## About

91

Publications

10,249

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

3,985

Citations

Introduction

**Skills and Expertise**

## Publications

Publications (91)

In this paper we study a continuous time equilibrium model of limit order book (LOB) in which the liquidity dynamics follows a non-local, reflected mean-field stochastic differential equation (SDE) with evolving intensity. Generalizing the basic idea of Ma et al. (2015), we argue that the frontier of the LOB (e.g., the best asking price) is the val...

In this paper we consider a class of {\it conditional McKean-Vlasov SDEs} (CMVSDE for short). Such an SDE can be considered as an extended version of McKean-Vlasov SDEs with common noises, as well as the general version of the so-called {\it conditional mean-field SDEs} (CMFSDE) studied previously by the authors [1, 14], but with some fundamental d...

We introduce a new notion of conditional nonlinear expectation under probability distortion. Such a distorted nonlinear expectation is not subadditive in general, so it is beyond the scope of Peng’s framework of nonlinear expectations. A more fundamental problem when extending the distorted expectation to a dynamic setting is time inconsistency, th...

We study fully nonlinear second-order (forward) stochastic PDEs. They can also be viewed as forward path-dependent PDEs and will be treated as rough PDEs under a unified framework. For the most general fully nonlinear case, we develop a local theory of classical solutions and then define viscosity solutions through smooth test functions. Our notion...

In this paper, we establish an analytic framework for studying Set-Valued Backward Stochastic Differential Equations (SVBSDE for short), motivated largely by the current studies of dynamic set-valued risk measures for multi-asset or network-based financial models. Our framework will be based on the notion of Hukuhara difference between sets, in ord...

In this paper we study a continuous time equilibrium model of limit order book (LOB) in which the liquidity dynamics follows a non-local, reflected mean-field stochastic differential equation (SDE) with evolving intensity. Generalizing the basic idea of Ma et.al. (2015), we argue that the frontier of the LOB (e.g., the best asking price) is the val...

In this paper we continue investigating the optimal dividend and investment problems under the Sparre Andersen model. More precisely, we assume that the claim frequency is a renewal process instead of a standard compound Poisson process, whence semi-Markovian. Building on our previous work \cite{BaiMa17}, where we established the dynamic programmin...

We introduce a new notion of conditional nonlinear expectation where the underlying probability scale is distorted by a weight function. Such a distorted nonlinear expectation is not sub-additive in general, so is beyond the scope of Peng's framework of nonlinear expectations. A more fundamental problem when extending the distorted expectation to a...

In this paper, we are interested in the well-posedness of a class of fully coupled forward-backward SDE (FBSDE) in which the forward drift coefficient is allowed to be discontinuous with respect to the backward component of the solution. Such an FBSDE is motivated by a practical issue in regime-switching term structure interest rate models, and the...

In this paper we study the Kyle-Back strategic insider trading equilibrium model in which the insider has an instantaneous information on an asset, assumed to follow an Ornstein-Uhlenback-type dynamics that allows possible influence by the market price. Such a model exhibits some further interplay between insider's information and the market price,...

In this paper we study viscosity solutions for a fairly large class of fully nonlinear (forward) stochastic partial differential equations (SPDEs). These SPDEs can also be viewed as forward path dependent PDEs (PPDEs), and will be treated under a unified framework, as rough partial differential equations (RPDEs). Our definition of viscosity solutio...

In this paper we are interested in a new type of {\it mean-field}, non-Markovian stochastic control problems with partial observations. More precisely, we assume that the coefficients of the controlled dynamics depend not only on the paths of the state, but also on the conditional law of the state, given the observation to date. Our problem is stro...

In this paper we study the optimal control problem for a class of general mean-field stochastic differential equations, in which the coefficients depend, nonlinearly, on both the state process as well as of its law. In particular, we assume that the control set is a general open set that is not necessary convex, and the coefficients are only contin...

In this paper we study a class of optimal dividend and investment problems assuming that the underlying reserve process follows the Sparre Andersen model, that is, the claim frequency is a "renewal" process, rather than a standard compound Poisson process. The main feature of such problems is that the underlying reserve dynamics, even in its simple...

In this paper we investigate possible approaches to study general time-inconsistent optimization problems without assuming the existence of optimal strategy. This leads immediately to the need to refine the concept of time-consistency as well as any method that is based on Pontryagin's Maximum Principle. The fundamental obstacle is the dilemma of h...

In this paper, we study the well-posedness of the Forward-Backward Stochastic
Differential Equations (FBSDE) in a general non-Markovian framework. The main
purpose is to find a unified scheme which combines all existing methodology in
the literature, and to address some fundamental longstanding problems for
non-Markovian FBSDEs. An important device...

Professor Xunjing Li was born in Qingdao, Shandong Province, China, on the 13th June 1935. Shandong is a province with a rich culture that has nurtured a great number of influential intellectuals during its long history, including Confucius. Immediately after his graduation from the Department of Mathematics at Shandong University in 1956, Professo...

In this paper we propose a new notion of pathwise viscosity solution for a
fairly large class of fully nonlinear (forward) Stochastic PDEs whose diffusion
term is allowed to depend on both the solution and its gradient. With the help
of the newly developed pathwise analysis in the sense of Dupire \cite{Dupire},
we shall identify a forward SPDE as a...

This paper provides a large deviation principle for Non-Markovian, Brownian
motion driven stochastic differential equations with random coefficients.
Similar to Gao and Liu \cite{GL}, this extends the corresponding results
collected in Freidlin and Wentzell \cite{FreidlinWentzell}. However, we use a
different line of argument, adapting the PDE meth...

In this paper we propose a dynamic model of Limit Order Book (LOB). The main
feature of our model is that the shape of the LOB is determined endogenously by
an expected utility function via a competitive equilibrium argument. Assuming
zero resilience, the resulting equilibrium density of the LOB is random,
nonlinear, and time inhomogeneous. Consequ...

In this paper we establish the pathwise Taylor expansions for random fields
that are "regular" in the spirit of Dupire's path-derivatives \cite{Dupire}.
Our result is motivated by but extends the recent result of Buckdahn-Bulla-Ma
\cite{BBM}, when translated into the language of pathwise calculus. We show
that with such a language the pathwise Tayl...

In this paper, we establish an equivalence relationship between the wellposedness of forward–backward SDEs (FBSDEs) with random coefficients and that of backward stochastic PDEs (BSPDEs). Using the notion of the “decoupling random field”, originally observed in the well-known Four Step Scheme (Ma et al., 1994 [13]) and recently elaborated by Ma et...

We consider a model of correlated defaults in which the default times of multiple entities depend not
only on common and specific factors, but also on the extent of past defaults in the market, via the average
loss process, including the average number of defaults as a special case. The paper characterizes the average
loss process when the number o...

In this paper we study a class of stochastic differential equations with
additive noise that contains a fractional Brownian motion and a Poisson point
process of class (QL). The differential equation of this kind is motivated by
the reserve processes in a general insurance model, in which the long term
dependence between the claim payment and the p...

In this paper we study an optimal portfolio selection problem under
instantaneous price impact. Based on some empirical analysis in the literature,
we model such impact as a concave function of the trading size when the trading
size is small. The price impact can be thought of as either a liquidity cost or
a transaction cost, but the concavity natu...

In this paper we continue exploring the notion of weak solution of forward–backward stochastic differential equations (FBSDEs) and associated forward–backward martingale problems (FBMPs). The main purpose of this work is to remove the constraints on the martingale integrands in the uniqueness proofs
in our previous work (Ma etal. in Ann Probab 36(6...

In this paper we study the wellposedness of the forward-backward stochastic
differential equations (FBSDE) in a general non-Markovian framework. The main
purpose is to find a unified scheme which combines all existing methodology in
the literature, and to overcome some fundamental difficulties that have been
longstanding problems for non-Markovian...

In this paper we use an intensity-based framework to analyze and compute the correlated default probabilities, both in finance and actuarial sciences, following the idea of "change of measure" initiated by Collin-Dufresne et al. (2004). Our method is based on a representation theorem for joint survival probability among an arbitrary number of defau...

In this paper we study the {\it pathwise stochastic Taylor expansion}, in the sense of our previous work \cite{Buckdahn_Ma_02}, for a class of It\^o-type random fields in which the diffusion part is allowed to contain both the random field itself and its spatial derivatives. Random fields of such an "self-exciting" type particularly contains the fu...

This paper studies a class of forward-backward stochastic differential equations (FBSDE) in a general Markovian framework.
The forward SDE represents a large class of strong Markov semimartingales, and the backward generator requires only mild regularity
assumptions. The authors show that the Four Step Scheme introduced by Ma, et al. (1994) is stil...

This note gives a brief history of the “martingale representation theorem”, one of the most powerful and fundamental theorems in stochastic calculus. Various forms of the theorem are presented so that a broad picture of the theorem can be seen from different perspectives. Some new extensions of the theorem, especially those connected to mathematica...

We consider a class of backward stochastic differential equations (BSDEs) driven by Brownian motion and Poisson random measure, and subject to constraints on the jump component. We prove the existence and uniqueness of the minimal solution for the BSDEs by using a penalization approach. Moreover, we show that under mild conditions the minimal solut...

Motivated by the so-called shortfall risk minimization problem,
we consider Merton's portfolio optimization problem in a non-Markovian
market driven by a Lévy process, with a bounded state-dependent utility
function. Following the usual dual variational approach, we show that the
domain of the dual problem enjoys an explicit “parametrization,” buil...

In this paper we study an optimal portfolio selection problem under general trans-action cost. We consider a simplified financial market that consists of a risk free asset and a risky asset, but the admissible portfolios are only allowed to have piecewise con-stant paths, reflecting a more practical perspective. The problem is then reduced to an im...

In this paper, we study the existence and uniqueness of a class of stochastic differential equations driven by fractional Brownian motions with arbitrary Hurst parameter $H\in (0,1)$. In particular, the stochastic integrals appearing in the equations are defined in the Skorokhod sense on fractional Wiener spaces, and the coefficients are allowed to...

In this paper the utility optimization problem for a general insurance model is studied. The reserve process of the insurance company is described by a stochastic differential equation driven by a Brownian motion and a Poisson random measure, representing the randomness from the financial market and the insurance claims, respectively. The random sa...

In this article we extend the notion of g-evaluation, in particular g-expectation, of Peng [88.
Peng , S. 1997 . Backward SDE and Related g-Expectation, Pitman Res. Notes Math. Ser., Vol. 364. Longman, Harlow . View all references, 99.
Peng , S. 2004 . Nonlinear Expectations, Nonlinear Evaluations and Risk Measures , Lecture Notes in Math. , Vol....

In this paper, we propose a new notion of Forward--Backward Martingale Problem (FBMP), and study its relationship with the weak solution to the forward--backward stochastic differential equations (FBSDEs). The FBMP extends the idea of the well-known (forward) martingale problem of Stroock and Varadhan, but it is structured specifically to fit the n...

We revisit Merton's portfolio optimization problem under boun-ded state-dependent utility functions, in a market driven by a L\'evy process $Z$ extending results by Karatzas et. al. (1991) and Kunita (2003). The problem is solved using a dual variational problem as it is customarily done for non-Markovian models. One of the main features here is th...

We study a new type of reflected backward stochastic differential
equations (RBSDEs), where the reflecting process enters the drift in a nonlinear manner. This
type of the reflected BSDEs is based on a variance of the Skorohod problem studied recently by
Bank and El Karoui (2004), and is hence named the “Variant Reflected BSDEs” (VRBSDE) in this pa...

In this paper we extend the notion of “filtration-consistent nonlinear expectation” (or “F-consistent nonlinear expectation”) to the case when it is allowed to be dominated by a g-expectation that may have a quadratic growth. We show that for such a nonlinear expectation many fundamental properties of a martingale can still make sense, including th...

In this paper we study a class of stochastic control problems in which the control of the jump size is essential. Such a model is a generalized version for various applied problems ranging from optimal reinsurance selections for general insurance models to queueing theory. The main novel point of such a control problem is that by changing the jump...

In this paper we study a class of pathwise stochastic control problems in which the optimality is allowed to depend on the paths of exogenous noise (or information). Such a phenomenon can be illustrated by considering a particular investor who wants to take advantage of certain extra information but in a completely legal manner. We show that such a...

In this paper we study the pricing problem for a class of Universal Variable Life (UVL) insurance products, using the idea of “principle of equivalent utility”. The main features of the UVL products include the varying (death) benefit based on both tradable and non-tradable investment incomes and “multiple decrement” cases. We formulate the pricing...

In this paper we study the pricing problem for a class of universal variable life (UVL) insurance products, using the idea of principle of equivalent utility. As the main features of UVL products we allow the (death) benefit to depend on certain indices or assets that are not necessarily tradable (e.g., pension plans), and we also consider the “mul...

In this paper we propose a new notion of Forward-Backward Martingale Problem (FBMP), and study its relationship with the weak solution to the backward stochastic differential equations. The FBMP extends the idea of the well-known (forward) mar-tingale problem of Stroock and Varadhan, but it is structured specifically to fit the nature of a forward-...

In this paper we study a class of backward stochastic differential equations with reflections (BSDER, for short). Three types of discretization procedures are introduced in the spirit of the so-called Bermuda Options in finance, so as to first establish a Feynman-Kac type formula for the martingale integrand of the BSDER, and then to derive the con...

In this paper, we study a class of multi-dimensional backward stochastic differential equations (BSDEs, for short) in which the terminal values and the generators are allowed to be "discrete-functionals" of a forward diffusion. We first establish some new types of Feynman-Kac formulas related to such BSDEs under various regularity conditions, and t...

In this note we study a class of forward - backward stochastic differential equations (FBSDE for short) with functional-type terminal conditions. In the case when the time duration and the coefficients are "compatible" (e.g., the time duration is small), we prove the existence and uniqueness of the strong adapted solution in the usual sense. In the...

We consider the problem of computing hedging portfolios for options that may have discontinuous payoffs, in the framework of diffusion models in which the number of factors may be larger than the number of Brownian motions driving the model. Extending the work of Fournié et al. (1999), as well as Ma and Zhang (2000), using integration by parts of M...

In this paper we study convolution formulae for the independent sum of a normal random variable and several power exponential distributed random variables. This problem is motivated by the numerical simulation for pricing financial derivatives (such as options) when the underlying assets follow a jump-diffusion model in which the logarithm of the j...

Jin Ma, Xiaodong Sun. Ruin probabilities for insurance models involv-ing investments. Scand. Actuarial J. 2003; 3: 217 –237. In this paper we study the ruin problem for insurance models that involve investments. Our risk reserve process is an extension of the classical Cramér – Lundberg model, which will contain stochastic interest rates, reserve-d...

In this paper we study the ruin problem for insurance models that involve investments. Our risk reserve process is an extension of the classical Cramér-Lundberg model, which will contain stochastic interest rates, reserve-dependent expense loading, diffusion perturbed models, and many others as special cases. By introducing a new type of exponentia...

In this paper we investigate a class of backward stochastic differential equations (BSDE) whose terminal values are allowed to depend on the history of a forward diffusion. We first establish a probabilistic representation for the spatial gradient of the viscosity solution to a quasilinear parabolic PDE in the spirit of the Feynman--Kac formula, wi...

In this paper we study a class of one-dimensional, degenerate, semilinear backward stochastic partial differential equations
(BSPDEs, for short) of parabolic type. By establishing some new a priori estimates for both linear and semilinear BSPDEs,
we show that the regularity and uniform boundedness of the adapted solution to the semilinear BSPDE ca...

In this paper we study a new type of "Taylor expansion" for Itô-type random fields, up to the second order. We show that an Itô-type random field with reasonably regular "integrands" can be expanded, up to the second order, to the linear combination of increments of temporal and spatial variables, as well as the driven Brownian motion, around even...

In this paper we study the path regularity of the adpated solutions to a class of backward stochastic differential equations (BSDE, for short) whose terminal values are allowed to be functionals of a forward diffusion. Using the new representation formula for the adapted solutions established in our previous work [7], we are able to show, under the...

We propose a method for numerical approximation of backward stochastic differential equations. Our method allows the final condition of the equation to be quite general and simple to implement. It relies on an approximation of Brownian motion by simple random walk.

The solvability of forward—backward stochastic differential equations (FBSDEs for short) has been studied extensively in
recent years. To guarantee the existence and uniqueness of adapted solutions, many different conditions, some quite restrictive,
have been imposed. In this paper we propose a new notion: the approximate solvability of FBSDEs, bas...

This paper, together with the accompanying work (Part II, Stochastic Process. Appl. 93 (2001) 205-228) is an attempt to extend the notion of viscosity solution to nonlinear stochastic partial differential equations. We introduce a definition of stochastic viscosity solution in the spirit of its deterministic counterpart, with special consideration...

This paper is a continuation of our previous work (Part I, Stochastic Process. Appl. 93 (2001) 181-204), with the main purpose of establishing the uniqueness of the stochastic viscosity solution introduced there. We shall prove a comparison theorem between a stochastic viscosity solution and an [omega]-wise stochastic viscosity solution, which will...

Let X be the solution of a stochastic differential equation driven by a Wiener process and a compensated Poisson random measure, such that X is an L
2 martingale. If H = Φ(X
s
; 0 ≤ s ≤ T) is in L
2, then H = α + ∫
0T
ξ
s
dX
s
+ N
T
, where N is an L
2 martingale orthogonal to X (the Kunita-Watanabe decomposition). We give sufficient conditions on...

In this paper we study a class of forward-backward stochastic differential
equations with reflecting boundary conditions (FBSDER for short). More
precisely, we consider the case in which the forward component of the
FBSDER is restricted to a fixed, convex region, and the backward component will stay, at each fixed time, in a convex region that m...

In this paper we study a general multidimensional diffusion-type stochastic control problem. Our model contains the usual
regular control problem, singular control problem and impulse control problem as special cases. Using a unified treatment
of dynamic programming, we show that the value function of the problem is a viscosity solution of certain...

In this paper we study the well-posedness and regularity of the adapted solutions to a class of linear, degenerate backward stochastic partial differential equations (BSPDE, for short). We establish new a priori estimates for the adapted solutions to BSPDEs in a general setting, based on which the existence, uniqueness, and regularity of adapted so...

Motivated by some problems from mathematical finance and
stochastic maximum principle, the solvability of forward-backward
stochastic differential equations (FBSDE, for short) has been studied
extensively in recent years. To guarantee the existence and uniqueness
of adapted solutions, many different conditions, some are quite
restrictive, have been...

In Chapter 3, in order to derive the stochastic maximum principle as a set of necessary conditions for optimal controls, we encountered the problem of finding adapted solutions to the adjoint equations. Those are terminal value problems of (linear) stochastic differential equations involving the Itô stochastic integral. We call them backward stocha...

An anticipating stochastic integral is proposed for 'normal martingales'. It agrees with the Skorohod integral in the Brownian case. A variational derivative of Malliavin type is also defined. An integration by parts formula is given which has some subtle and important differences from the formula in the Brownian case. The existence and uniqueness...

This note is concerned with the asymptotic behavior of adapted solutions to forward-backward SDEs when the forward diffusion contains small noises. We prove the sample path Large Deviation Principle (LDP) for the adapted solutions to the FBSDEs under appropriate conditions stated in terms of a certain type of convergence of the solutions to the ass...

In this paper we prove the existence and uniqueness, as well as the regularity, of the adapted solution to a class of degenerate linear backward stochastic partial differential equations (BSPDE) of parabolic type. We apply the results to a class of forward-backward stochastic differential equations (FBSDE) with random coefficients, and establish in...

. In this paper, the solvability of a class of forward-backward stochastic differential equations (SDEs for short) over an arbitrarily prescribed time duration is studied. We design a stochastic relaxed control problem, with both drift and diffusion all being controlled, so that the solvability problem is converted to a problem of finding the nodal...

In this paper we study a class of forward-backward stochastic differential equations with reflecting boundary conditions (FBSDER for short). More precisely, we consider the case in which the forward component of the FBSDER is restricted to a fixed, convex region, and the backward component will stay, at each fixed time, in a convex region that may...

. This paper confirms a version of a conjecture by Fischer Black regarding consol rate models for the term structure of interest rates. A consol rate model is one in which the stochastic behavior of the short rate is influenced by the consol rate. Since the consol rate is itself determined, via the usual discounted present value formula, by the sho...

In this paper we study numerical methods to approximate the adapted solutions to a class of forward-backward stochastic differential equations (FBSDE's). The almost sure uniform convergence as well as the weak convergence of the scheme are proved, and the rate of convergence is proved to be as good as the approximation for the corresponding forward...

In this paper we present some results concerning the solvability of the adapted solutions to a class of forward-backward stochastic differential equations over an arbitrarily prescribed time duration, and several applications of such equations in mathematical finance. In particular, we introduce a direct scheme (called “four step scheme”) initiated...

In the classical continuous-time financial market model, stock prices have been understood as solutions to linear stochastic differential equations, and an important problem to solve is the problem of hedging options (functions of the stock price values at the expiration date). In this paper we consider the hedging problem not only with a price mod...

In this paper we investigate the nature of the adapted solutions to a class of forward-backward stochastic differential equations (SDEs for short) in which the forward equation is non-degenerate. We prove that in this case the adapted solution can always be sought in an ordinary sense over an arbitrarily prescribed time duration, via a direct Four...

In this paper, we continue to study a diffusion-type, finite-fuel singular stochastic control problem and the related stochastic differential equations with discontinuous paths and reflecting boundary conditions as denned in the previous work of the author [15]. The measurable dependence of the solution with respect to the initial state and the und...

We continue to study a diffusion-type, finite-fuel singular stochastic control problem and the related stochastic differential equations with discontinuous paths and reflecting boundary conditions as defined in the previous work of the author [ibid. 44, No. 3/4, 225-252 (1993; Zbl 0791.60067)]. The measurable dependence of the solution with respect...

We study two kinds of Discontinuous Reflecting Problem (DRP for short), defined by Chaleyat-Maurel et al. [3] and Dupuis and Ishii [5] (reduced to the one-dimensional case) and the related Stochastic Differential Equations with Discontinuous Paths and Reflecting Boundary Conditions (SDEDR for short). We compare the properties of the solutions to th...

This paper considers the principle of smooth fit for a class of one-dimensional singular stochastic control problems allowing the system to be of nonlinear diffusion type. The existence and the uniqueness of a convex C2-solution to the corresponding variational inequality are obtained. It is proved that this solution gives the value function of the...

Abstract This paper, together with the accompanying work (Part II, Stochastic Process. Appl. 93 (2001) 205–228) is an attempt to extend the notion of viscosity solution to nonlinear stochastic partial di#erential equations. We introduce a de7nition of stochastic viscosity solution in the spirit of its deterministic counterpart, with special conside...

In this paper, we study a class of multi-dimensional backward stochastic di7erential equa- tions (BSDEs, for short) in which the terminal values and the generators are allowed to be "discrete-functionals" of a forward di7usion. We

The classical Merton's problem of utility maximization was recently solved in [2] in a market consisting of a bond with constant interest rate, a stock that follows a geometric Lévy model, and certain "fictitious" stocks called power-jump assets. Using their previous work [3] on the completeness of such a market and the martingale method, it was pr...