Abstract In this Note we consider a quadratic backward stochastic differential equation (BSDE) driven by a continuous martingale M and whose generator is a deterministic function. We prove (in Theorem 2.1) that if M is a strong homogeneous Markov process and if the BSDE has the form (1.2) then the unique solution (Y, Z,N) of the BSDE is reduced to (Y, Z), i.e. the orthogonal martingale N is equ...

Abstract. It is well-known that well-posedness of a martingale problem in the class of continuous (or r.c.l.l.) solutions enables one to construct the associated transition probability functions. We extend this result to the case when the martingale problem is well-posed in the class of solutions which are continuous in probability. This extension is used to improve on a criterion for a probabi...

We consider the problem of valuation of certain Asian options in the geometric jump-diffusion models with continuously dividend-paying assets. With the sources of diffusion risks and two primitive tradeable assets, the market in this model is, in general, incomplete, and so, there are more than one equivalent martingale measures and no-arbitrage prices. For this jump-diffusion model, we adopt t...

We consider the problem of producing lower bounds on the optimal cost-to-go function of a Markov decision problem. We present two approaches to this problem: one based on the methodology of approximate linear programming (ALP) and another based on the so-called martingale duality approach. We show that these two approaches are intimately connected. Exploring this connection leads us to the prob...

We prove the max-martingale conjecture given in Ob lój and Yor [5]. We show that for a continuous local martingale (Nt : t ≥ 0) and a function H : R×R+ → R, H(Nt, sups≤t Ns) is a local martingale if and only if there exists a locally integrable function f such that H(x, y) = ∫ y 0 f(s)ds− f(y)(x− y) +H(0, 0). This implies readily, via Lévy’s equivalence theorem, an analogous result with the max...

Let (Mt)0≤t≤1 be a continuous martingale with initial lawM0 ∼ μ0, and terminal lawM1 ∼ μ1, and let S = sup 0≤t≤1 Mt. In this paper we prove that there exists a greatest lower bound with respect to stochastic ordering of probability measures, on the law of S. We give an explicit construction of this bound. Furthermore a martingale is constructed which attains this minimum by solving a Skorokhod ...

is a martingale w.r.t. {Ft∧Tn : t ≥ 0}. The stopping times {Tn} are said to reduce X . Remarks: 1. I brooded over why we set XT t = 0 in the definition, and this is the only explanation I could find: If we defined X t = XT∧t on {T ≥ 0}, thenXT 0 = X0, so according to the above definition of a local martingale,X T t a martingale implies E[XT 0 ] = E[X0] <∞. So, with the above definition of XT t ...