Kabanov Conic Model: Geometric Market Framework

• The Kabanov conic model is a numeraire-free financial framework that represents portfolios as vectors and orders them using a cone-induced partial order.
• It encodes proportional transaction costs and market solvency through bid-ask matrices and closed convex cones, ensuring no-arbitrage conditions.
• The model supports vector-valued optimization and duality, addressing challenges in utility maximization and filtration-sensitive strategy admissibility.

Searching arXiv for the cited Kabanov-model-related papers to ground the article in the current literature. The Kabanov conic model is a numeraire-free formulation of financial markets with proportional transaction costs in which portfolios are represented as vectors of physical asset holdings and compared through a cone-induced partial order rather than by liquidation into a single scalar wealth variable. In the literature considered here, this geometry is realized in several closely related ways: as a discrete-time bid-ask and solvency-cone model for foreign exchange markets, as an abstract vector-valued market model ordered by a cone, and as a simplified continuous-time conic model used to study utility maximization under filtration-sensitive admissibility constraints (Bassett et al., 2016, Sikic, 2015, Sidorenko, 4 Sep 2025).

1. Geometric core of the conic formulation

A defining feature of the Kabanov framework is that a portfolio is a vector in $\mathbb{R}^d$, interpreted as holdings in physical units of $d$ assets, rather than a scalar wealth process expressed in a chosen numéraire. This is the explicit starting point of the multistage portfolio paper, which contrasts the conic model with liquidation-based formulations and emphasizes that, with transaction costs, liquidation is numeraire-dependent and may be unrealistic (Bassett et al., 2016).

In the discrete-time foreign-exchange formulation, proportional transaction costs are encoded by a bid-ask matrix $\Pi=(\pi^{ij})$ satisfying

$$\pi^{ij}>0,\qquad \pi^{ii}=1,\qquad \pi^{ij}\le \pi^{ik}\pi^{kj}.$$

The solvency cone is then

$$K(\Pi)=\operatorname{cone}\big(\{e^i\}_{i=1}^d \cup \{\pi^{ij}e^i-e^j:\ 1\le i,j\le d\}\big).$$

Economically, $K(\Pi)$ is the cone of portfolios that can be reduced to the zero portfolio by admissible trades and disposal, while $-K(\Pi)$ is the cone of positions obtainable from zero at zero net cost (Bassett et al., 2016).

The 2025 study uses a simplified but recognizably Kabanov-style conic structure. It fixes a closed proper convex cone

$$K\subset \mathbb R^d,\qquad \mathbb R^d_{++}\subset K,\qquad K\cap(-K)=\{0\},$$

and defines the dual cone by

$$K^*=\{y\in \mathbb R^d:\ xy\ge 0\ \forall x\in K\},$$

with

$$K^*\setminus\{0\}\subset \mathbb R^d_{++},\qquad K^{**}=K.$$

This version does not use a random time-dependent solvency cone $d$0, and the source explicitly notes that it is “a somewhat simplified conic market model compared with the most general Kabanov framework” (Sidorenko, 4 Sep 2025).

The common geometric principle is that trading and solvency are encoded by cones, and portfolio comparison is only partial. This distinguishes the conic model from scalar wealth formulations and explains why vector optimization, dual cones, and cone-ordered admissibility recur throughout the literature.

2. Trading, self-financing, and admissibility

In the finite-state discrete-time formulation, the market evolves on a filtered probability space

$d$1

with an adapted bid-ask process $d$2 and induced cone-valued process $d$3. A portfolio process $d$4 is self-financing when its increments

$d$5

satisfy

$d$6

with $d$7 by convention. The cone of portfolios attainable from initial endowment $d$8 is

$d$9

and from a nonzero initial endowment $\Pi=(\pi^{ij})$0 the feasible set is

$\Pi=(\pi^{ij})$1

Thus self-financing is expressed geometrically by cone constraints on increments rather than by a scalar stochastic integral identity (Bassett et al., 2016).

The continuous-time paper formulates the market as

$\Pi=(\pi^{ij})$2

where $\Pi=(\pi^{ij})$3 is generated by the null sets of $\Pi=(\pi^{ij})$4 and an $\Pi=(\pi^{ij})$5-valued càdlàg process $\Pi=(\pi^{ij})$6, and

$\Pi=(\pi^{ij})$7

is $\Pi=(\pi^{ij})$8-valued, adapted, continuous, with

$\Pi=(\pi^{ij})$9

A path $$\pi^{ij}>0,\qquad \pi^{ii}=1,\qquad \pi^{ij}\le \pi^{ik}\pi^{kj}.$$0 is called $$\pi^{ij}>0,\qquad \pi^{ii}=1,\qquad \pi^{ij}\le \pi^{ik}\pi^{kj}.$$1-increasing if

$$\pi^{ij}>0,\qquad \pi^{ii}=1,\qquad \pi^{ij}\le \pi^{ik}\pi^{kj}.$$2

and $$\pi^{ij}>0,\qquad \pi^{ii}=1,\qquad \pi^{ij}\le \pi^{ik}\pi^{kj}.$$3-decreasing if it is $$\pi^{ij}>0,\qquad \pi^{ii}=1,\qquad \pi^{ij}\le \pi^{ik}\pi^{kj}.$$4-increasing. Let $$\pi^{ij}>0,\qquad \pi^{ii}=1,\qquad \pi^{ij}\le \pi^{ik}\pi^{kj}.$$5 denote the set of all $$\pi^{ij}>0,\qquad \pi^{ii}=1,\qquad \pi^{ij}\le \pi^{ik}\pi^{kj}.$$6-decreasing càdlàg paths. Any adapted process $$\pi^{ij}>0,\qquad \pi^{ii}=1,\qquad \pi^{ij}\le \pi^{ik}\pi^{kj}.$$7 whose paths lie in $$\pi^{ij}>0,\qquad \pi^{ii}=1,\qquad \pi^{ij}\le \pi^{ik}\pi^{kj}.$$8 is a strategy (Sidorenko, 4 Sep 2025).

For initial endowment $$\pi^{ij}>0,\qquad \pi^{ii}=1,\qquad \pi^{ij}\le \pi^{ik}\pi^{kj}.$$9, the portfolio wealth in physical units is

$$K(\Pi)=\operatorname{cone}\big(\{e^i\}_{i=1}^d \cup \{\pi^{ij}e^i-e^j:\ 1\le i,j\le d\}\big).$$0

with componentwise Riemann–Stieltjes integration. Wealth in numéraire units is

$$K(\Pi)=\operatorname{cone}\big(\{e^i\}_{i=1}^d \cup \{\pi^{ij}e^i-e^j:\ 1\le i,j\le d\}\big).$$1

A strategy is admissible if

$$K(\Pi)=\operatorname{cone}\big(\{e^i\}_{i=1}^d \cup \{\pi^{ij}e^i-e^j:\ 1\le i,j\le d\}\big).$$2

The same source proves, via Appendix Lemma $$K(\Pi)=\operatorname{cone}\big(\{e^i\}_{i=1}^d \cup \{\pi^{ij}e^i-e^j:\ 1\le i,j\le d\}\big).$$3, that $$K(\Pi)=\operatorname{cone}\big(\{e^i\}_{i=1}^d \cup \{\pi^{ij}e^i-e^j:\ 1\le i,j\le d\}\big).$$4-decreasing paths have bounded variation and their Radon–Nikodym derivative with respect to total variation lies in $$K(\Pi)=\operatorname{cone}\big(\{e^i\}_{i=1}^d \cup \{\pi^{ij}e^i-e^j:\ 1\le i,j\le d\}\big).$$5, validating the pathwise Riemann–Stieltjes wealth integral and giving a geometric self-financing interpretation (Sidorenko, 4 Sep 2025).

A central technical point is that admissibility in the conic model is filtration-dependent. In the 2025 setup the filtration is generated by $$K(\Pi)=\operatorname{cone}\big(\{e^i\}_{i=1}^d \cup \{\pi^{ij}e^i-e^j:\ 1\le i,j\le d\}\big).$$6, so admissible strategies are adapted functionals of $$K(\Pi)=\operatorname{cone}\big(\{e^i\}_{i=1}^d \cup \{\pi^{ij}e^i-e^j:\ 1\le i,j\le d\}\big).$$7. This becomes decisive in probability-space transfer arguments.

3. Ordering, optimization, and duality

The conic model is naturally associated with optimization problems over vector-valued terminal portfolios. In the continuous-time utility paper, the primal problem is

$$K(\Pi)=\operatorname{cone}\big(\{e^i\}_{i=1}^d \cup \{\pi^{ij}e^i-e^j:\ 1\le i,j\le d\}\big).$$8

or, when the model must be specified,

$$K(\Pi)=\operatorname{cone}\big(\{e^i\}_{i=1}^d \cup \{\pi^{ij}e^i-e^j:\ 1\le i,j\le d\}\big).$$9

Here $K(\Pi)$0 is simply “interpreted as the utility function.” The source explicitly does not impose standard assumptions such as concavity, monotonicity, upper semicontinuity, boundedness from above, Inada conditions, asymptotic elasticity, or differentiability. It also emphasizes that the formulation remains on the primal geometric side of the conic framework: there is no liquidation into numéraire at terminal time, no support function of $K(\Pi)$1, and no dual optimization problem with consistent price systems (Sidorenko, 4 Sep 2025).

The same paper calls $K(\Pi)$2 the “Bellman function,” but it also states that there is no recursive dynamic programming principle, no family $K(\Pi)$3, and no Bellman equation. In that context, “Bellman functional” means only the value function (Sidorenko, 4 Sep 2025). This addresses a common terminological misunderstanding.

In the finite-state multistage paper, the optimization problem is embedded into set-valued optimization. The utility is componentwise,

$K(\Pi)$4

with each $K(\Pi)$5 strictly concave, strictly increasing, differentiable, and satisfying the stated Inada-type limits. The objective map is

$K(\Pi)$6

and the primal problem is

$K(\Pi)$7

This converts expected utility maximization into minimization in the ordered family $K(\Pi)$8, preserving the vector character of terminal portfolios rather than scalarizing by liquidation (Bassett et al., 2016).

The same work derives a dual problem over

$K(\Pi)$9

with dual objective

$-K(\Pi)$0

and proves strong duality: $-K(\Pi)$1 The paper identifies convexity, Slater’s condition, and the conic no-arbitrage geometry as the structural basis of the result (Bassett et al., 2016).

4. No-arbitrage, recession analysis, and abstraction beyond concavity

A different line of development treats the Kabanov model as a special case of a broader theory of market models. In that framework, a market model is a mapping

$-K(\Pi)$2

or, in the vector case,

$-K(\Pi)$3

defined on adapted strategies

$-K(\Pi)$4

The two axioms are A1 (normalization), $-K(\Pi)$5, and A2 (locality), which formalizes nonanticipativity under time-local modifications of the strategy (Sikic, 2015).

Under upper-semicontinuity, the scalar market model admits a normal-integrand representation: $-K(\Pi)$6 for every adapted strategy. In the vector-valued case, exact recovery is weaker: one obtains a measurable map

$-K(\Pi)$7

such that

$-K(\Pi)$8

where the order is induced by a random closed convex cone $-K(\Pi)$9 (Sikic, 2015).

This abstract formulation is explicitly designed to go “beyond the concave case.” In particular, the source says that it studies market models without concavity assumptions and extends the theory to vector-valued outcomes, “an example of which is the Kabanov model of currency markets” (Sikic, 2015).

The no-arbitrage concept is correspondingly recession-based. In the scalar case, the recession market model is

$$K\subset \mathbb R^d,\qquad \mathbb R^d_{++}\subset K,\qquad K\cap(-K)=\{0\},$$0

and no-arbitrage is

$$K\subset \mathbb R^d,\qquad \mathbb R^d_{++}\subset K,\qquad K\cap(-K)=\{0\},$$1

In the vector case, using the positive polar

$$K\subset \mathbb R^d,\qquad \mathbb R^d_{++}\subset K,\qquad K\cap(-K)=\{0\},$$2

no-arbitrage becomes

$$K\subset \mathbb R^d,\qquad \mathbb R^d_{++}\subset K,\qquad K\cap(-K)=\{0\},$$3

The paper states that in the positively homogeneous case this reduces to

$$K\subset \mathbb R^d,\qquad \mathbb R^d_{++}\subset K,\qquad K\cap(-K)=\{0\},$$4

and explicitly identifies this with the classical efficient no-arbitrage condition of Kabanov-type conic models (Sikic, 2015).

The consequences are structural. Under no-arbitrage, the superhedging set is closed in probability, hedging strategies are bounded in probability, and the paper proves an existence result for utility maximization that does not rely on concavity of utility or of the market model (Sikic, 2015). This places the Kabanov model inside a broader nonconcave, nonhomogeneous theory while preserving its cone-ordered logic in the homogeneous case.

5. Consistent pricing, attainable portfolios, and the finite-state FTAP

In the multistage conic portfolio paper, the no-arbitrage condition is stated directly in terms of attainable portfolios: $$K\subset \mathbb R^d,\qquad \mathbb R^d_{++}\subset K,\qquad K\cap(-K)=\{0\},$$5 This means one cannot start from zero and end with a nonnegative, nonzero portfolio (Bassett et al., 2016).

Dual pricing objects are introduced through the positive polar cone

$$K\subset \mathbb R^d,\qquad \mathbb R^d_{++}\subset K,\qquad K\cap(-K)=\{0\},$$6

and a consistent price system for a bid-ask matrix $$K\subset \mathbb R^d,\qquad \mathbb R^d_{++}\subset K,\qquad K\cap(-K)=\{0\},$$7 is any nonzero

$$K\subset \mathbb R^d,\qquad \mathbb R^d_{++}\subset K,\qquad K\cap(-K)=\{0\},$$8

At the process level, an adapted process $$K\subset \mathbb R^d,\qquad \mathbb R^d_{++}\subset K,\qquad K\cap(-K)=\{0\},$$9 with values in $$K^*=\{y\in \mathbb R^d:\ xy\ge 0\ \forall x\in K\},$$0 is a consistent pricing process if it is a martingale and satisfies

$$K^*=\{y\in \mathbb R^d:\ xy\ge 0\ \forall x\in K\},$$1

for every $$K^*=\{y\in \mathbb R^d:\ xy\ge 0\ \forall x\in K\},$$2. The paper cites Kabanov–Stricker’s finite-state version of the Fundamental Theorem of Asset Pricing: the bid-ask process satisfies no arbitrage if and only if there exists a consistent pricing process (Bassett et al., 2016).

This dual structure is not merely ancillary to optimization. The same source proves the equivalent dual feasibility condition

$$K^*=\{y\in \mathbb R^d:\ xy\ge 0\ \forall x\in K\},$$3

and uses it in the proof that the primal value is proper. In the resulting duality theory, the variable $$K^*=\{y\in \mathbb R^d:\ xy\ge 0\ \forall x\in K\},$$4 supports the attainable set, while $$K^*=\{y\in \mathbb R^d:\ xy\ge 0\ \forall x\in K\},$$5 scalarizes the ordering cone (Bassett et al., 2016).

The paper’s two-asset, three-period example makes this geometry explicit. On

$$K^*=\{y\in \mathbb R^d:\ xy\ge 0\ \forall x\in K\},$$6

the bid-ask matrix process is

$$K^*=\{y\in \mathbb R^d:\ xy\ge 0\ \forall x\in K\},$$7

which yields solvency cones

$$K^*=\{y\in \mathbb R^d:\ xy\ge 0\ \forall x\in K\},$$8

or equivalently

$$K^*=\{y\in \mathbb R^d:\ xy\ge 0\ \forall x\in K\},$$9

The example is used to show concretely how a bid-ask process determines solvency cones, how attainable portfolios are formed by sums of these cones, and how dual feasibility is computed from the intersection of polar cones (Bassett et al., 2016).

6. Filtration sensitivity, Skorokhod transfer, and scope of current formulations

A foundational issue arises when one studies utility maximization in conic market models by weak convergence and Skorokhod representation methods. The 2025 paper examines exactly this point. Because the filtration is generated by the process $$K^*\setminus\{0\}\subset \mathbb R^d_{++},\qquad K^{**}=K.$$0, admissible strategies are $$K^*\setminus\{0\}\subset \mathbb R^d_{++},\qquad K^{**}=K.$$1-adapted functionals of $$K^*\setminus\{0\}\subset \mathbb R^d_{++},\qquad K^{**}=K.$$2, and law preservation of $$K^*\setminus\{0\}\subset \mathbb R^d_{++},\qquad K^{**}=K.$$3 does not automatically preserve adaptedness of a transferred strategy (Sidorenko, 4 Sep 2025).

The paper states the obstacle in explicit terms: Skorokhod representation does not necessarily preserve adaptedness. A process $$K^*\setminus\{0\}\subset \mathbb R^d_{++},\qquad K^{**}=K.$$4 adapted to $$K^*\setminus\{0\}\subset \mathbb R^d_{++},\qquad K^{**}=K.$$5 on the original space may become a process $$K^*\setminus\{0\}\subset \mathbb R^d_{++},\qquad K^{**}=K.$$6 with the same law on a new space, while $$K^*\setminus\{0\}\subset \mathbb R^d_{++},\qquad K^{**}=K.$$7 fails to be measurable with respect to $$K^*\setminus\{0\}\subset \mathbb R^d_{++},\qquad K^{**}=K.$$8. In the conic setting this is serious because admissibility combines filtration-adaptedness, pathwise $$K^*\setminus\{0\}\subset \mathbb R^d_{++},\qquad K^{**}=K.$$9-decreasing self-financing structure, and the cone-valued solvency condition

$d$00

The law of the control alone is not sufficient (Sidorenko, 4 Sep 2025).

The paper does not present a new Skorokhod theorem. Instead, it proves value-function consistency across probability spaces. Under Assumption H, namely that $d$01 has independent increments, it first shows that enlarging the filtration by an independent uniform random variable $d$02 does not change the Bellman functional: $d$03 It then proves the main equal-law invariance result: $d$04 Finally, it removes Assumption H by using the prediction process and shows that

$d$05

with a further extension involving

$d$06

The paper discusses two routes from the literature—one requiring independent increments and one based on the prediction process and extended weak convergence—and mentions Chau–Rásonyi, Aldous, Jakubowski–Słomiński, Bayraktar–Dolinsky–Dolinsky, and Hoover in this connection (Sidorenko, 4 Sep 2025).

These results have a sharply delimited meaning. They establish value-function invariance under law-preserving transfer of the relevant market data, but they do not establish existence of optimal strategies, uniqueness, direct pathwise transfer of an optimizer, or dual characterization by consistent price systems (Sidorenko, 4 Sep 2025). This suggests that, within current formulations, Skorokhod-type arguments are reliable for Bellman-functional statements only when filtration information is explicitly controlled.

Taken together, the cited works show that the Kabanov conic model is best understood as a cone-geometric, vector-portfolio approach to markets with transaction costs rather than as a single canonical specification. What remains stable across the formulations is the use of cones to encode feasibility, solvency, and order; what varies is the surrounding analytic machinery—bid-ask matrices and consistent pricing processes in finite-state discrete time, abstract normal-integrand and recession analysis beyond concavity, or probability-space invariance results for filtration-sensitive utility maximization (Bassett et al., 2016, Sikic, 2015, Sidorenko, 4 Sep 2025).