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David Criens

Publications and source records attributed to David Criens.

At least 19 recordsLinked to original sources
arXiv2026

On the hedging problem in general 1D diffusion markets

We develop a PDE-based methodology for pricing and hedging European contingent claims in general one-dimensional diffusion markets characterized solely by their scale function and speed measure, possibly without a classical SDE representation, and with constant interest rate. We derive a hedging equation whose solution generates a self-financing hedging strategy and provide sufficient conditions on scale, speed, and interest rate, under which this strategy achieves the minimal hedging capital, expressed through the no free lunch with vanishing risk (NFLVR) condition. We further prove necessary and sufficient conditions for NFLVR and characterize the class of equivalent local martingale measures through an auxiliary diffusion whose scale and speed characteristics are determined by those of the real-world diffusion and by the interest rate. When the NFLVR condition fails, the framework may produce multiple hedging equations corresponding to non-minimal strategies, whose associated prices can exceed the minimal hedging capital. We illustrate both the effectiveness and limitations of the approach through numerical experiments involving diffusion models with irregular features.

q-fin.MF↗
arXiv2026

A Small-Noise Analysis of Controlled Functional Differential Equations with Gaussian Noise

We study small-noise asymptotics for controlled functional differential equations driven by additive Gaussian noise. The Gaussian noise is modeled on an abstract Wiener space, covering both classical Brownian perturbations and non-Markovian perturbations such as fractional Brownian motion. The drift coefficient is assumed to be non-anticipative, Lipschitz continuous in the state path, and of linear growth. For bounded uniformly continuous cost functionals, we prove game-theoretic lower and upper bounds for the small-noise logarithmic value functions and identify their limit whenever the associated deterministic zero-sum game has a value. In the limiting game, one player chooses the drift control, while the other selects a Cameron--Martin shift of the Gaussian noise, penalized by the corresponding quadratic energy cost. We further provide sufficient Fan-type convexity and concavity conditions under which the game has a value, thereby obtaining a full small-noise Laplace principle. The proof combines the Bou\'e--Dupuis variational representation on abstract Wiener spaces with pathwise stability of the controlled solution map and adapted finite-dimensional approximations of Cameron--Martin shifts.

math.PR↗
arXiv2026

Risk-sensitive exit-time control for stochastic differential equations with path-dependent coefficients

In this work, we study small-noise asymptotics of risk-sensitive exit-time control problems governed by stochastic differential equations with path-dependent coefficients. Our main result establishes the convergence of the $\log$-transformed exit-time problem to a deterministic control problem with path-dependent coefficients. For its proof, we first derive a novel variational representation for general $\log$-transformed stochastic control problems with path-dependent coefficients, combining tools from the theory of path-dependent partial differential equations and convex expectations on path spaces. In a second step, we use probabilistic methods to analyze the convergence of the resulting variational formulas. To illustrate the scope of our analysis, we consider a computable example for a stochastic differential equation with memory and characterize the limiting problem and associated control strategies.

math.OC↗
arXiv2025

On the structure of increasing profits in a 1D general diffusion market with interest rates

In this paper, we investigate a financial market model consisting of a risky asset, modeled as a general diffusion parameterized by a scale function and a speed measure, and a bank account process with a constant interest rate. This flexible class of financial market models allows for features such as reflecting boundaries, skewness effects, sticky points, and slowdowns on fractal sets. For this market model, we study the structure of a strong form of arbitrage opportunity called increasing profits. Our main contributions are threefold. First, we characterize the existence of increasing profits in terms of an auxiliary deterministic signed measure $\nu$ and a canonical trading strategy $\theta$, both of which depend only on the deterministic parametric characteristics of our model, namely the scale function, the speed measure, and the interest rate. More precisely, we show that an increasing profit exists if and only if $\nu$ is nontrivial, and that this is equivalent to $\theta$ itself generating an increasing profit. Second, we provide a precise characterization of the entire set of increasing profits in terms of $\nu$ and $\theta$, and moreover characterize the value processes associated with increasing profits. Finally, we establish novel connections between no-arbitrage theory and the general theory of stochastic processes. Specifically, we relate the failure of the representation property for general diffusions to the existence of certain types of increasing profits whose value processes are dominated by the quadratic variation measure of a space-transformed version of the asset price process.

q-fin.MF↗
arXiv2025

On weak notions of no-arbitrage in a 1D general diffusion market with interest rates

We establish deterministic necessary and sufficient conditions for the no-arbitrage notions "no increasing profit" (NIP), "no strong arbitrage" (NSA) and "no unbounded profit with bounded risk" (NUPBR) in one-dimensional general diffusion markets. These are markets with one risky asset, which is modeled as a regular continuous strong Markov process that is also a semimartingale, and a riskless asset that grows exponentially at a constant rate $r\in \mathbb{R}$. All deterministic criteria are provided in terms of the scale function and the speed measure of the risky asset process. Our study reveals a variety of surprising effects. For instance, irrespective of the interest rate, NIP is not excluded by reflecting boundaries or an irregular scale function. In the case of non-zero interest rates, it is even possible that NUPBR holds in the presence of reflecting boundaries and/or skew thresholds. In the zero interest rate regime, we also identify NSA as the minimal no arbitrage notion that excludes reflecting boundaries and that forces the scale function to be continuously differentiable with strictly positive absolutely continuous derivative, meaning that it is of the same form as for a stochastic differential equation.

q-fin.MF↗
arXiv2025

Representation Theorems for Convex Expectations and Semigroups on Path Space

The objective of this paper is to investigate the connection between penalty functions from stochastic optimal control, convex semigroups from analysis and convex expectations from probability theory. Our main result provides a one-to-one relation between these objects. As an application, we use the representation via penality functions and duality arguments to show that convex expectations are determined by their finite dimensional distributions. To illustrate this structural result, we show that Hu and Peng's axiomatic description of $G$-L\'evy processes in terms of finite dimensional distributions extends uniquely to the control approach introduced by Neufeld and Nutz. Finally, we show that convex expectations with a Markovian structure are fully determined by their one-dimensional distributions, which give rise to a classical semigroup on the state space. As an application of this result, we establish a Laplace principle for entropic risk measures associated to controlled diffusions.

math.OC↗
arXiv2024

No arbitrage and the existence of ACLMMs in general diffusion models

In a seminal paper, F. Delbaen and W. Schachermayer proved that the classical NA ("no arbitrage") condition implies the existence of an "absolutely continuous local martingale measure" (ACLMM). It is known that in general the existence of an ACLMM alone is not sufficient for NA. In this paper we investigate how close these notions are for single asset general diffusion market models. We show that NA is equivalent to the existence of an ACLMM plus a mild regularity condition on the scale function and the absence of reflecting boundaries. For infinite time horizon scenarios, the regularity assumption and the requirement on the boundaries can be dropped, showing equivalence between NA and the existence of an ACLMM. By means of counterexamples, we show that our characterization of NA for finite time horizons is sharp in the sense that neither the regularity condition on the scale function nor the absence of reflecting boundaries can be dropped.

q-fin.MF↗
arXiv2024

On the representation property for 1D general diffusion semimartingales

A general diffusion semimartingale is a one-dimensional path-continuous semimartingale that is also a regular strong Markov process. We say that a continuous semimartingale has the representation property if all local martingales w.r.t. its canonical filtration have an integral representation w.r.t. its continuous local martingle part. The representation property is of fundamental interest in the field of mathematical finance, where it is strongly connected to market completeness. The main result from this paper shows that the representation property holds for a general diffusion semimartingale (that is not started in an absorbing boundary point) if and only if its scale function is (locally) absolutely continuous on the interior of the state space. As an application of our main theorem, we deduce that the laws of general diffusion semimartingales with such scale functions are extreme points of their semimartingale problems, and, moreover, we construct a general diffusion semimartingale whose law is no extreme point of its semimartingale problem. These observations contribute to a solution of problems posed by J. Jacod and M. Yor, and D. W. Stroock and M. Yor, on the extremality of strong Markov solutions.

math.PR↗
arXiv2024

Stochastic optimal control problems with measurable coefficients and $L_d$-drift

We consider controlled stochastic differential equations (SDEs) with measurable coefficients, a uniformly elliptic diffusion coefficient and an $L_d$-drift. No space-regularity will be assumed for the coefficients. In this framework we investigate the relation of value functions, partial differential equations (PDEs) and operator semigroups. First, for a cost with infinite time horizon on a bounded domain, we identify the value function as $L_{d_0}$-viscosity solution to a Hamilton-Jacobi-Bellman equation and we establish quantitative regularity estimates. The constant $d_0 \in (d/2, d)$ only depends on the space dimension $d$, the ellipticity constants of the diffusion coefficient and the $L_d$-bound of the drift. To illustrate applications of these results, we provide a uniqueness theorem under an additional assumption on the diffusion coefficient, showing a stochastic representation, and we discuss stability of value functions. Second, we consider a cost with a finite time horizon, terminal and running terms. We show that the value function indexed over the terminal cost is a nonlinear semigroup on $C_b (\mathbb{R}^d)$ and we establish a regularization by noise effect, which shows that the semigroup regularizes lower semicontinuity to local H\"older continuity. Lastly, we relate the semigroup to a parabolic PDE, showing that it is an $L_{d + 1}$-viscosity solution, and we establish local in time and global in space quantitative regularity estimates. Our proofs for the regularity of the value functions, the $C_b$-Feller property of the semigroup and its regularization by noise effects are based on a strong Markov selection principle and analytic estimates for linear diffusions that were recently established by N. V. Krylov in a series of papers. We highlight that our method covers frameworks without uniqueness of the controlled SDEs, as well as the associated PDEs.

math.AP↗
arXiv2024

Robust Market Convergence: From Discrete to Continuous Time

Continuous time financial market models are often motivated as scaling limits of discrete time models. The objective of this paper is to establish such a connection for a robust framework. More specifically, we consider discrete time models that are parameterized by Markovian transition kernels, and a continuous time framework with drift and volatility uncertainty, again parameterized in a Markovian way. Our main result is a limit theory that establishes convergence of the uncertainty sets in the Hausdorff metric topology and weak convergence of the associated worst-case expectations. Furthermore, we discuss a structure preservation property of certain approximations. Namely, we establish the convergence of discrete to continuous time robust superhedging prices for some complete robust market models. As illustration of our main results, we use the idea of Kushner's Markov chain approximation method and provide a recursive algorithm for the computation of continuous time robust superhedging prices.

math.PR↗
arXiv2023

Set-valued propagation of chaos for controlled path-dependent McKean-Vlasov SPDEs

We develop a limit theory for controlled path-dependent mean field stochastic partial differential equations (SPDEs) within the semigroup approach of Da Prato and Zabczyk. More precisely, we prove existence results for mean field limits and particle approximations, and we establish set-valued propagation of chaos in the sense that we show convergence of sets of empirical distributions to sets of mean field limits in the Hausdorff metric topology. Furthermore, we discuss consequences of our results to stochastic optimal control. As another application, we deduce a propagation of chaos result for Peng's $G$-Brownian motion with drift interaction.

math.PR↗
arXiv2023

A stochastic representation theorem for sublinear semigroups with non-local generators

In this paper we investigate sublinear semigroups whose pointwise generators are given by non-local Hamilton-Jacobi-Bellman operators. Our main result provides a stochastic representation in terms of a family of sublinear (conditional) expectations that can be understood as a nonlinear Markov family with uncertain local characteristics. The proofs are based on viscosity methods.

math.PR↗
arXiv2023

Nonlinear Semimartingales and Markov Processes with Jumps

In this paper we study a family of nonlinear (conditional) expectations that can be understood as a semimartingale with uncertain local characteristics. Here, the differential characteristics are prescribed by a time and path-dependent set-valued function. We show that the associated control problem coincides with both its weak and relaxed counterparts. Furthermore, we establish regularity properties of the value function and discuss their relation to Feller properties of nonlinear semigroups. In the Markovian case we provide conditions that allow us to identify the corresponding semigroup as the unique viscosity solution to a nonlinear Hamilton-Jacobi-Bellman equation. To illustrate our results we discuss a random $G$-double exponential L\'evy setting.

math.PR↗
arXiv2023

A limit theory for controlled McKean-Vlasov SPDEs

We develop a limit theory for controlled mean field stochastic partial differential equations in a variational framework. More precisely, we prove existence results for mean field limits and particle approximations, and we establish a set-valued propagation of chaos result which shows that sets of empirical distributions converge to sets of mean field limits in the Hausdorff metric topology. Further, we discuss limit theorems related to stochastic optimal control theory. To illustrate our findings, we apply them to a controlled interacting particle system of stochastic porous media equations.

math.PR↗
arXiv2023

Criteria for the absence of arbitrage in general diffusion markets

We establish deterministic necessary and sufficient conditions for the no-arbitrage notions NA ("no arbitrage"), NUPBR ("no unbounded profit with bounded risk") and NFLVR ("no free lunch with vanishing risk") in general diffusion market models with finite and infinite time horizons. These are single asset models whose (discounted) asset price process $Y$ is a regular continuous strong Markov process that is also a semimartingale. We further characterize the existence of an equivalent martingale measure in such models. All deterministic criteria are provided in terms of the scale function and the speed measure of $Y$.

q-fin.MF↗
arXiv2022

Separating Times for One-Dimensional General Diffusions

The separating time for two probability measures on a filtered space is an extended stopping time which captures the phase transition between equivalence and singularity. More specifically, two probability measures are equivalent before their separating time and singular afterwards. In this paper, we investigate the separating time for two laws of general one-dimensional regular continuous strong Markov processes, so-called general diffusions, which are parameterized via scale functions and speed measures. Our main result is a representation of the corresponding separating time as (loosely speaking) a hitting time of a deterministic set which is characterized via speed and scale. As hitting times are fairly easy to understand, our result gives access to explicit and easy-to-check sufficient and necessary conditions for two laws of general diffusions to be (locally) absolutely continuous and/or singular. Most of the related literature treats the case of stochastic differential equations. In our setting we encounter several novel features, which are due to general speed and scale on the one hand, and to the fact that we do not exclude (instantaneous or sticky) reflection on the other hand. These new features are discussed in a variety of examples. As an application of our main theorem, we investigate the no arbitrage concept no free lunch with vanishing risk (NFLVR) for a single asset financial market whose (discounted) asset is modeled as a general diffusion which is bounded from below (e.g., non-negative). More precisely, we derive deterministic criteria for NFLVR and we identify the (unique) equivalent local martingale measure as the law of a certain general diffusion on natural scale.

math.PR↗
arXiv2022

Stochastic Processes under Parameter Uncertainty

In this paper we study a family of nonlinear (conditional) expectations that can be understood as a stochastic process with uncertain parameters. We develop a general framework which can be seen as a version of the martingale problem method of Stroock and Varadhan with parameter uncertainty. To illustrate our methodology, we explain how it can be used to model nonlinear L\'evy processes in the sense of Neufeld and Nutz, and we introduce the new class of stochastic partial differential equations under parameter uncertainty. Moreover, we study properties of the nonlinear expectations. We prove the dynamic programming principle, i.e., the tower property, and we establish conditions for the (strong) $\textit{USC}_b$-Feller property and a strong Markov selection principle.

math.PR↗