BSDE on an infinite horizon and elliptic PDEs in infinite dimension

In this paper, we study the existence and uniqueness of mild solutions to a possibly degenerate elliptic partial differential equation in Hilbert spaces. Our aim is, in the case in which ψ(·, 0, 0) is bounded, to drop the assumptions on the size of λ needed in [11]. The main tool will be existence, uniqueness and regular dependence on parameters of a bounded solution to a suitable backward stochastic differential equation with infinite horizon. Finally we apply the result to study an optimal control problem.      

Asset Pricing with Stochastic Volatility

In this paper we study the asset pricing problem when the volatility is random. First, we derive a PDE for the risk-minimizing price of any contingent claim. Secondly, we assume that the volatility process \si _t is observed through an observation process Y _t subject to random error. A price formula and a PDE are then derived regarding the stock price S _t and the observation process Y _t as parameters. Finally, we assume that S _t is observed. In this case we have a complete market and any contingent claim is then priced by an arbitrage argument instead of by risk-minimizing. Accepted 15 August 2000. Online publication 8 December 2000.     

On Jensen’s inequality for g-expectation and for nonlinear expectation

In this paper, we give a necessary and sufficient condition for g under which Jensen’s inequality holds for g-expectation. In particular, we show that if Jensen’s inequality holds for g-expectation, then g is independent of y and g is superhomogeneous. We also establish a necessary and sufficient condition under which Jensen’s inequality holds for a general filtration-consistent nonlinear expectation. Received: 18 January 2005     

The law of the Euler scheme for stochastic differential equations

Summary We study the approximation problem ofE f(X _T ) byE f(X _T ⁿ ), where (X _t ) is the solution of a stochastic differential equation, (X _T ⁿ ) is defined by the Euler discretization scheme with stepT/n, andf is a given function. For smoothf's, Talay and Tubaro have shown that the errorE f(X _T ) –f(X _T ⁿ ) can be expanded in powers of 1/n, which permits to construct Romberg extrapolation precedures to accelerate the convergence rate. Here, we prove that the expansion exists also whenf is only supposed measurable and bounded, under an additional nondegeneracy condition of Hörmander type for the infinitesimal generator of (X _t ): to obtain this result, we use the stochastic variations calculus. In the second part of this work, we will consider the density of the law ofX _T ⁿ and compare it to the density of the law ofX _T .     

The Ornstein-Uhlenbeck Equation and a Related Malliavin Calculus

We consider two different Brownian motions, B and B ^a ; each of them produces a Wiener-It? chaos representation and therefore it defines a Malliavin derivative, D and D ^a , and a Skorohod integral, δ and δ ^a , respectively. Our aim is to rewrite the differential operators D ^a and δ ^a in terms of D and δ.     

The behavior of solutions of stochastic differential inequalities

Summary LetX andZ be ^d -valued solutions of the stochastic differential inequalities dX _t a(t,X _t )dt+(t,X _t )dW _t andb(t, Z _t )dt+(t, Z _t )dW _t dZ _t , respectively, with a fixed ^m -valued Wiener processW. In this paper we give conditions ona, b and under which the relationX ₀Z ₀ of the initial values leads to the same relation between the solutions with probability one. Further we discuss whether in general our conditions can be weakened or not. Then we deal with notions like maximal/minimal solution of a stochastic differential inequality. Using the comparison result we derive a sufficient condition for the existence of such solutions as well as some Gronwall-type estimates.     

Large Deviation Principle for Optimal Filtering

Abstract. We derive a large deviation principle for the optimal filter where the signal and the observation processes take values in conuclear spaces. The approach follows from the framework established by the author in an earlier paper. The key is the verification of the exponential tightness for the optimal filtering process and the exponential continuity of the coefficients in the Zakai equation.     

On Interacting Systems of Hilbert-Space-Valued Diffusions

A nonlinear Hilbert-space-valued stochastic differential equation where L ^-1 (L being the generator of the evolution semigroup) is not nuclear is investigated in this paper. Under the assumption of nuclearity of L ^-1 , the existence of a unique solution lying in the Hilbert space H has been shown by Dawson in an early paper. When L ^-1 is not nuclear, a solution in most cases lies not in H but in a larger Hilbert, Banach, or nuclear space. Part of the motivation of this paper is to prove under suitable conditions that a unique strong solution can still be found to lie in the space H itself. Uniqueness of the weak solution is proved without moment assumptions on the initial random variable. A second problem considered is the asymptotic behavior of the sequence of empirical measures determined by the solutions of an interacting system of H -valued diffusions. It is shown that the sequence converges in probability to the unique solution Λ ₀ of the martingale problem posed by the corresponding McKean—Vlasov equation. Accepted 4 April 1996     

Large Deviation Principle for Optimal Filtering

   Abstract. We derive a large deviation principle for the optimal filter where the signal and the observation processes take values in conuclear spaces. The approach follows from the framework established by the author in an earlier paper. The key is the verification of the exponential tightness for the optimal filtering process and the exponential continuity of the coefficients in the Zakai equation.     

Optimal control of the insurance company with proportional reinsurance policy under solvency constraints

This paper considers the optimal control problem of the insurance company with proportional reinsurance policy under solvency constraints. The management of the company controls the reinsurance rate and dividends payout processes to maximize the expected present value of the dividend until the time of bankruptcy. This is a mixed singular-regular control problem. However, the optimal dividend payout barrier may be too low to be acceptable. The company may be prohibited to pay dividend according to external reasons because this low dividend payout barrier will result in bankruptcy soon. Therefore, some constraints on the insurance company’s dividend policy will be imposed. One reasonable and normal constraint is that if b is the minimum dividend barrier, then the bankrupt probability should not be larger than some predetermined ε within the time horizon T. This paper is to work out the optimal control policy of the insurance company under the solvency constraints.     

Linear forward-backward stochastic differential equations with random coefficients

Solvability of linear forward-backward stochastic differential equations (FBSDEs, for short) with random coefficients is studied. A decoupling reduction method is introduced via which a large class of linear FBSDEs with random or deterministic time-varying coefficients is proved to be solvable. On the other hand, by means of Four Step Scheme, a Riccati backward stochastic equation (BSDE, for short) for (m×n) matrix-valued processes is derived. Global solvability of such Riccati BSDEs is discussed for some special (but nontrivial) cases, which leads to the solvability of the corresponding linear FBSDEs. This work is supported in part by the NSFC, under grant 10131030, the Chinese Education Ministry Science Foundation under grant 2000024605, the Cheung Kong Scholars Programme, and Shanghai Commission of Science and Technology under grant 02DJ14063.     

An inverse first-passage problem for one-dimensional diffusions with random starting point

We consider an inverse first-passage time (FPT) problem for a homogeneous one-dimensional diffusion X(t), starting from a random position η. Let S(t) be an assigned boundary, such that P(η≥S(0))=1, and F an assigned distribution function. The problem consists of finding the distribution of η such that the FPT of X(t) below S(t) has distribution F. We obtain some generalizations of the results of Jackson et al., 2009, which refer to the case when X(t) is Brownian motion and S(t) is a straight line across the origin.     

Regularity of the Local Time for the d-dimensional Fractional Brownian Motion with N-parameters

ABSTRACT

We give the Wiener–Ito? chaotic decomposition for the local time of the d-dimensional fractional Brownian motion with N-parameters and study its smoothness in the Sobolev–Watanabe spaces.     

Optimal Control of Point Processes with Noisy Observations: The Maximum Principle

This paper studies the optimal control problem for point processes with Gaussian white-noised observations. A general maximum principle is proved for the partially observed optimal control of point processes, without using the associated filtering equation . Adjoint flows—the adjoint processes of the stochastic flows of the optimal system—are introduced, and their relations are established. Adjoint vector fields , which are observation-predictable, are introduced as the solutions of associated backward stochastic integral-partial differential equtions driven by the observation process. In a heuristic way, their relations are explained, and the adjoint processes are expressed in terms of the adjoint vector fields, their gradients and Hessians, along the optimal state process. In this way the adjoint processes are naturally connected to the adjoint equation of the associated filtering equation . This shows that the conditional expectation in the maximum condition is computable through filtering the optimal state, as usually expected. Some variants of the partially observed stochastic maximum principle are derived, and the corresponding maximum conditions are quite different from the counterpart for the diffusion case. Finally, as an example, a quadratic optimal control problem with a free Poisson process and a Gaussian white-noised observation is explicitly solved using the partially observed maximum principle. Accepted 8 August 2001. Online publication 17 December, 2001.     

Quadratic BSDEs with convex generators and unbounded terminal conditions

In Briand and Hu (Probab Theory Relat Fields 136(4):604–618, 2006), the authors proved an existence result for BSDEs with quadratic generators with respect to the variable z and with unbounded terminal conditions. However, no uniqueness result was stated in that work. The main goal of this paper is to fill this gap. In order to obtain a comparison theorem for this kind of BSDEs, we assume that the generator is convex with respect to the variable z. Under this assumption of convexity, we are also able to prove a stability result in the spirit of the a priori estimates stated in Karoui et al. (Math Finance 7(1):1–71, 1997). With these tools in hands, we can derive the nonlinear Feynman–Kac formula in this context.     

The noncentral Wishart as an exponential family, and its moments

While the noncentral Wishart distribution is generally introduced as the distribution of the random symmetric matrix where Y₁,…,Y_n are independent Gaussian rows in with the same covariance, the present paper starts from a slightly more general definition, following the extension of the chi-square distribution to the gamma distribution. We denote by γ(p,a;σ) this general noncentral Wishart distribution: the real number p is called the shape parameter, the positive definite matrix σ of order k is called the shape parameter and the semi-positive definite matrix a of order k is such that the matrix ω=σaσ is called the noncentrality parameter. This paper considers three problems: the derivation of an explicit formula for the expectation of when Xγ(p,a,σ) and h₁,…,h_m are arbitrary symmetric matrices of order k, the estimation of the parameters (a,σ) by a method different from that of Alam and Mitra [K. Alam, A. Mitra, On estimated the scale and noncentrality matrices of a Wishart distribution, Sankhyā, Series B 52 (1990) 133–143] and the determination of the set of acceptable p’s as already done by Gindikin and Shanbag for the ordinary Wishart distribution γ(p,0,σ).     

Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients

We prove a general theorem that the -valued solution of an infinite horizon backward doubly stochastic differential equation, if exists, gives the stationary solution of the corresponding stochastic partial differential equation. We prove the existence and uniqueness of the -valued solutions for backward doubly stochastic differential equations on finite and infinite horizon with linear growth without assuming Lipschitz conditions, but under the monotonicity condition. Therefore the solution of finite horizon problem gives the solution of the initial value problem of the corresponding stochastic partial differential equations, and the solution of the infinite horizon problem gives the stationary solution of the SPDEs according to our general result.     

Quasi sure analysis and Stratonovich anticipative stochastic differential equations

Summary Consider a stochastic differential equation on ^d with smooth and bounded coefficients. We apply the techniques of the quasi-sure analysis to show that this equation can be solved pathwise out of a slim set. Furthermore, we can restrict the equation to the level sets of a nondegenerate and smooth random variable, and this provides a method to construct the solution to an anticipating stochastic differential equation with smooth and nondegenerate initial condition.