Showing papers on "Semimartingale published in 2022"

Applications of weak transport theory

Abstract: Motivated by applications to geometric inequalities, Gozlan, Roberto, Samson, and Tetali (J. Funct. Anal. 273 (2017) 3327–3405) introduced a transport problem for ‘weak’ cost functionals. Basic results of optimal transport theory can be extended to this setup in remarkable generality. In this article, we collect several problems from different areas that can be recast in the framework of weak transport theory, namely: the Schrodinger problem, the Brenier–Strassen theorem, optimal mechanism design, linear transfers, semimartingale transport. Our viewpoint yields a unified approach and often allows to strengthen the original results.

Applications of weak transport theory

Abstract: Motivated by applications to geometric inequalities, Gozlan, Roberto, Samson, and Tetali (J. Funct. Anal. 273 (2017) 3327–3405) introduced a transport problem for ‘weak’ cost functionals. Basic results of optimal transport theory can be extended to this setup in remarkable generality. In this article, we collect several problems from different areas that can be recast in the framework of weak transport theory, namely: the Schrödinger problem, the Brenier–Strassen theorem, optimal mechanism design, linear transfers, semimartingale transport. Our viewpoint yields a unified approach and often allows to strengthen the original results.

Stochastic energy balance climate models with Legendre weighted diffusion and an additive cylindrical Wiener process forcing

Abstract: <p style='text-indent:20px;'>We consider a class of one-dimensional nonlinear stochastic parabolic problems associated to Sellers and Budyko diffusive energy balance climate models with a Legendre weighted diffusion and an additive cylindrical Wiener processes forcing. Our results use in an important way that, under suitable assumptions on the Wiener processes, a suitable change of variables leads the problem to a pathwise random PDE, hence an essentially "deterministic" formulation depending on a random parameter. Two applications are also given: the stability of solutions when the Wiener process converges to zero and the asymptotic behaviour of solutions for large time.</p>

Simplified stochastic calculus via semimartingale representations

Abstract: We develop a stochastic calculus that makes it easy to capture a variety of predictable transformations of semimartingales such as changes of variables, stochastic integrals, and their compositions. The framework offers a unified treatment of real-valued and complex-valued semimartingales. The proposed calculus is a blueprint for the derivation of new relationships among stochastic processes with specific examples provided below.

Non‐geometric rough paths on manifolds

Abstract: We provide a theory of manifold-valued rough paths of bounded 3 > p $3 > p$ -variation, which we do not assume to be geometric. Rough paths are defined in charts, relying on the vector space-valued theory of Friz and Hairer (A course on rough paths, 2014), and coordinate-free (but connection-dependent) definitions of the rough integral of cotangent bundle-valued controlled paths, and of rough differential equations driven by a rough path valued in another manifold, are given. When the path is the realisation of semimartingale, we recover the theory of Itô integration and stochastic differential equations on manifolds (Émery, Stochastic calculus in manifolds, 1989). We proceed to present the extrinsic counterparts to our local formulae, and show how these extend the work in Cass et al. (Proc. Lond. Math. Soc. (3) 111 (2015) 1471–1518) to the setting of non-geometric rough paths and controlled integrands more general than 1-forms. In the last section, we turn to parallel transport and Cartan development: the lack of geometricity leads us to make the choice of a connection on the tangent bundle of the manifold T M $TM$ , which figures in an Itô correction term in the parallelism rough differential equation; such connection, which is not needed in the geometric/Stratonovich setting, is required to satisfy properties which guarantee well-definedness, linearity, and optionally isometricity of parallel transport. We conclude by providing a few examples that explore the additional subtleties introduced by our change in perspective.

Unified signature cumulants and generalized Magnus expansions

Abstract: Abstract The signature of a path can be described as its full non-commutative exponential. Following T. Lyons, we regard its expectation, the expected signature , as a path space analogue of the classical moment generating function. The logarithm thereof, taken in the tensor algebra, defines the signature cumulant . We establish a universal functional relation in a general semimartingale context. Our work exhibits the importance of Magnus expansions in the algorithmic problem of computing expected signature cumulants and further offers a far-reaching generalization of recent results on characteristic exponents dubbed diamond and cumulant expansions with motivations ranging from financial mathematics to statistical physics. From an affine semimartingale perspective, the functional relation may be interpreted as a type of generalized Riccati equation.

Kernel estimation of spot volatility with microstructure noise using pre-averaging

Abstract: We revisit the problem of estimating the spot volatility of an Itô semimartingale using a kernel estimator. A central limit theorem (CLT) with an optimal convergence rate is established for a general two-sided kernel. A new pre-averaging/kernel estimator for spot volatility is also introduced to handle the microstructure noise of ultra high-frequency observations. A CLT for the estimation error of the new estimator is obtained, and the optimal selection of the bandwidth and kernel function is subsequently studied. It is shown that the pre-averaging/kernel estimator’s asymptotic variance is minimal for two-sided exponential kernels, hence justifying the need of working with kernels of unbounded support. Feasible implementation of the proposed estimators with optimal bandwidth is developed as well. Monte Carlo experiments confirm the superior performance of the new method.

High-dimensional estimation of quadratic variation based on penalized realized variance

Abstract: In this paper, we develop a penalized realized variance (PRV) estimator of the quadratic variation (QV) of a high-dimensional continuous Itô semimartingale. We adapt the principle idea of regularization from linear regression to covariance estimation in a continuous-time high-frequency setting. We show that under a nuclear norm penalization, the PRV is computed by soft-thresholding the eigenvalues of realized variance (RV). It therefore encourages sparsity of singular values or, equivalently, low rank of the solution. We prove our estimator is minimax optimal up to a logarithmic factor. We derive a concentration inequality, which reveals that the rank of PRV is—with a high probability—the number of non-negligible eigenvalues of the QV. Moreover, we also provide the associated non-asymptotic analysis for the spot variance. We suggest an intuitive data-driven subsampling procedure to select the shrinkage parameter. Our theory is supplemented by a simulation study and an empirical application. The PRV detects about three–five factors in the equity market, with a notable rank decrease during times of distress in financial markets. This is consistent with most standard asset pricing models, where a limited amount of systematic factors driving the cross-section of stock returns are perturbed by idiosyncratic errors, rendering the QV—and also RV—of full rank.

Log-Optimal Portfolio without NFLVR: Existence, Complete Characterization, and Duality

Abstract: This paper addresses the log-optimal portfolio, which is the portfolio with finite expected log-utility that maximizes the expected logarithm utility from terminal wealth, for an arbitrary general semimartingale model. The most advanced literature on this topic elaborates existence and characterization of this portfolio under the no-free-lunch-with-vanishing-risk (NFLVR for short) assumption, while there are many financial models violating NFLVR and admitting the log-optimal portfolio. In this paper, we provide a complete and explicit characterization of the log-optimal portfolio and its associated optimal deflator, give necessary and sufficient conditions for their existence, and elaborate their duality no matter what the market model. Furthermore, our characterization gives an explicit and direct relationship between log-optimal and numéraire portfolios without changing the probability or the numéraire.

Almost sure stability of stochastic neutral Cohen–Grossberg neural networks with Lévy noise and time‐varying delays

Abstract: The almost sure stability for the stochastic neutral Cohen–Grossberg neural networks (SNCGNNs) with Lévy noise, time‐varying delays, and Markovian switching would be deliberated in this article. By means of the nonnegative semimartingale convergence theorem (NSCT), the neutral Itô formula, M‐matrix method, and selecting appropriate Lyapunov function, several almost sure stability criterions for the SNCGNNs could be derived. Moreover, according to the M‐matrix theory, the upper bounds of the coefficients at any mode are given. Finally, two examples and numerical simulations verify the correctness of theoretical analysis for the stability criterions proposed in the article.

The Itô-Tanaka Trick : a non-semimartingale approach

Abstract: In this paper we provide an Ito-Tanaka-Wentzell trick in a non semimartingale context. We apply this result to the study of a fractional SDE with irregular drift coefficient.

Solving equations with semimartingale noise

Abstract: Abstract In this work we focus on a method for solving equations with a coefficient formally given in terms of the derivative of a continuous semimartingale. This generalizes the case of coefficients being the white noise. The idea for solving the equation is to find explicitly the inverse of the ill-posed differential operator, which boils down to finding the associated Green kernel. To find the kernel we give explicitly two homogeneous solutions in terms of the so-called Dolean–Dade exponential. The general idea to define rigorously differential operators lies on dealing with them through bilinear forms. We give several examples with explicit calculations.

Almost Sure Exponential Stability of Numerical Solutions for Stochastic Pantograph Differential Equations with Poisson Jumps

Abstract: The stability analysis of the numerical solutions of stochastic models has gained great interest, but there is not much research about the stability of stochastic pantograph differential equations. This paper deals with the almost sure exponential stability of numerical solutions for stochastic pantograph differential equations interspersed with the Poisson jumps by using the discrete semimartingale convergence theorem. It is shown that the Euler–Maruyama method can reproduce the almost sure exponential stability under the linear growth condition. It is also shown that the backward Euler method can reproduce the almost sure exponential stability of the exact solution under the polynomial growth condition and the one-sided Lipschitz condition. Additionally, numerical examples are performed to validate our theoretical result.

Semimartingale price systems in models with transaction costs beyond efficient friction

Abstract: A standing assumption in the literature on proportional transaction costs is efficient friction. Together with robust no free lunch with vanishing risk, it rules out strategies of infinite variation, as they usually appear in frictionless markets. In this paper, we show how the models with and without transaction costs can be unified. The bid and the ask price of a risky asset are given by c\'adl\'ag processes which are locally bounded from below and may coincide at some points. In a first step, we show that if the bid-ask model satisfies "no unbounded profit with bounded risk" for simple strategies, then there exists a semimartingale lying between the bid and the ask price process. In a second step, under the additional assumption that the zeros of the bid-ask spread are either starting points of an excursion away from zero or inner points from the right, we show that for every bounded predictable strategy specifying the amount of risky assets, the semimartingale can be used to construct the corresponding self-financing risk-free position in a consistent way. Finally, the set of most general strategies is introduced, which also provides a new view on the frictionless case.

Efficient Integrated Volatility Estimation in the Presence of Infinite Variation Jumps via Debiased Truncated Realized Variations

Abstract: Statistical inference for stochastic processes based on high-frequency observations has been an active research area for more than two decades. One of the most well-known and widely studied problems is the estimation of the quadratic variation of the continuous component of an It\^o semimartingale with jumps. Several rate- and variance-efficient estimators have been proposed in the literature when the jump component is of bounded variation. However, to date, very few methods can deal with jumps of unbounded variation. By developing new high-order expansions of the truncated moments of a locally stable L\'evy process, we construct a new rate- and variance-efficient volatility estimator for a class of It\^o semimartingales whose jumps behave locally like those of a stable L\'evy process with Blumenthal-Getoor index $Y\in (1,8/5)$ (hence, of unbounded variation). The proposed method is based on a two-step debiasing procedure for the truncated realized quadratic variation of the process. Our Monte Carlo experiments indicate that the method outperforms other efficient alternatives in the literature in the setting covered by our theoretical framework.

Shadow Price Approximation for the Fractional Black Scholes Model

Abstract: In this work, we used Tran Hung Thao’s approximation of fractional Brownian motion to approximate the shadow price of the fractional Black Scholes model. In the case to maximize expectation of the utility function in a portfolio optimization problem under transaction cost, the shadow price is approximated by a Markovian process and semimartingale.

Information Jumps, Liquidity Jumps, and Market Efficiency

Abstract: We formulate a measure of information efficiency in a general, no-arbitrage semimartingale model of the price process. The market quality measure is applied to a high-frequency dataset from the interdealer FX market to identify changes in market efficiency after a decimalization of tick size.

Rate-optimal estimation of mixed semimartingales

Abstract: Consider the sum $Y=B+B(H)$ of a Brownian motion $B$ and an independent fractional Brownian motion $B(H)$ with Hurst parameter $H\in(0,1)$. Surprisingly, even though $B(H)$ is not a semimartingale, Cheridito proved in [Bernoulli 7 (2001) 913--934] that $Y$ is a semimartingale if $H>3/4$. Moreover, $Y$ is locally equivalent to $B$ in this case, so $H$ cannot be consistently estimated from local observations of $Y$. This paper pivots on a second surprise in this model: if $B$ and $B(H)$ become correlated, then $Y$ will never be a semimartingale, and $H$ can be identified, regardless of its value. This and other results will follow from a detailed statistical analysis of a more general class of processes called mixed semimartingales, which are semiparametric extensions of $Y$ with stochastic volatility in both the martingale and the fractional component. In particular, we derive consistent estimators and feasible central limit theorems for all parameters and processes that can be identified from high-frequency observations. We further show that our estimators achieve optimal rates in a minimax sense. The estimation of mixed semimartingales with correlation is motivated by applications to high-frequency financial data contaminated by rough noise.

Duality for optimal consumption under no unbounded profit with bounded risk

Abstract: We give a definitive treatment of duality for optimal consumption over the infinite horizon, in a semimartingale incomplete market satisfying no unbounded profit with bounded risk (NUPBR). Rather than base the dual domain on (local) martingale deflators, we use a class of supermartingale deflators such that deflated wealth plus cumulative deflated consumption is a supermartingale for all admissible consumption plans. This yields a strong duality, because the enlarged dual domain of processes dominated by deflators is naturally closed, without invoking its closure. In this way, we automatically reach the bipolar of the set of deflators. We complete this picture by proving that the set of processes dominated by local martingale deflators is dense in our dual domain, confirming that we have identified the natural dual space. In addition to the optimal consumption and deflator, we characterise the optimal wealth process. At the optimum, deflated wealth is a supermartingale and a potential, while deflated wealth plus cumulative deflated consumption is a uniformly integrable martingale. This is the natural generalisation of the corresponding feature in the terminal wealth problem, where deflated wealth at the optimum is a uniformly integrable martingale. We use no constructions involving equivalent local martingale measures. This is natural, given that such measures typically do not exist over the infinite horizon and that we are working under NUPBR, which does not require their existence. The structure of the duality proof reveals an interesting feature compared with the terminal wealth problem. There, the dual domain is L1-bounded, but here the primal domain has this property, and hence many steps in the duality proof show a marked reversal of roles for the primal and dual domains, compared with the proofs of Kramkov and Schachermayer (Ann. Appl. Probab. 9 (1999) 904–950; Ann. Appl. Probab. 13 (2003) 1504–1516).

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 FIPP. In a semimartingale factor model, we show that the FIPP can be recovered as a triplet of processes which 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 (HJB) equation. We develop explicit constructions for the class of time-monotone FIPPs, generalizing existing results from Brownian to semimartingale market models.

Stochastic Integrals and Two Filtrations

Abstract: In the definition of the stochastic integral, apart from the integrand and the integrator, there is an underlying filtration that plays a role. Thus, it is natural to ask: Does the stochastic integral depend upon the filtration? In other words, if we have two filtrations, and , a process X that is semimartingale under both filtrations and a process f that is predictable for both filtrations, then are the two stochastic integrals— $$Y=\int f\,dX$$ , with filtration and $$Z=\int f\,dX$$ , with filtration the same? When f is left continuous with right limits, then the answer is yes. We give sufficient conditions under which Y = Z. It was proven by Slud that the quadratic variation of the process Y − Z is zero. When one filtration is an enlargement of the other, it is known that the two integrals are equal if f is bounded (or locally bounded) but this may not be the case when f is unbounded. Interestingly, if we extend the definition of the stochastic integral (named improper stochastic integral) then the equality always holds when one filtration is an enlargement of the other.

Malliavin calculus and its application to robust optimal portfolio for an insider

Abstract: Insider information and model uncertainty are two unavoidable problems for the portfolio selection theory in reality. This paper studies the robust optimal portfolio strategy for an investor who owns general insider information under model uncertainty. On the aspect of the mathematical theory, we improve some properties of the forward integral and use Malliavin calculus to derive the anticipating It\^{o} formula . Then we use forward integrals to formulate the insider-trading problem with model uncertainty. We give the half characterization of the robust optimal portfolio and obtain the semimartingale decomposition of the driving noise $W$ with respect to the insider information filtration, which turns the problem turns to the nonanticipative stochastic differential game problem. We give the total characterization by the stochastic maximum principle. When considering two typical situations where the insider is `small' and `large', we give the corresponding BSDEs to characterize the robust optimal portfolio strategy, and derive the closed form of the portfolio and the value function in the case of the small insider by the Donsker $\delta$ functional. We present the simulation result and give the economic analysis of optimal strategies under different situations.

Full classification of dynamics for one-dimensional continuous-time Markov chains with polynomial transition rates

Abstract: Abstract This paper provides a full classification of the dynamics for continuous-time Markov chains (CTMCs) on the nonnegative integers with polynomial transition rate functions and without arbitrary large backward jumps. Such stochastic processes are abundant in applications, in particular in biology. More precisely, for CTMCs of bounded jumps, we provide necessary and sufficient conditions in terms of calculable parameters for explosivity, recurrence versus transience, positive recurrence versus null recurrence, certain absorption, and implosivity. Simple sufficient conditions for exponential ergodicity of stationary distributions and quasi-stationary distributions as well as existence and nonexistence of moments of hitting times are also obtained. Similar simple sufficient conditions for the aforementioned dynamics together with their opposite dynamics are established for CTMCs with unbounded forward jumps. Finally, we apply our results to stochastic reaction networks, an extended class of branching processes, a general bursty single-cell stochastic gene expression model, and population processes, none of which are birth–death processes. The approach is based on a mixture of Lyapunov–Foster-type results, the classical semimartingale approach, and estimates of stationary measures.

Doob Decomposition, Dirichlet Processes, and Entropies on Wiener Space

Abstract: As an extension of the Doob-Meyer decomposition of a semimartingale and the Fukushima representation of a Dirichlet process, we introduce a general Doob decomposition in continuous time, where a square-integrable process is represented as the sum of a martingale and a process with “vanishing local risk”. For a probability measure Q on Wiener space, we discuss how entropy conditions on Q formulated with respect to Wiener measure P are connected with the Doob decomposition of the coordinate process W under Q. The situation is well understood if the relative entropy H(Q|P) is finite; in this case the decomposition is classical and yields an immediate proof of Talagrand’s transport inequality on Wiener space. To go beyond this restriction, we consider the specific relative entropy h(Q|P) on Wiener space that was introduced by Gantert in [11]. We discuss its interplay with the Doob decomposition of W under Q and a corresponding version of Talagrand’s inequality, with special emphasis on the case where W is a Dirichlet process under Q.