arXiv subjects

Juan Li

Publications and source records attributed to Juan Li.

At least 163 records · Page 9Linked to original sources

arXiv2013

$L^p$ estimates for fully coupled FBSDEs with jumps

In this paper we study useful estimates, in particular $L^p$-estimates, for fully coupled forward-backward stochastic differential equations (FBSDEs) with jumps. These estimates are proved at one hand for fully coupled FBSDEs with jumps under the monotonicity assumption for arbitrary time intervals and on the other hand for such equations on small time intervals. Moreover, the well-posedness of this kind of equation is studied and regularity results are obtained.

math.PR↗

arXiv2013

Stochastic differential games for fully coupled FBSDEs with jumps

This paper is concerned with stochastic differential games (SDGs) defined through fully coupled forward-backward stochastic differential equations (FBSDEs) which are governed by Brownian motion and Poisson random measure. For SDGs, the upper and the lower value functions are defined by the controlled fully coupled FBSDEs with jumps. Using a new transformation introduced in [6], we prove that the upper and the lower value functions are deterministic. Then, after establishing the dynamic programming principle for the upper and the lower value functions of this SDGs, we prove that the upper and the lower value functions are the viscosity solutions to the associated upper and the lower Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations, respectively. Furthermore, for a special case (when $σ,\ h$ do not depend on $y,\ z,\ k$), under the Isaacs' condition, we get the existence of the value of the game.

math.OC↗

arXiv2012

Reflected Mean-Field Backward Stochastic Differential Equations. Approximation and Associated Nonlinear PDEs

Mathematical mean-field approaches have been used in many fields, not only in Physics and Chemistry, but also recently in Finance, Economics, and Game Theory. In this paper we will study a new special mean-field problem in a purely probabilistic method, to characterize its limit which is the solution of mean-field backward stochastic differential equations (BSDEs) with reflections. On the other hand, we will prove that this type of reflected mean-field BSDEs can also be obtained as the limit equation of the mean-field BSDEs by penalization method. Finally, we give the probabilistic interpretation of the nonlinear and nonlocal partial differential equations with the obstacles by the solutions of reflected mean-field BSDEs.

math.PR↗

arXiv2012

Multi-component Wronskian solution to the Kadomtsev-Petviashvili equation

It is known that the Kadomtsev-Petviashvili (KP) equation can be decomposed into the first two members of the coupled Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy by the binary nonlinearization of Lax pairs. In this paper, we construct the N-th iterated Darboux transformation (DT) for the second- and third-order m-coupled AKNS systems. By using together the N-th iterated DT and Cramer's rule, we find that the KPII equation has the unreduced multi-component Wronskian solution and the KPI equation admits a reduced multi-component Wronskian solution. In particular, based on the unreduced and reduced two-component Wronskians, we obtain two families of fully-resonant line-soliton solutions which contain arbitrary numbers of asymptotic solitons as y->\mp\infty to the KPII equation, and the ordinary N-soliton solution to the KPI equation. In addition, we find that the KPI line solitons propagating in parallel can exhibit the bound state at the moment of collision.

math-ph↗

arXiv2012

Note on stochastic control problems related with general fully coupled forward-backward stochastic differential equations

In this paper we study stochastic optimal control problems of general fully coupled forward-backward stochastic differential equations (FBSDEs). In Li and Wei [8] the authors studied two cases of diffusion coefficients $σ$ of FSDEs, in one case when $σ$\ depends on the control and does not depend on the second component of the solution $(Y, Z)$ of the BSDE, and in the other case $σ$ depends on $Z$ and doesn't depend on the control. Here we study the general case when $σ$ depends on both $Z$ and the control at the same time. The recursive cost functionals are defined by controlled general fully coupled FBSDEs, then the value functions are given by taking the essential supremum of the cost functionals over all admissible controls. We give the formulation of related generalized Hamilton-Jacobi-Bellman (HJB) equations, and prove the value function is its viscosity solution.

math.PR↗

arXiv2012

Regularity properties for general HJB equations. A BSDE method

In this work we investigate regularity properties of a large class of Hamilton-Jacobi-Bellman (HJB) equations with or without obstacles, which can be stochastically interpreted in form of a stochastic control system which nonlinear cost functional is defined with the help of a backward stochastic differential equation (BSDE) or a reflected BSDE (RBSDE). More precisely, we prove that, firstly, the unique viscosity solution $V(t,x)$ of such a HJB equation over the time interval $[0,T],$ with or without an obstacle, and with terminal condition at time $T$, is jointly Lipschitz in $(t,x)$, for $t$ running any compact subinterval of $[0,T)$. Secondly, for the case that $V$ solves a HJB equation without an obstacle or with an upper obstacle it is shown under appropriate assumptions that $V(t,x)$ is jointly semiconcave in $(t,x)$. These results extend earlier ones by Buckdahn, Cannarsa and Quincampoix [1]. Our approach embeds their idea of time change into a BSDE analysis. We also provide an elementary counter-example which shows that, in general, for the case that $V$ solves a HJB equation with a lower obstacle the semi-concavity doesn't hold true.

math.PR↗

arXiv2012

Value Function of Differential Games without Isaacs Conditions. An Approach with Non-Anticipative Mixed Strategies

In the present paper we investigate the problem of the existence of a value for differential games without Isaacs condition. For this we introduce a suitable concept of mixed strategies along a partition of the time interval, which are associated with classical nonanticipative strategies (with delay). Imposing on the underlying controls for both players a conditional independence property, we obtain the existence of the value in mixed strategies as the limit of the lower as well as of the upper value functions along a sequence of partitions which mesh tends to zero. Moreover, we characterize this value in mixed strategies as the unique viscosity solution of the corresponding Hamilton-Jacobi-Isaacs equation.

math.OC↗

arXiv2011

Large-Scale Kinematics, Astrochemistry and Magnetic Field Studies of Massive Star-forming Regions through HC3N, HNC and C2H Mappings

We have mapped 27 massive star-forming regions associated with water masers using three dense gas tracers: HC3N 10-9, HNC 1-0 and C2H 1-0. The FWHM sizes of HNC clumps and C2H clumps are about 1.5 and 1.6 times higher than those of HC3N, respectively, which can be explained by the fact that HC3N traces more dense gas than HNC and C2H. We found evidence for increase in optical depth of C2H with `radius' from center to outer regions in some targets, supporting the chemical model of C2H. The C2H optical depth is found to decline as molecular clouds evolve to later stage, suggesting that C2H might be used as "chemical clock" for molecular clouds. Large-scale kinematic structure of clouds was investigated with three molecular lines. All these sources show significant velocity gradients. The magnitudes of gradient are found to increase towards the inner region, indicating differential rotation of clouds. Both the ratio of rotational to gravitational energy and specific angular momentum seem to decrease toward the inner region, implying obvious angular momentum transfer, which might be caused by magnetic braking. The average magnetic field strength and number density of molecular clouds is derived using the uniformly magnetic sphere model. The derived magnetic field strengths range from 3 to 88 μG, with a median value of 13 μG. The mass-to-flux ratio of molecular cloud is calculated to be much higher than critical value with derived parameters, which agrees well with numerical simulations.

astro-ph.SR↗

arXiv2010

ATCA observations of SiO masers in the Galactic center

We present the Australia Telescope Compact Array (ATCA) observations of the SiO masers in the Galactic center in transitions of v=1, J=2--1 at 86 GHz and v=1, J=1--0 at 43 GHz. Two 86-GHz SiO masers were detected within the central parsec, and they are associated with IRS 10EE and IRS 15NE, respectively. We detected eighteen 43-GHz SiO masers within a projected separation of <2 pc from Sagittarius A* (Sgr A*), among which seven masers are newly discovered from our observations. This raises the total number of 43-GHz SiO masers within the central 4 parsecs of the GC region to 22. Simultaneous observations at 86 and 43 GHz showed that the intensity of 43-GHz SiO maser is ~3 times higher than that of 86-GHz maser in IRS 10EE (an OH/IR star), while the integrated flux of the SiO maser emission at 43 GHz is comparable with that at 86~GHz in IRS~15NE (an ordinary Mira variable). These results are consistent with previous observations of massive late-type stars in the Galaxy in which the 86-GHz SiO maser is in general weaker than the 43-GHz SiO maser in OH/IR stars, while the two transitions are comparably strong in Mira stars.

astro-ph.GA↗

arXiv2010

Integral-Partial Differential Equations of Isaacs' Type Related to Stochastic Differential Games with Jumps

In this paper we study zero-sum two-player stochastic differential games with jumps with the help of theory of Backward Stochastic Differential Equations (BSDEs). We generalize the results of Fleming and Souganidis [10] and those by Biswas [3] by considering a controlled stochastic system driven by a d-dimensional Brownian motion and a Poisson random measure and by associating nonlinear cost functionals defined by controlled BSDEs. Moreover, unlike the both papers cited above we allow the admissible control processes of both players to depend on all events occurring before the beginning of the game. This quite natural extension allows the players to take into account such earlier events, and it makes even easier to derive the dynamic programming principle. The price to pay is that the cost functionals become random variables and so also the upper and the lower value functions of the game are a priori random fields. The use of a new method allows to prove that, in fact, the upper and the lower value functions are deterministic. On the other hand, the application of BSDE methods [18] allows to prove a dynamic programming principle for the upper and the lower value functions in a very straight-forward way, as well as the fact that they are the unique viscosity solutions of the upper and the lower integral-partial differential equations of Hamilton-Jacobi-Bellman-Isaacs' type, respectively. Finally, the existence of the value of the game is got in this more general setting if Isaacs' condition holds.

math.OC↗

arXiv2010

ATCA Detection of SiO Masers in the Inner Parsecs of the Galactic Center

We present sensitive Australia Telescope Compact Array (ATCA) observations of SiO masers in the inner parsecs of the Galactic center (GC). We detected five SiO J=2-1, v=1 (86 GHz) masers in the innner 25 arcsec (1 pc) of the GC. All the detected 86 GHz SiO masers are previously known SiO J=1-0, v=1 (43 GHz) masers and associated with late-type stars. Eighteen 43 GHz SiO masers were detected within 50 arcsec (2 pc) of Sagittarius A*. Among them, seven are detected for the first time, which brings the total number of 43 GHz SiO masers to 22 in this region.

astro-ph.GA↗

arXiv2009

On the transcendence of some infinite sums

In this paper we investigate the infinite convergent sum $T=\sum_{n=0}^\infty\frac{P(n)}{Q(n)}$, where $P(x)\in\bar{\mathbb{Q}}[x]$, $Q(x)\in\mathbb{Q}[x]$ and $Q(x)$ has only simple rational zeros. N. Saradha and R. Tijdeman have obtained sufficient and necessary conditions for the transcendence of $T$ if the degree of $Q(x)$ is 3. In this paper we give sufficient and necessary conditions for the transcendence of $T$ if the degree of $Q(x)$ is 4 and $Q(x)$ is reduced.

math.NT↗

arXiv2009

Mean-field backward stochastic differential equations: A limit approach

Mathematical mean-field approaches play an important role in different fields of Physics and Chemistry, but have found in recent works also their application in Economics, Finance and Game Theory. The objective of our paper is to investigate a special mean-field problem in a purely stochastic approach: for the solution $(Y,Z)$ of a mean-field backward stochastic differential equation driven by a forward stochastic differential of McKean--Vlasov type with solution $X$ we study a special approximation by the solution $(X^N,Y^N,Z^N)$ of some decoupled forward--backward equation which coefficients are governed by $N$ independent copies of $(X^N,Y^N,Z^N)$. We show that the convergence speed of this approximation is of order $1/\sqrt{N}$. Moreover, our special choice of the approximation allows to characterize the limit behavior of $\sqrt{N}(X^N-X,Y^N-Y,Z^N-Z)$. We prove that this triplet converges in law to the solution of some forward--backward stochastic differential equation of mean-field type, which is not only governed by a Brownian motion but also by an independent Gaussian field.

math.PR↗

arXiv2009

The variability of Sagittarius A* at 3 millimeter

We have performed monitoring observations of the 3-mm flux density toward the Galactic Center compact radio source Sgr A* with the Australia Telescope Compact Array since 2005 October. Careful calibrations of both elevation-dependent and time-dependent gains have enabled us to establish the variability behavior of Sgr A*. Sgr A* appeared to undergo a high and stable state in 2006 June session, and a low and variable state in 2006 August session. We report the results, with emphasis on two detected intra-day variation events during its low states. One is on 2006 August 12 when Sgr A* exhibited a 33% fractional variation in about 2.5 hr. The other is on 2006 August 13 when two peaks separated by about 4 hr, with a maximum variation of 21% within 2 hr, were seen. The observed short timescale variations are discussed in light of two possible scenarios, i.e., the expanding plasmon model and the sub-Keplerian orbiting hot spot model. The fitting results indicate that for the adiabatically expanding plasmon model, the synchrotron cooling can not be ignored, and a minimum mass-loss rate of 9.7*10^{-10}M_sun /yr is obtained based on parameters derived for this modified expanding plasmon model. Simultaneous multi-wavelength observation is crucial to our understanding the physical origin of rapid radio variability in Sgr A*.

astro-ph.GA↗

arXiv2008

Stochastic Differential Games with Reflection and Related Obstacle Problems for Isaacs Equations

In this paper we first investigate zero-sum two-player stochastic differential games with reflection with the help of theory of Reflected Backward Stochastic Differential Equations (RBSDEs). We will establish the dynamic programming principle for the upper and the lower value functions of this kind of stochastic differential games with reflection in a straight-forward way. Then the upper and the lower value functions are proved to be the unique viscosity solutions of the associated upper and the lower Hamilton-Jacobi-Bellman-Isaacs equations with obstacles, respectively. The method differs heavily from those used for control problems with reflection, it has its own techniques and its own interest. On the other hand, we also prove a new estimate for RBSDEs being sharper than that in El Karoui, Kapoudjian, Pardoux, Peng and Quenez [7], which turns out to be very useful because it allows to estimate the $L^p$-distance of the solutions of two different RBSDEs by the $p$-th power of the distance of the initial values of the driving forward equations. We also show that the unique viscosity solution of the approximating Isaacs equation which is constructed by the penalization method converges to the viscosity solution of the Isaacs equation with obstacle.

math.PR↗

arXiv2008

Probabilistic Interpretation for Systems of Isaacs Equations with Two Reflecting Barriers

In this paper we investigate zero-sum two-player stochastic differential games whose cost functionals are given by doubly controlled reflected backward stochastic differential equations (RBSDEs) with two barriers. For admissible controls which can depend on the whole past and so include, in particular, information occurring before the beginning of the game, the games are interpreted as games of the type "admissible strategy" against "admissible control", and the associated lower and upper value functions are studied. A priori random, they are shown to be deterministic, and it is proved that they are the unique viscosity solutions of the associated upper and the lower Bellman-Isaacs equations with two barriers, respectively. For the proofs we make full use of the penalization method for RBSDEs with one barrier and RBSDEs with two barriers. For this end we also prove new estimates for RBSDEs with two barriers, which are sharper than those in [18]. Furthermore, we show that the viscosity solution of the Isaacs equation with two reflecting barriers not only can be approximated by the viscosity solutions of penalized Isaacs equations with one barrier, but also directly by the viscosity solutions of penalized Isaacs equations without barrier.

math.OC↗

arXiv2007

Mean-Field Backward Stochastic Differential Equations and Related Partial Differential Equations

In [5] the authors obtained Mean-Field backward stochastic differential equations (BSDE) associated with a Mean-field stochastic differential equation (SDE) in a natural way as limit of some highly dimensional system of forward and backward SDEs, corresponding to a large number of ``particles'' (or ``agents''). The objective of the present paper is to deepen the investigation of such Mean-Field BSDEs by studying them in a more general framework, with general driver, and to discuss comparison results for them. In a second step we are interested in partial differential equations (PDE) whose solutions can be stochastically interpreted in terms of Mean-Field BSDEs. For this we study a Mean-Field BSDE in a Markovian framework, associated with a Mean-Field forward equation. By combining classical BSDE methods, in particular that of ``backward semigroups" introduced by Peng [14], with specific arguments for Mean-Field BSDEs we prove that this Mean-Field BSDE describes the viscosity solution of a nonlocal PDE. The uniqueness of this viscosity solution is obtained for the space of continuous functions with polynomial growth. With the help of an example it is shown that for the nonlocal PDEs associated to Mean-Field BSDEs one cannot expect to have uniqueness in a larger space of continuous functions.

math.PR↗

arXiv2007

Symbolic-computation study of integrable properties for the (2+1)-dimensional Gardner equation with the two-singular-manifold method

The singular manifold method from the Painleve analysis can be used to investigate many important integrable properties for the nonlinear partial differential equations.In this paper, the two-singular-manifold method is applied to the (2+1)-dimensional Gardner equation with two Painleve expansion branches to determine the Hirota bilinear form, Backlund transformation, Lax pairs and Darboux transformation. Based on the obtained Lax pairs, the binary Darboux transformation is constructed and the N N Grammian solution is also derived by performing the iterative algorithm Ntimes with symbolic computation.

nlin.SI↗