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We assume that the filtration 픽 is generated by a d-dimensional Brownian motion W = (W1, ..., Wd)' as well as an integer-valued random measure μ(du, dy). The random variable ... is the default time and L is the default loss. Let 픾 = {...t; t ≥ 0} be the progressive enlargement of 픽 by (..., L); that is, 픾 is the smallest filtration including 픽 such that ... is a 픾-stopping time and L is ......-measurable. We mainly consider the forward CDS with loss in the framework of stochastic interest rates whose term structures are modeled by the Heath-Jarrow-Morton approach with jumps under the general conditional density hypothesis. We describe the dynamics of the defaultable bond in 픾 and the forward CDS with random loss explicitly by the BSDEs method. [ABSTRACT FROM AUTHOR]

We construct a market of bonds with jumps driven by a general marked point process as well as by a n-valued Wiener process based on Bjork et al. [6], in which there exists at least one equivalent martingale measure Q0. Then we consider the mean-variance hedging of a contingent claim H ∈ L2(FT0) based on the self-financing portfolio based on the given maturities T1,..., Tn with T0 < T1 < ...

We consider the dynamic convex valuation (DCV) in an incomplete market of m stocks S = (S1,..., Sm) in the Brownian motion setting. In this framework, we continue our work in Xiong and Kohlmann [17] on S-related DCV by now considering the S-related DCV generated by a conditional g-expectation under an equivalent martingale measure Q0 for a given function g(t, y, z) satisfying a Lipschitz condition. We give a sufficient and necessary condition for g so that Eg is an S-related DCV. We mainly study the dynamics of an [image omitted]-dominated S-related DCV C = {(Ct(ξ)); ξ ∈L∞(FT)}. By applying Theorem 5 of Delbaen et al. [4], it is seen that the penalty functional α of C satisfies [image omitted] for a function [image omitted] with [image omitted], where k is a positive constant. Under the assumption that [image omitted] is continuous with respect to l, we prove that {Ct(ξ); t ∈ [0, T]} is the unique bounded solution of a BSDE generated by the function g(t, z2) with quadratic growth in z2. This main result generalizes Theorem 7.1 of Coquet et al. [2] about the “Eμ-dominated F-consistent expectation.” [ABSTRACT FROM AUTHOR]

We consider an incomplete market with general jumps in the given price process S of a risky asset. We define the S-related dynamic convex valuation (S-related DCV) which is time-consistent. We discuss the representation for a given S-related DCV C in terms of a 'penalty functional' α and give some characteristics of α, which are the sufficient conditions for a given C to be an S-related DCV. Finally, we give two special forms of α satisfying those conditions to describe the dynamics of the corresponding S-related DCV by a backward semimartingale equation. [ABSTRACT FROM AUTHOR]

In this article, we consider a new valuation, which we call dynamic q-valuation [image omitted] of a contingent claim in a semimartingale model with a general continuous filtration. We prove that this valuation has the properties of a convex risk valuation and by making use of the (p, B)-optimal martingale measure introduced in Mania et al. [8] we obtain a backward semimartingale equation (BSE) to characterize the dynamic q-valuation. We prove the convexity of this q-valuation its time-consistency property. Given q and [image omitted], we consider the ∈ f-convolution of [image omitted] and [image omitted]. This new risk valuation is shown to have an explicitly stated representation as a backward semimartingale equation (BSE). Furthermore, we discuss the convergence of a sequence of ∈ f-valuation of [image omitted]. So, starting from the q-valuation we derive several new risk-measures which allow for an explicit representation. [ABSTRACT FROM AUTHOR]

We consider the mean-variance hedging of a defaultable claim in a general stochastic volatility model. By introducing a new measure Q0, we derive the martingale representation theorem with respect to the investors' filtration . We present an explicit form of the optimal-variance martingale measure by means of a stochastic Riccati equation (SRE). For a general contingent claim, we represent the optimal strategy and the optimal cost of the mean-variance hedging by means of another backward stochastic differential equation (BSDE). For the defaultable option, especially when there exists a random recovery rate we give an explicit form of the solution of the BSDE. [ABSTRACT FROM AUTHOR]

In a market with a discontinuous filtration, whose price is influenced by a random factor, we study an optimization problem of an investor who is facing a sequence of losses driven by a Cox process. We give a form of variance-optimal martingale measure by changing the filtration. By using the solutions of the stochastic Riccati equation and another associated backward stochastic equation, we obtain a solution of the optimization problem of the investor. [ABSTRACT FROM AUTHOR]

A utility optimization problem for continuous time financial markets is examined in the presence of additional information. Three cases, including “side information known in advance,” “information disclosure at the market-known time,” and “information disclosure at the market-unknown time,” are discussed. The martingale representation theorems for each case are obtained by using stochastic filtering technique. In the case of logarithmic utility, the analytic forms of optimal solutions are obtained and the effect of these kinds of additional information to investor's strategies are revealed. [ABSTRACT FROM AUTHOR]