Showing papers on "Semimartingale published in 2023"

A trajectorial approach to relative entropy dissipation of McKean–Vlasov diffusions: Gradient flows and HWBI inequalities

Abstract: We formulate a trajectorial version of the relative entropy dissipation identity for McKean–Vlasov diffusions, extending recent results which apply to non-interacting diffusions. Our stochastic analysis approach is based on time-reversal of diffusions and Lions’ differential calculus over Wasserstein space. It allows us to compute explicitly the rate of relative entropy dissipation along every trajectory of the underlying diffusion via the semimartingale decomposition of the corresponding relative entropy process. As a first application, we obtain a new interpretation of the gradient flow structure for the granular media equation, generalizing a formulation developed recently for the linear Fokker–Planck equation. Secondly, we show how the trajectorial approach leads to a new derivation of the HWBI inequality, which relates relative entropy (H), Wasserstein distance (W), barycenter (B) and Fisher information (I). MSC 2020 subject classifications: Primary 60H30, 60G44; secondary 60J60, 94A17

Power Forward Performance in Semimartingale Markets with Stochastic Integrated Factors

Abstract: We study the forward investment performance process (FIPP) in an incomplete semimartingale market model with closed and convex portfolio constraints, when the investor’s risk preferences are of the power form. We provide necessary and sufficient conditions for the existence of such a FIPP. In a semimartingale factor model, we show that the FIPP can be recovered as a triplet of processes that admit an integral representation with respect to semimartingales. Using an integrated stochastic factor model, we relate the factor representation of the triplet of processes to the smooth solution of an ill-posed partial integro-differential Hamilton–Jacobi–Bellman equation. We develop explicit constructions for the class of time-monotone FIPPs, generalizing existing results from Brownian to semimartingale market models. Funding: L. Bo was supported by the National Natural Science Foundation of China (NSFC) [Grant 11971368] and National Center for Applied Mathematics in Shaanxi (NCAMS). A. Capponi was supported in part by the National Science Foundation [Grant DMS-1716145]. C. Zhou was supported by the Singapore Ministry of Education Academic Research Fund [Grant R-146-000-271-112] and NSFC [Grant 11871364].

No-arbitrage conditions and pricing from discrete-time to continuous-time strategies

Abstract: In this paper, a general framework is developed for continuous-time financial market models defined from simple strategies through conditional topologies that avoid stochastic calculus and do not necessitate semimartingale models. We then compare the usual no-arbitrage conditions of the literature, e.g. the usual no-arbitrage conditions NFL, NFLVR and NUPBR and the recent AIP condition. With appropriate pseudo-distance topologies, we show that they hold in continuous time if and only if they hold in discrete time. Moreover, the super-hedging prices in continuous time coincide with the discrete-time super-hedging prices, even without any no-arbitrage condition.

Numeraire-Invariant Quadratic Hedging and Mean–Variance Portfolio Allocation

Abstract: The paper investigates quadratic hedging in a semimartingale market that does not necessarily contain a risk-free asset. An equivalence result for hedging with and without numeraire change is established. This permits direct computation of the optimal strategy without choosing a reference asset and/or performing a numeraire change. New explicit expressions for optimal strategies are obtained, featuring the use of oblique projections that provide unified treatment of the case with and without a risk-free asset. The analysis yields a streamlined computation of the efficient frontier for the pure investment problem in terms of three easily interpreted processes. The main result advances our understanding of the efficient frontier formation in the most general case in which a risk-free asset may not be present. Several illustrations of the numeraire-invariant approach are given.

Asymptotic Expansions for High-Frequency Option Data

Abstract: We derive a nonparametric higher-order asymptotic expansion for small-time changes of conditional characteristic functions of It\^o semimartingale increments. The asymptotics setup is of joint type: both the length of the time interval of the increment of the underlying process and the time gap between evaluating the conditional characteristic function are shrinking. The spot semimartingale characteristics of the underlying process as well as their spot semimartingale characteristics appear as leading terms in the derived asymptotic expansions. The analysis applies to a general class of It\^o semimartingales that includes in particular L\'evy-driven SDEs and time-changed L\'evy processes. The asymptotic expansion results are of direct use for constructing nonparametric estimates pertaining to the stochastic volatility dynamics of an asset from high-frequency data of options written on the underlying asset.

Stochastic Approximation Procedures for Lévy-Driven SDEs

Abstract: Abstract We consider a continuous-time Robbins–Monro-type stochastic approximation procedure for a system described by a (multidimensional) stochastic differential equation driven by a general Lévy process, and we find sufficient conditions for its convergence in terms of Lyapunov functions. While the jump part of the noise may spoil convergence to the root of the drift in some cases, we show that by a suitable choice of noise coefficients we obtain convergence under hypotheses on the drift weaker than those used in the diffusion case or convergence to a selected root in the case of multiple roots of the drift.

General theory of stochastic processes in applications

Abstract: The main goal of this chapter is to show how the general theory developed before can be applied to mathematical finance and statistics of random processes. In the area of mathematical finance a semimartingale financial market model is introduced. Applying to this general model the technique of stochastic exponents the fundamental questions of arbitrage and completeness of such a market are studied. These results have a number of corollaries for modeling and option pricing (Black-Scholes model and formula, Cox-Ross-Rubinstein model and formula etc). In the area of statistics of random processes the technique developed above gives a possibility to introduce semimartingale models. It is shown that classical discrete time and continuous time models of stochastic approximation are embedded in a semimartingale scheme. Moreover, it is proved that semimartingale stochastic approximation procedures are strong consistent and asymptotically normal under very wide conditions. In case of semimartingale regression the structural least-squared estimates are strong consistent and their sequential versions satisfy the important Fixed accuracy property fixed accuracy property (see [3], [4], [11], [13], [18], [23], [30], [31], [32], [34], and [43]).

An anticipating tangent to a semimartingale

Abstract: We construct a perturbation of identity type mapping on an abstract Wiener space where the Cameron-Martin space has an orthonormal basis indexed by the jumps a one dimensional semimartingale. We then derive a change of variables formula and a degree type result for this map.

Tool Degradation Prediction Based on Semimartingale Approximation of Linear Fractional Alpha-Stable Motion and Multi-Feature Fusion

Abstract: Tool wear will reduce workpieces’ quality and accuracy. In this paper, the vibration signals of the milling process were analyzed, and it was found that historical fluctuations still have an impact on the existing state. First of all, the linear fractional alpha-stable motion (LFSM) was investigated, along with a differential iterative model with it as the noise term is constructed according to the fractional-order Ito formula; the general solution of this model is derived by semimartingale approximation. After that, for the chaotic features of the vibration signal, the time-frequency domain characteristics were extracted using principal component analysis (PCA), and the relationship between the variation of the generalized Hurst exponent and tool wear was established using multifractal detrended fluctuation analysis (MDFA). Then, the maximum prediction length was obtained by the maximum Lyapunov exponent (MLE), which allows for analysis of the vibration signal. Finally, tool condition diagnosis was carried out by the evolving connectionist system (ECoS). The results show that the LFSM iterative model with semimartingale approximation combined with PCA and MDFA are effective for the prediction of vibration trends and tool condition. Further, the monitoring of tool condition using ECoS is also effective.

General theory of stochastic processes under “usual conditions”

Abstract: Chapter 10General theory of stochastic processes is devoted to a systematic exposition of a continuous time version of stochastic analysis under “usual conditions” with its standard notions like a stochastic basis, filtration, stopping times, random sets, predictable and optional sigma-algebras etc. It is shown how the discrete time martingale theory as well as a pure continuous time theory of diffusion processes are generalized for so-called cadlag processes. Using the predictable notion of a compensator the fundamental Doob-Meyer theorem is formulated for the class of sub- and supermartingales ofClass D class D. The full version of stochastic integration of predictable processes with respect to square-integrable martingale is developed. Moreover, different decompositions of such martingales are proved as well as the Kunuta-Watanabe inequality. It is shown how the theory can be extended with the help of localization procedures (local martingales, processes with locally integrable variation, semimartingales). The Ito formula is proved for semimartingales. SDEs with respect to semimartingales are studied including the existence and uniqueness of solutions of such equations with the Lipschitz coefficients (see [2], [8], [9], [16], [18], [20], [26], [33], [36], and [37]).

Regularity analysis for SEEs with multiplicative fBms and strong convergence for a fully discrete scheme

Abstract: One of the open problems in the study of stochastic differential equations is regularity analysis and approximations to stochastic partial differential equations driven by multiplicative fractional Brownian motions (fBms), especially for the case $H\in (0,\frac {1}{2})$. In this paper, we address this problem by considering a class of stochastic evolution equations (SEEs) driven by multiplicative fBms. We analyze the well-posedness and regularity of mild solutions to such equations with $H\in (0,\frac {1}{2})$ and $H\in (\frac {1}{2},1)$ under the Lipschitz conditions and linear growth conditions. The two cases are treated separately. Compared with the standard Brownian motion case, the main difficulty is that fBm is neither a Markov process nor a semimartingale such that some classical stochastic calculus theories are unavailable. As a consequence, we need to explore some new strategies to complete the existence and uniqueness and regularity analysis of the solutions. Especially for the case $H\in (0,\frac {1}{2})$, we utilize some delicate techniques to overcome the difficulties from the singularity of the covariance of fBms. In addition, we give a fully discrete scheme for such equations, carried out by the spectral Galerkin method in space and a time-stepping method in time. The obtained regularity results of the equations help us to examine the strong convergence of the discrete scheme. In final, several numerical examples are done to substantiate the theoretical findings.

Restoration of Well-Posedness of Infinite-Dimensional Singular ODE’s via Noise

Abstract: Abstract In this paper we aim at generalizing the results of A. K. Zvonkin ( That removes the drift , 22 , 129, 41) and A. Y. Veretennikov ( Theory Probab. Appl. , 24 , 354, 39) on the construction of unique strong solutions of stochastic differential equations with singular drift vector field and additive noise in the Euclidean space to the case of infinite-dimensional state spaces. The regularizing driving noise in our equation is chosen to be a locally non-Hölder continuous Hilbert space valued process of fractal nature, which does not allow for the use of classical construction techniques for strong solutions from PDE or semimartingale theory. Our approach, which does not resort to the Yamada-Watanabe principle for the verification of pathwise uniqueness of solutions, is based on Malliavin calculus.

Criteria for NUPBR, NFLVR and the existence of EMMs in integrated diffusion markets

Abstract: Consider a single asset financial market whose discounted asset price process is a stochastic integral with respect to a continuous regular strong Markov semimartingale (a so-called general diffusion semimartingale) that is parameterized by a scale function and a speed measure. In a previous paper, we established a characterization of the no free lunch with vanishing risk (NFLVR) condition for a canonical framework of such a financial market in terms of the scale function and the speed measure. Ioannis Karatzas (personal communication) asked us whether it is also possible to prove a characterization for the weaker no unbounded profit with bounded risk (NUPBR) condition, which is the main question we treat in this paper. Here, we do not restrict our attention to canonical frameworks but we allow a general setup with a general filtration that preserves the strong Markov property. Our main results are precise characterizations of NUPBR and NFLVR which only depend on the scale function and the speed measure. In particular, we prove that NUPBR forces the scale function to be continuously differentiable with absolutely continuous derivative. The latter extends our previous result, that, in the canonical framework, NFLVR implies such a property, in two directions (a weaker no-arbitrage notion and a more general framework). We also make the surprising observation that NUPBR and NFLVR are equivalent whenever finite boundary points are accessible for the driving diffusion.

Information loss in volatility measurement with flat price trading

Abstract: A model of financial asset price determination is proposed that incorporates flat trading features into an efficient price process. The model involves the superposition of a Brownian semimartingale process for the efficient price and a Bernoulli process that determines the extent of flat price trading. The approach is related to sticky price modeling and the Calvo pricing mechanism in macroeconomic dynamics. A limit theory for the conventional realized volatility (RV) measure of integrated volatility is developed. The results show that RV is still consistent but has an inflated asymptotic variance that depends on the probability of flat trading. Estimated quarticity is similarly affected, so that both the feasible central limit theorem and the inferential framework suggested in Barndorff-Nielsen and Shephard (J Royal Stat Soc Ser B (Stat Methodol) 64:253–280, 2002) remain valid under flat price trading even though there is information loss due to flat trading effects. The results are related to work by Jacod (J Financ Econom 16:526–569, 2018) and Mykland and Zhang (Ann Stat 34:1931–1963, 2006) on realized volatility measures with random and intermittent sampling, and to ACD models for irregularly spaced transactions data. Extensions are given to include models with microstructure noise. Some simulation results are reported. Empirical evaluations with tick-by-tick data indicate that the effect of flat trading on the limit theory under microstructure noise is likely to be minor in most cases, thereby affirming the relevance of existing approaches.

Bias reduction estimation for drift coefficient in diffusion models with jumps

Abstract: In this paper, we reconstruct the local linear threshold estimator for the drift coefficient of a semimartingale with jumps. Under mild conditions, we provide the asymptotic normality of our estimator in the presence of finite activity jumps whether the underlying process is Harris recurrent or positive recurrent. Simulation studies for different models show that our estimator performs better than previous research in finite samples, which can correct the boundary bias automatically. Finally, the estimator is illustrated empirically through the stock index from Shanghai Stock Exchange in China under 15-minute high sampling frequency.

Rough semimartingales and p-variation estimates for martingale transforms

Abstract: We establish a new scale of $p$-variation estimates for martingale paraproducts, martingale transforms, and It\^o integrals, of relevance in rough paths theory, stochastic, and harmonic analysis. As an application, we introduce rough semimartingales, a common generalization of classical semimartingales and (controlled) rough paths, and their integration theory.

Relative performance criteria of multiplicative form in complete markets

Abstract: We consider existence and uniqueness of Nash equilibria in an $N$-player game of utility maximization under relative performance criteria of multiplicative form in complete semimartingale markets. For a large class of players' utility functions, a general characterization of Nash equilibria for a given initial wealth vector is provided in terms of invertibility of a map from $\mathbb{R}^N$ to $\mathbb{R}^N$. As a consequence of the general theorem, we derive existence and uniqueness of Nash equilibria for an arbitrary initial wealth vector, as well as their convergence, if either (i) players' utility functions are close to CRRA, or (ii) players' competition weights are small and relative risk aversions are bounded away from infinity.

Convergence Uniform on Compacts in Probability and Semimartingale Topology on Duals of Nuclear Spaces

Abstract: Let $\Phi'$ denote the strong dual of a nuclear space $\Phi$. In this paper we introduce sufficient conditions for the convergence uniform on compacts in probability for a sequence of $\Phi'$-valued processes with continuous or c\'{a}dl\`{a}g paths. We use these results to introduce a topology on the space of $\Phi'$-valued semimartingales which are good integrators. This is done under the assumption that $\Phi$ is either a Fr\'{e}chet nuclear space or the strict inductive limit of Fr\'{e}chet nuclear spaces. In particular, we show that this topology is complete and that the stochastic integral mapping is continuous on the integrators.

Stochastic optimal transport with at most quadratic growth cost

Abstract: We consider a class of stochastic optimal transport, SOT for short, with given two endpoint marginals in the case where a cost function exhibits at most quadratic growth. We first study the upper and lower estimates and the short-time asymptotics of the SOT. As a by-product, we characterize the finiteness of SOT by that of the Monge-Kantorovich problem with the same two endpoint marginals. As an application, we show the existence of a continuous semimartingale, with given initial and terminal distributions, of which the drift vector is $r$th integrable for $r\in (1,2)$. We also consider the same problem and the long-time asymptotics for Schr\"odinger's problem where $r=2$. This paper is a continuation of our previous work.

Calibration of Local Volatility Models with Stochastic Interest Rates using Optimal Transport

Abstract: We develop a non-parametric, optimal transport driven, calibration methodology for local volatility models with stochastic interest rate. The method finds a fully calibrated model which is the closest to a given reference model. We establish a general duality result which allows to solve the problem via optimising over solutions to a non-linear HJB equation. We then apply the method to a sequential calibration setup: we assume that an interest rate model is given and is calibrated to the observed term structure in the market. We then seek to calibrate a stock price local volatility model with volatility coefficient depending on time, the underlying and the short rate process, and driven by a Brownian motion which can be correlated with the randomness driving the rates process. The local volatility model is calibrated to a finite number of European options prices via a convex optimisation problem derived from the PDE formulation of semimartingale optimal transport. Our methodology is analogous to Guo, Loeper, and Wang, 2022 and Guo, Loeper, Obloj, et al., 2022a but features a novel element of solving for discounted densities, or sub-probability measures. We present numerical experiments and test the effectiveness of the proposed methodology.

On two-dimensional extensions of Bougerol’s identity in law

Abstract: Let B={Bt}t≥0 be a one-dimensional standard Brownian motion and denote by At,t≥0, the quadratic variation of semimartingale eBt,t≥0. The celebrated Bougerol’s identity in law (1983) asserts that, if β={βt}t≥0 is another Brownian motion independent of B, then βAt has the same law as sinhBt for every fixed t>0. Bertoin, Dufresne and Yor (2013) obtained a two-dimensional extension of the identity involving, as the second coordinates, the local times of B and β at level zero. In this paper, we present a generalization of their extension in a situation that the levels of those local times are not restricted to zero. Our argument provides a short elementary proof of the original extension and sheds new light on that subtle identity.

On the Gaussian Volterra processes with power-type kernels

Abstract: We consider the Gaussian Volterra process Xθ={Xθ(t), t∈[0,T]}, θ=(α,β,γ) introduced in Mishura and Shklyar [Theory and Applications 2022a, 431–452] We specify the parameters θ for which Xθ is non Markovian, semimartingale, and exhibits long-range dependence. Finally, by using its Paley–Wiener–Zygmund representation we establish its continuity in θ, uniformly in t.

On the nth-order subfractional Brownian motion

Abstract: In the present work, we introduce the nth-Order subfractional Brownian motion (S_H^n (t), t ≥ 0) with Hurst index H ∈ (n − 1,n) and order n ≥ 1; then we examine some of its basic properties: self-similarity, long-range dependence, non Markovian nature and semimartingale property. A local law of iterated logarithm for S_H^n (t) is also established.

Joint calibration to SPX and VIX options with signature-based models

Abstract: We consider a stochastic volatility model where the dynamics of the volatility are described by linear functions of the (time extended) signature of a primary underlying process, which is supposed to be some multidimensional continuous semimartingale. Under the additional assumption that this primary process is of polynomial type, we obtain closed form expressions for the VIX squared, exploiting the fact that the truncated signature of a polynomial process is again a polynomial process. Adding to such a primary process the Brownian motion driving the stock price, allows then to express both the log-price and the VIX squared as linear functions of the signature of the corresponding augmented process. This feature can then be efficiently used for pricing and calibration purposes. Indeed, as the signature samples can be easily precomputed, the calibration task can be split into an offline sampling and a standard optimization. For both the SPX and VIX options we obtain highly accurate calibration results, showing that this model class allows to solve the joint calibration problem without adding jumps or rough volatility.

Inference on the intraday spot volatility from high-frequency order prices with irregular microstructure noise

Abstract: We consider estimation of the spot volatility in a stochastic boundary model with one-sided microstructure noise for high-frequency limit order prices. Based on discrete, noisy observations of an It\^o semimartingale with jumps and general stochastic volatility, we present a simple and explicit estimator using local order statistics. We establish consistency and stable central limit theorems as asymptotic properties. The asymptotic analysis builds upon an expansion of tail probabilities for the order statistics based on a generalized arcsine law. In order to use the involved distribution of local order statistics for a bias correction, an efficient numerical algorithm is developed. We demonstrate the finite-sample performance of the estimation in a Monte Carlo simulation.