Ying Hu ∗ and Shanjian Tang †
November 25, 2013
This paper is concerned with the switching game of a one-dimensional backwardstochasticdifferentialequation (BSDE). The associated Bellman-Isaacs equation is a system of matrix-valued BSDEs living in a special unbounded convex domain with reflection on the boundary along an oblique direction. In this paper, we show the existence of an adapted solution to this system of BSDEs with oblique reflection by the penalization method, the monotone convergence, and the a priori estimates.
allowance price equals the marginal abatement cost, and market participants implement all the abate- ment measures whose costs are not greater than the cost of compliance (i.e. the equilibrium price of an allowance).
The next section puts together the economic activities of a large number of producers and search for the existence of an equilibrium price for the emissions allowances. Such a problem leads naturally to a forward stochasticdifferentialequation (SDE) for the aggregate emissions in the economy, and a backwardstochasticdifferentialequation (BSDE) for the allowance price. However, these equa- tions are ”coupled” since a nonlinear function of the price of carbon (i.e. the price of an emission allowance) appears in the forward equation giving the dynamics of the aggregate emissions. This feedback of the emission price in the dynamics of the emissions is quite natural. For the purpose of option pricing, this approach was described in where it was called detailed risk neutral approach. Forward backwardstochasticdifferential equations (FBSDEs) of the type considered in this sec- tion have been studied for a long time. See for example , or . However, the FBSDEs we need to consider for the purpose of emission prices have an unusual pecularity: the terminal condition of the backwardequation is given by a discontinuous function of the terminal value of the state driven by the forward equation. We use our first model to prove that this lack of continuity is not an issue when the forward dynamics are strongly elliptic, in other words when the volatility of the forward SDE is bounded from below. However, using our second equilibrium model, we also show that when the forward dynamics are degenerate (even if they are hypoelliptic), discontinuities in the terminal con- dition and lack of uniform ellipticity in the forward dynamics can conspire to produce point masses in the terminal distribution of the forward component, at the locations of the discontinuities. This implies that the terminal value of the backward component is not given by a deterministic function of the forward component, for the forward scenarios ending at the locations of jumps in the terminal condition, and justifies relaxing the definition of a solution of the FBSDE.
In this paper, we are concerned with reflected (forward or backward) StochasticDifferential Equations (SDE) in the case where the constraint is on the law of the solution rather than on its paths. This kind of equations have been introduced in their backward form in  in the scalar case and when the constraint is of the form R h dµ ≥ 0 for some map h : R → R satisfying suitable assumptions and where µ denotes the law of the considering process. Such a system being reflected according to the mean of (a functional of) the process, the authors called it a Mean Reflected BackwardStochasticDifferentialEquation (MR BSDE). In , the authors studied the forward version (hence called MR SDE) in the same setting as well as its approximation by an appropriate interacting particle system and numerical schemes. In , weak solution to related forward equations with constraint 1 are built . In the same framework, let us also mention the work  where the approximation of MR BSDE by an interacting particle system is studied,  where MR BSDE with quadratic generator are investigated and  where MR SDE with jumps are considered .
differentiable with bounded derivatives. The mapping φ is an at least Lipschitz function from R d to R whose precise regularity is given below.
McKean Vlasov processes may be regarded as a limit approximation for interacting systems with large number of particles. They appeared initially in statistical mechanics, but are now used in many fields because of the wide range of applications requiring large populations interactions. For example, they are used in finance, as factor stochastic volatility models [ Ber09 ] or uncertain volatility models [ GHL11 ]; in economics, in the theory of “mean field games” recently developed by J.M. Lasry and P.L. Lions in a series of papers [ LL06a , LL06b , LL07a , LL07b ] (see also [ CDL12 , CD12a , CD12b ] for the probabilistic counterpart) and also in physics, neuroscience, biology, etc. In section 5 , we present a class of control problems in which equation ( 1.1 ) explicitly appears.
Recently, jump-constrained BSDE has attracted some interested in relation with quasivari- ational inequality with a representation which suggest numerical scheme based on penalized BSDE [15, 5].
In the present work we propose to approximate the solution of a BSDEJ driven by a Brown- ian Motion and an independent compensated Poisson process, through the solution of a discrete backwardequation, following the approach proposed for BSDE by P. Briand, B. Delyon and J. Mémin in . The algorithm to compute this approximation is simple.
The extension to fully nonlinear PDE, motivated in particular by uncertain volatility model and more generally by stochastic control problem where control can affect both drift and diffusion terms of the state process, generated important recent developments. Soner, Touzi and Zhang  introduced the notion of second order BSDEs (2BSDEs), whose basic idea is to require that the solution verifies the equation P α a.s. for every pro- bability measure in a non dominated class of mutually singular measures. This theory is closely related to the notion of nonlinear and G-expectation of Peng . Alternatively, Kharroubi and Pham , following , introduced the notion of BSDE with nonposi- tive jumps. The basic idea was to constrain the jumps-component solution to the BSDE driven by Brownian motion and Poisson random measure, to remain nonpositive, by ad- ding a nondecreasing process in a minimal way. A key feature of this class of BSDEs is its formulation under a single probability measure in contrast with 2BSDEs, thus avoiding technical issues in quasi-sure analysis, and its connection with fully nonlinear HJB equa- tion when considering a Markovian framework with a simulatable regime switching dif- fusion process, defined as a randomization of the controlled state process. This approach opens new perspectives for probabilistic scheme for fully nonlinear PDEs as currently investigated in .
(s,x)∈[0,T ]×E corresponds to the laws (for diﬀerent starting times s
and starting points x) of an underlying forward Markov process with time index [0, T ], taking values in a Polish state space E. Indeed this Markov process is supposed to solve a martingale problem with respect to a given deterministic operator a, which is the natural generalization of stochastic diﬀerential equation in law. (1.2) will be naturally associated with a deterministic problem involving a, which will be called Pseudo-PDE, being of the type
Second order BSDEs were introduced by Cheredito, Soner, Touzi and Victoir in . In a Markovian framework, they show that there exists a connection between 2BSDEs and fully nonlinear PDEs while standard BS- DEs induce quasi-linear PDEs. However, except in the case where the PDEs admits sufficiently regular solutions, they do not provide a general existence result. In , Denis and Martini generalized the uncertain volatility model introduced in  or  to a family of martingale measures thanks to the quasi-sure analysis. The uncertain volatility model is directly linked to the Black-Scholes-Barrenblat equation which is fully nonlinear. This problem is strongly linked to the problem of G-integration theory studied mainly by Peng (see , ) for the definition of the main properties. Denis, Hu and Peng in  established connections between  and  while Soner, Touzi and Zhang in  provide a martingale representation theorem for the G-martingale which corresponds to a hedging strategy in the uncer- tain volatility model. Inspired by this quasi-sure framework, Soner, Touzi and Zhang study in  the second order stochastic target problem whose solution solves a 2BSDE and prove existence and uniqueness for general 2BSDEs in  with an undominated family of mutually singular martin- gale measures. Recently, Possamai and Zhou extend their results for a one dimensional 2BSDE with bounded terminal condition and continuous gen- erator with quadratic growth in the z variable (see Possamai , Possamai and Zhou ). This result allow to solve second order reflected BSDEs and utility maximization problem under volatility uncertainty as we can see in
of the adjoint equation is not provided by standard results on BackwardStochasticDifferential Equa- tions (BSDEs) as the distributions of the solution processes (more precisely their joint distributions with the control and state processes α and X) appear in the coefficients of the equation. However, a slight modification of the original existence and uniqueness result of Pardoux and Peng  shows that existence and uniqueness still hold in our more general setting. The main lines of the proof are given in , Proposition 3.1 and Lemma 3.1. However, Lemma 3.1 in  doesn’t apply directly since the coefficients (∂ µ b(t, ˜ X t , P X t , ˜ α t )(X t ) ˜ Y t ) 0≤t≤T and (∂ µ σ(t, ˜ X t , P X t , ˜ α t )(X t ) ˜ Z t ) 0≤t≤T
field particle interpretation of a class of nonlinear diffusions equipped with an interacting sample covariance matrix functional.
We underline that the Itô-Alekseev-Gröbner formula (4.6) discussed in  is an extension of the interpolation formula (1.10) to stochastic diffusion flows in matrix spaces. In this context the unperturbed model is given by the flow of a deterministic matrix Riccati differentialequation and the random perturbations are described by matrix-valued diffusion martingales. The corresponding Itô-Alekseev-Gröbner formulae can be seen as a matrix version of theorem 1.2 in the present article when σ “ 0. These stochastic interpolation formulae were used in  to quantify the fluctuation of the stochastic flow around the limiting deterministic Riccati equation, at any order. We will briefly discuss the analog of these Taylor type expansions in section 7.1 in the context of Euclidian diffusions.
McKean Vlasov processes may be regarded as a limit approximation for interacting systems with large number of particles. They appeared initially in statistical mechanics, but are now used in many fields because of the wide range of applications requiring large populations interactions. For example, they are used in finance, as factor stochastic volati- lity models [ Ber09 ] or uncertain volatility models [ GHL11 ] ; in economics, in the theory of “mean field games” recently developed by J.M. Lasry and P.L. Lions in a series of papers [ LL06b , LL06a , LL07 ] (see also [ CDL12 , CD12b , CD12a ] for the probabilistic counterpart) and also in physics, neuroscience, biology, etc. In section 2.4 , we present a class of control problems in which equation ( 2.0.7 ) explicitly appears.
it is supposed to point outward on the boundary of the set on which we work. We give in Subsection 1.4 the main result (Theorem 1.4.1) which sums up the results obtained. In Section 5, we extend the results to random time intervals [0; τ ], where τ is successively a bounded stopping time (Theorem 5.3.1) and a stopping time verifying an exponential integrability condition (Theorem 5.3.2); then to conclude this paper, we give some generations and applications to the theory of PDEs, as well as the variational problem related to equation (M + D) 0 .
Our goal in this paper is thus to prove existence and uniqueness for the RBSDEs (1.1) for generic H and convex domain D without imposing any structural dependence condition on the driver f of the equation. In this direction, we are able to obtain very general existence result in a Markovian setting, assuming only a weak domination prop- erty of the forward process, see Section 4. We also discuss there the non-uniqueness issue. In the general case of P-measurable random coefficients f , H and terminal con- dition ξ, we need to impose some smoothness assumptions on the domain, on H, which depend then only on the time and y variables and on the terminal condition ξ. In this case, we obtain both existence and uniqueness for the solution of (1.1). Let us remark that our new results on Obliquely Reflected BSDEs allow us to treat some new optimal switching problems called “randomised switching problems” and introduced in the Sec- tion 5 of .
Since the pioneering work of E. Pardoux and S. Peng , backwardstochastic dif- ferential equations (BSDEs) have been intensively studied during the two last decades. Indeed, this notion has been a very useful tool to study problems in many areas, such as mathematical finance, stochastic control, partial differential equations; see e.g.  where many applications are described. Discretization schemes for BSDEs have been studied by several authors. The first papers on this topic are that of V.Bally  and D.Chevance . In his thesis, J.Zhang made an interesting contribution which was the starting point of intense study among, which the works of B. Bouchard and N.Touzi , E.Gobet, J.P. Lemor and X. Warin,... The notion of BSDE has been generalized by E. Pardoux and S. Peng  to that of Backward Doubly StochasticDifferentialEquation (BDSDE) as follows. Let (Ω, F, P) be a probability space, T denote some fixed terminal time which will be used throughout the paper, (W t ) 0≤t≤T and (B t ) 0≤t≤T be two independent standard
(21) α(t, x, y, µ) ∈ argmin ˆ α∈R d H(t, x, y, µ, α), t ∈ [0, T ], x, y ∈ R d , µ ∈ P 2 (R d ).
The existence of such a function was proven in  under specific assumptions on the drift b and the running cost function f used there. However, the major difference with the mean field game problem comes from the form of the adjoint equation which now involves differentiation of the Hamiltonian with respect to the measure parameter. A special form of adjoint equation was introduced, and a new stochastic maximum principle was proven in . Once this new form of adjoint equation is coupled with the forward dynamical equation through the plugged-in optimal control feedback ˆ α defined in (21), the associated McKean-Vlasov FBSDE takes the form
Since the pioneering work of E. Pardoux and S. Peng [PP92], backwardstochasticdifferential equations (BSDEs) have been intensively studied during the two last decades. Indeed, this notion has been a very useful tool to study problems in many areas, such as mathematical finance, stochastic control, partial differential equations; see e.g. [MY99] where many applications are described. Discretization schemes for BSDEs have been introduced and studied by several authors. The first papers on this topic are that of V.Bally [Ba97] and D.Chevance [Ch97]. In his thesis, Zhang made an interesting contribution which was the starting point of intense study among which the works of B. Bouchard and N.Touzi [BT04], E.Gobet, J.P. Lemor and X. Warin[GLW05],... The notion of BSDE has been generalized by E. Pardoux and S. Peng [PP94] to that of Backward Doubly StochasticDifferentialEquation (BDSDE) as follows. Let (Ω, F , P) be a probability space, T denote some fixed terminal time which will be used throughout the paper, (Wt) 0≤t≤T and (Bt) 0≤t≤T be two independent standard Brownian motions defined on (Ω, F , P) and with values in R d , and R respectively. On this space we will deal with two families of σ-algebras:
After the pioneer work of Von Neumann and Morgenstern [ 109 ], Merton rst studied portfolio selection with utility maximization by stochastic optimal control in the seminal paper [ 81 ]. Kramkov and Schachermayer solved the problem of maximizing utility of nal wealth in a general semimartingale model by means of duality in [ 64 ]. Later, El Karoui and Rouge [ 38 ] considered the indierence pricing problem via exponential utility maximization by means of the BSDE theory. Their strategy set is supposed to be closed and convex, and the problem is solved using BSDEs with quadratic growth generators. In [ 54 ], with a similar approach, Hu, Imkeller and Müller studied three important types of utility function with only closed admissible strategies set within incomplete market and found that the maximization problem is linked to quadratic BSDEs. They also showed a deep link between quadratic growth and the BMO spaces. Morlais [ 82 ] extended results in [ 54 ] to more general continuous ltration, for this purpose, proved existence and uniqueness of the solution to a particular type quadratic BSDEs driven by a continuous martingale. In a more recent paper [ 57 ], Jeanblanc, Matoussi and Ngoupeyou studied the indierence price of an unbounded claim in an incomplete jump-diusion model by considering the risk aversion represented by an exponential utility function. Using the dynamic programming equation, they found the price of an unbounded credit derivatives as a solution of a quadratic BSDE with jumps.
The differentialequation, considered as a dynamical system, is de- scribed by its state equations and its initial value at time t 0 .
A generic expression of its generating series G t truncated at any
order k, of the output and its derivatives y (j) (t) expanded at any
Another interesting extension could be the addition of noise in the form of a geometric Brownian motion to (27) along the lines of equation (20).
4.4 Information Percolation in Finance
A related class of models arises when studying the distribution of information across individuals in an economy, e.g., beliefs about the value of a particular financial asset. These models are useful to understand the dynamics of asset prices and how these are affected when market participants do not share common beliefs about the “intrinsic” value of a financial asset. A simple example is provided by Duffie and Manso (2007) who consider the beliefs about the realization of a binary random variable. Individuals are initially endowed with a prior about this realization. Over time, individuals randomly meet at a constant Poisson rate α. Upon a meeting individuals exchange their information and update their beliefs. In their example they show that beliefs are characterize by a distribution over a sufficient statistic x, and the updating of beliefs after a meeting with an individual of belief x ′ is simply given by the sum of x and x ′ . The evolution of the