingale principle). Standard BSDE results do not apply due to the non-Markovian/non-semimartingale nature, leaving the load-bearing step unverified.

    Authors: We concur that providing a bespoke existence and uniqueness result for the Riccati BSDEJ is essential to fully substantiate the martingale optimality approach in this non-standard setting. The manuscript leverages the affine structure to reduce the BSDEJ to a system of deterministic multivariate Riccati-Volterra equations, whose well-posedness follows from standard results on Volterra integral equations under suitable regularity assumptions on the kernels and jump measures. Nevertheless, we did not include an explicit theorem statement or proof sketch in the current version. In the revised manuscript, we will insert a new subsection dedicated to the well-posedness of the Riccati BSDEJ, outlining the reduction to the Riccati-Volterra equations and referencing appropriate existence theorems for affine Volterra models. revision: yes

  2. Referee: The paper states that the candidate value process is a supermartingale with equality attained under the candidate control, but provides no detailed verification of this property or the associated integrability conditions (see the verification step following the BSDEJ construction). This is required to rigorously apply the martingale optimality principle and confirm the derived strategies are indeed optimal.

    Authors: Thank you for highlighting the need for more explicit verification. The supermartingale property is verified by applying a generalized Itô formula adapted to Volterra processes with jumps, where the BSDEJ is designed so that the drift term vanishes, and the remaining terms form a local martingale. Integrability is ensured by the exponential integrability assumptions on the jump measure and the boundedness of the optimal controls derived from the Riccati solutions. The current presentation condenses these steps. We will expand the verification section in the revision to provide a step-by-step derivation of the supermartingale property, including the precise integrability estimates. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds from model to BSDEJ without self-referential reduction

full rationale

The paper derives the Riccati BSDEJ directly from the dynamics of the multivariate affine Volterra-Heston model with jumps by applying the martingale optimality principle and constructing candidate supermartingale processes. Optimal strategies are then expressed in semi-closed form via solutions to the associated time-dependent multivariate Riccati-Volterra equations, with the value function given by the BSDEJ solution. No step reduces by construction to its own inputs (e.g., no parameter fitted to data then renamed as prediction, no ansatz smuggled via self-citation, and no uniqueness theorem imported from overlapping prior work as an external fact). The explicit assumption that the BSDEJ admits a unique solution is a standard well-posedness hypothesis rather than a circular definition, leaving the central claim independent of the fitted values or self-referential loops.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the existence of solutions to the newly introduced Riccati BSDEJ and the associated Volterra equations, which are not standard and require verification.

axioms (1)
  • domain assumption The multivariate Volterra-Heston model with jumps is well-defined and satisfies the necessary integrability conditions for the BSDE.
    Invoked to apply the martingale optimality principle.
invented entities (1)
  • Riccati BSDEJ no independent evidence
    purpose: To characterize the supermartingale processes for the optimality principle.
    Introduced as an original equation in the paper.

pith-pipeline@v0.9.0 · 5492 in / 1322 out tokens · 83028 ms · 2026-05-09T18:47:54.687831+00:00 · methodology

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