Quadratic Semimartingale BSDEs Under an Exponential Moments Condition

Abstract

In the present article we provide existence, uniqueness and stability results under an exponential moments condition for quadratic semimartingale backward stochastic differential equations (BSDEs) having convex generators. We show that the martingale part of the BSDE solution defines a true change of measure and provide an example which demonstrates that pointwise convergence of the drivers is not sufficient to guarantee a stability result within our framework.

BSDEs Driven by Multidimensional Martingales and Their Applications to Markets with Funding Costs

“…In the paper by Mocha and Westray [23], the comparison theorem is proven using the θ-technique under the postulate that m, the terminal value η, and solution Y 1 and Y 2 have exponential moments of all orders (see Theorem 5.1 in [23]). Tevzadze [32] examined the case when the terminal condition is bounded, and he focussed on bounded solutions Y complemented by BMO martingale component.…”

“…To this end, they used the Doléans exponential of a càdlàg martingale and they imposed the requirement it is a positive, uniformly integrable martingale and, in addition, satisfies some integrability conditions (see, in particular, Lemma 2.2 in [3] or equation (2.6) in Section 2 of this work), which are not easy to verify and may be too restrictive for applications). We stress in this regard that the results from [3,23,32] are not sufficient for the purposes studied in [1, 25,26], since the assumptions made in these papers fail to hold in the context of a typical financial model. Consequently, some extensions of the existing comparison theorems for BSDEs driven by multi-dimensional martingales are needed to demonstrate the existence of non-empty intervals for fair bilateral prices (or bilaterally profitable prices), as well as the monotonicity of prices with respect to the initial endowment of an agent.…”

BSDEs in utility maximization with BMO market price of risk

“…(i) Using the assumption δ = 0, we can argue similarly to the proof of [7] Theorem 2.1. That proof is given in a Brownian setting but translates correspondingly to our semimartingale model similarly to [25]. Withγ := max{1, γ} and based on (4.2), this yields the existence of a solution (Ψ, Z, N ) satisfying…”

“…for all t ∈ [0, T ] and z, z 1 , z 2 ∈ R d , where β F , γ are constants and η · M is a BMO martingale such that E exp γ T 0 B t η t 2 dA t < +∞ forγ := max{1, γ}. From [25] Theorem 4.1 (noting that convexity of F in z is not needed by [25] Remark 4.3), we obtain that there exists a solution (Ψ, Z, N ) to (6.7) which satisfies…”

“…Those existence results were subsequently extended by Morlais [26], Briand and Hu [5,6], Delbaen, Hu and Richou [7] as well as Barrieu and El Karoui [3]. In particular, the assumption that the mean-variance tradeoff process be bounded has been weakened to it having certain exponential moments, see Mocha and Westray [24,25].…”