Parametrix Method Research Articles

A stochastic interpretation of the parametrix method

UDC 519.21 We revisit, in a didactic manner and by using stochastic analysis, the parametrix method and its application to unbiased simulation. We consider, in particular, the case of one-dimensional diffusions without drift.

Upper bounds for the derivatives of the density associated to solutions of stochastic differential equations with jumps

In this article, we prove the existence and regularity of densities associated to stochastic differential equations driven by a jump process with infinite activity. Furthermore, we provide explicit bounds for the density and its derivatives. The main argument is based on two ideas: (1) The use of the regularity of the Brownian motion approximation of small jumps of an infinite activity process (the so-called Asmussen-Rosiński approximation); (2) The improvement of the approximation procedure using a Taylor like expansion of the law based on the so-called Levi parametrix method.

Optimal investment and reinsurance strategies under 4/2 stochastic volatility model

This paper studies a mean-variance investment-reinsurance problem under a new stochastic volatility model, namely the 4/2 stochastic volatility model. Solving this problem requires a deep understanding of a class of parabolic partial differential equations (PPDEs). By the parametrix method and the integral transform method, we derive explicit solutions to the PPDEs in several special cases. Through the Lie symmetry analysis, we obtain a four-parameter family of the 4/2 stochastic volatility models such that the corresponding PPDEs have closed-form solutions. The efficient strategy and the efficient frontier of the mean-variance problem are represented by using the closed-form solutions to PPDEs. Numerical examples for the obtained efficient frontier are provided by Monto Carlo method.

A Yosida's parametrix approach to Varadhan's estimates for a degenerate diffusion under the weak Hörmander condition

We adapt and extend Yosida's parametrix method, originally introduced for the construction of the fundamental solution to a parabolic operator on a Riemannian manifold, to derive Varadhan-type asymptotic estimates for the transition density of a degenerate diffusion under the weak Hörmander condition. This diffusion process, widely studied by Yor in a series of papers, finds direct application in the study of a class of path-dependent financial derivatives known as Asian options. We obtain the Varadhan formula−2log⁡p(t,x;T,y)Ψ(t,x;T,y)→1,as T−t→0+, where p denotes the transition density and Ψ denotes the optimal cost function of a deterministic control problem associated to the diffusion. We provide a partial proof of this formula, and present numerical evidence to support the validity of an intermediate inequality that is required to complete the proof. We also derive an asymptotic expansion of the cost function Ψ, expressed in terms of elementary functions, which is useful in order to design efficient approximation formulas for the transition density.

On weak solution of SDE driven by inhomogeneous singular Lévy noise

We study a time-inhomogeneous SDE in R d \mathbb {R}^d driven by a cylindrical Lévy process with independent coordinates which may have different scaling properties. Such a structure of the driving noise makes it strongly spatially inhomogeneous and complicates the analysis of the model significantly. We prove that the weak solution to the SDE is uniquely defined, is Markov, and has the strong Feller property. The heat kernel of the process is presented as a combination of an explicit ‘principal part’ and a ‘residual part’, subject to certain L ∞ ( d x ) ⊗ L 1 ( d y ) L^\infty (dx)\otimes L^1(dy) and L ∞ ( d x ) ⊗ L ∞ ( d y ) L^\infty (dx)\otimes L^\infty (dy) -estimates showing that this part is negligible in a short time, in a sense. The main tool of the construction is the analytic parametrix method, specially adapted to Lévy-type generators with strong spatial inhomogeneities.

Toward a Wong–Zakai Approximation for Big Order Generators

We give a new approximation with respect of the traditional parametrix method of the solution of a parabolic equation whose generator is of big order and under the Hoermander form. This generalizes to a higher order generator the traditional approximation of Stratonovitch diffusion which put in relation random ordinary differential equation (the leading process is random and of finite energy. When a trajectory of it is chosen, the solution of the equation is defined) and stochastic differential equation (the leading process is random and only continuous and we cannot choose a path in the solution which is only almost surely defined). We consider simple operators where the computations can be fully performed. This approximation fits with the semi-group only and not for the full path measure in the case of a stochastic differential equation.

$$L^{\alpha -1}$$ distance between two one-dimensional stochastic differential equations driven by a symmetric $$\alpha$$-stable process

In this article, we consider a coefficient stability problem for one-dimensional stochastic differential equations driven by an $$\alpha$$ -stable process with $$\alpha \in (1,2)$$ . More precisely, we find an upper bound for the $$L^{\alpha -1}(\varOmega ,{\mathbb {P}})$$ distance between two solutions in terms of the $$L^{\alpha }\left( {\mathbb {R}},\mu^{\alpha }_{x_0}\right)$$ distance of the coefficients for an appropriate measure $$\mu^{\alpha} _{x_0}$$ which characterizes symmetric stable laws and depends on the initial value of the stochastic differential equation. We obtain this result using the method introduced by Komatsu (Proc Jpn Acad Ser A Math Sci 58(8):353–356, 1982) which is used in the proof of uniqueness of solutions together with an upper bound for the transition density function of the solution of the stochastic differential equation obtained by Kulik (The parametrix method and the weak solution to an SDE driven by an $$\alpha$$ -stable noise. arXiv:1412.8732 , 2014).

Fundamental Solution for Cauchy Initial Value Problem for Parabolic PDEs with Discontinuous Unbounded First-Order Coefficient at the Origin. Extension of the Classical Parametrix Method

We prove the existence of a fundamental solution of the Cauchy initial boundary value problem on the whole space for a parabolic partial differential equation with discontinuous unbounded first-order coefficient at the origin. We establish also non-asymptotic, rapidly decreasing at infinity, upper and lower estimates for the fundamental solution. We extend the classical parametrix method of E.E. Levi.

The parametrix method for parabolic SPDEs

We consider the Cauchy problem for a linear stochastic partial differential equation. By extending the parametrix method for PDEs whose coefficients are only measurable with respect to the time variable, we prove existence, regularity in Hölder classes and estimates from above and below of the fundamental solution. This result is applied to SPDEs by means of the Itô–Wentzell formula, through a random change of variables which transforms the SPDE into a PDE with random coefficients.

Second order probabilistic parametrix method for unbiased simulation of stochastic differential equations

In this article, following the paradigm of bias–variance trade-off philosophy, we derive parametrix expansions of order two, based on the Euler–Maruyama scheme with random partitions, for the purpose of constructing an unbiased simulation method for multidimensional stochastic differential equations. These formulas lead to Monte Carlo simulation methods which can be easily parallelized. The second order method proposed here requires further regularity of coefficients in comparison with the first order method but achieves finite moments even when Poisson sampling is used for the partitions, in contrast to Andersson and Kohatsu-Higa (2017). Moreover, using an exponential scaling technique one achieves an unbiased simulation method which resembles a space importance sampling technique which significantly improves the efficiency of the proposed method. A hint of how to derive higher order expansions is also presented.

Parametrix method for the first hitting time of an elliptic diffusion with irregular coefficients

ABSTRACT In this article, we are interested in studying the transition density function of the couple given by the first hitting time of a fixed threshold by a one-dimensional uniformly elliptic diffusion process and the associated stopped process. Our strategy relies on the parametrix technique applied to the related semigroup. It notably allows us to extend standard PDEs results on Green and Poisson functions, see e.g. Garroni and Menaldi [Green functions for second order parabolic integro-differential problems, volume 275 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992], by establishing the infinite expansion of the corresponding transition density for irregular coefficients, here bounded measurable drift coefficient and bounded η-Hölder-continuous diffusion coefficient. As a by-product, we establish some Gaussian upper estimates.

Intrinsic compound kernel estimates for the transition probability density of Lévy-type processes and their applications

INTRINSIC COMPOUND KERNEL ESTIMATES FOR THE TRANSITION PROBABILITY DENSITY OF LÉVY-TYPE PROCESSES AND THEIR APPLICATIONSStarting with an integro-differential operator L, C2∞ ℜn, we prove that its C∞ℜn-closure is the generator of a Feller process X, which admits a transition probability density. To construct this transition probability density, we develop a version of the parametrix method and a verification procedure, which proves that the constructed object is the claimed one. As a part of the construction, we prove the intrinsic upper and lower estimates on the density. As an application of the constructed estimates we state the necessary and separately sufficient conditions under which a given Borel measure belongs to the Kato and Dynkin classes with respect to the constructed transition probability density.

Parametrix construction of the transition probability density of the solution to an SDE driven by $\alpha$-stable noise

Soit $L:=-a(x)(-\Delta)^{\alpha/2}+(b(x),\nabla)$, avec $\alpha\in(0,2)$, et $a:\mathbb{R}^{d}\to(0,\infty)$, $b:\mathbb{R}^{d}\to\mathbb{R}^{d}$. Sous certaines hypotheses de regularite des coefficients $a$ et $b$, nous associons a la $C_{\infty}(\mathbb{R}^{d})$-fermeture de $(L,C_{\infty}^{2}(\mathbb{R}^{d}))$ un processus de Markov fellerien $X$, possedant une densite de probabilite de transition $p_{t}(x,y)$. Afin de construire cette densite, et d’en obtenir des bornes superieures et inferieures, nous developpons une nouvelle version de la methode parametrix, qui permet meme de traiter le cas ou $0<\alpha\leq1$ et $b\neq0$, c’est-a-dire quand la partie de gradient du generateur n’est pas dominee par la partie de saut.

Stochastic formulations of the parametrix method

In this manuscript, we consider stochastic expressions of the parametrix method for solutions ofd-dimensional stochastic differential equations (SDEs) with drift coefficients which belong toLp(Rd),p&gt;d. We prove the existence and Hölder continuity of probability density functions for distributions of solutions at fixed points and obtain an explicit expansionvia(stochastic) parametrix methods. We also obtain Gaussian type upper and lower bounds for these probability density functions.

Unbiased simulation of stochastic differential equations

We propose an unbiased Monte-Carlo estimator for E[g(X t 1 , · · · , X tn)], where X is a diffusion process defined by a multi-dimensional stochastic differential equation (SDE). The main idea is to start instead from a well-chosen simulatable SDE whose coefficients are updated at independent exponential times. Such a simulatable process can be viewed as a regime-switching SDE, or as a branching diffusion process with one single living particle at all times. In order to compensate for the change of the coefficients of the SDE, our main representation result relies on the automatic differentiation technique induced by Bismu-Elworthy-Li formula from Malliavin calculus, as exploited by Fournie et al. [14] for the simulation of the Greeks in financial applications. In particular, this algorithm can be considered as a variation of the (infinite variance) estimator obtained in Bally and Kohatsu-Higa [3, Section 6.1] as an application of the parametrix method.

On quasianalytic classes of Gelfand–Shilov type. Parametrix and convolution

We develop a convolution theory for quasianalytic ultradistributions of Gelfand–Shilov type. We also construct a special class of ultrapolynomials, and use it as a base for the parametrix method in the study of new topological and structural properties of several quasianalytic spaces of functions and ultradistributions. In particular, our results apply to Fourier hyperfunctions and Fourier ultra-hyperfunctions.

Nonlinear trend exclusion procedure for models defined by stochastic differential and difference equations

We consider a diffusion process and its approximation with a Markov chain whose trends contain a nonlinear unbounded component. The usual parametrix method is inapplicable here since the trend is unbounded. We present a procedure that lets us exclude a nonlinear growing trend and pass to a stochastic differential equation with bounded drift and diffusion coefficients. A similar procedure is also considered for a Markov chain.

Singular SDEs with critical non-local and non-symmetric Lévy type generator

In this work, by using Levi’s parametrix method we first construct the fundamental solution of the critical non-local operator perturbed by gradient. Then, we use the obtained estimates to prove the pathwise uniqueness of strong solutions for stochastic differential equation driven by Markov process with irregular coefficients, whose generator is a non-local and non-symmetric Lévy type operator.

Strong existence and uniqueness for degenerate SDE with Hölder drift

In this paper, we prove pathwise uniqueness for stochastic degenerate systems with a H{\"o}lder drift, for a H{\"o}lder exponent larger than the critical value 2/3. This work extends to the degenerate setting the earlier results obtained by Zvonkin, Veretennikov, Krylov and R{\"o}ckner from non-degenerate to degenerate cases. The existence of a threshold for the H{\"o}lder exponent in the degenerate case may be understood as the price to pay to balance the degeneracy of the noise. Our proof relies on regularization properties of the associated PDE, which is degenerate in the current framework and is based on a parametrix method.

Regularity of the density of a stable-like driven SDE with Hölder continuous coefficients

In this article, we use the backward parametrix method in order to prove the existence and regularity of the the transition density associated to the solution process of a stable-like driven stochastic differential equation (SDE) with Hölder continuous coefficients. The method of proof uses the parametrix method on the Gaussian component of a subordinated Brownian motion. This analysis which can be generalized also provides a stochastic representation of the density which is potentially useful for other applications.Abbrevations: B: Brownian motion; V: α-stable-like subordinator independent of B; μ: Lévy measure of the subordinator V; m(·): positive concave increasing function; ; δy(dx): Dirac measure with unit mass at ; ψ: Lévy exponent of Z; q(M, x): Gaussian density with covariance matrix M and ; ϕ: a regular varying function; b: drift coefficient of X; σ: coefficient of associated with the driving Lévy process Z ≔ BV; ζ: coefficient associated with the diffusion (if X is a jump diffusion process); : generator of process X, solution of the stochastic differential equation (1); pt(x, y): density of the process X; : generator of the parametrix or frozen process ; : density of the parametrix ; k: Hölder exponential of a ≔ σσT; and : uniform upper and lower bounds for the eigenvalues of a(·); k′: Hölder exponential of e ≔ ζζT; η and (ηi)i: auxiliary Bernoulli used to link the drift part and the jump part; and : auxiliary Bernoulli used to link the small and large jumps; : ordered jump times of an Poisson process; , where w ∈ (α, 1∧(α + λ)) and λ > 0; ; ; x+ ≔ x∨0 for ; ; ; Int(y, x) ≔ ∫t0…∫tn − 10dtn…dt1 K(tn, …, t1, t; x, y); ; ; ; ; ; ; ;