The authors combine results on Brownian backward stochastic differential equations (BSDEs) and progressive enlargement of filtrations to prove a general existence and uniqueness result for the solutions to BDSEs with random marked jumps. The methodology is illustrated by applying it to the pricing and hedging of European options and the problem of utility maximization in markets with jumps.
Let \((\Omega,\mathcal G, \mathbb P)\) be a probability space, and fix a right-continuous filtration \(\mathbb F = (\mathcal F _{t})_{t \geq 0}\) generated by a \(d\)-dimensional Brownian motion \(W\). Consider a finite sequence \((\tau_{k}, \zeta_{k})_{1\leq k \leq n}\), where \((\tau_{k})_{1\leq k \leq n}\) is a nondecreasing sequence of random times and \((\zeta_{k})_{1\leq k \leq n}\) is a sequence of random marks valued in some Borel subset \(E\) of \(\mathbb R^{m}\). The random measure \(\mu\) associated with the sequence \((\tau_{k}, \zeta_{k})_{1\leq k \leq n}\) is defined by \[ \mu([0,t]\times B) = \sum_{k=1}^{n} I_{[\tau_{k} \leq t, \zeta_{k} \in B]}. \]
The progressive enlargement of \(\mathbb F\) is the smallest right-continuous filtration \(\mathbb G = (\mathcal G _{t})_{t \geq 0}\) containing \(\mathbb F\) such that for each \(k = 1, \dots, n\), \(\tau_{k}\) is a \(\mathbb G\)-stopping time and \(\zeta_{k}\) is \(\mathcal G_{\tau_{k}}\)-measurable. The \(\sigma\)-algebra generated by the left-continuous \(\mathbb G\)-progressively measurable processes is denoted by \(\mathcal P \mathcal M (\mathbb G)\); the space of essentially bounded \(\mathbb R\)-valued \(\mathcal P \mathcal M (\mathbb G)\)-measurable processes defined on \([0,T]\) is denoted by \(\mathcal S ^{\infty}_{\mathbb G}[0,T]\).
The authors consider the following type of BSDEs: find \((Y,Z,U) \in \mathcal S ^{\infty}_{\mathbb G}[0,T] \times L^{2}_{\mathbb G}[0,T] \times L^{2}(\mu)\) such that for all \(0 \leq t \leq T\), \[ Y_{t} = \xi + \int_{(t,T]}f(s,Y_{s},Z_{s},U_{s})\, ds-\int_{(t,T]}Z_{s} d W_{s} - \int_{(t,T]\times E}U_{s}(e)\,d \mu, \] where \(\xi\) is a \(\mathcal G_{T}\)-measurable random variable of the form \[ \xi = \sum_{k=0}^{n}\xi^{k}((\tau_{i})_{i\leq k},(\zeta_{i})_{i\leq k}) \cdot I_{[\tau_{k}\leq T < \tau_{k+1}]}, \] and \(f : [0,T] \times \Omega \times \mathbb R \times \mathbb R^{d} \times\mathrm{Bor}(E,\mathbb R) \rightarrow \mathbb R\) is sufficiently measurable; \(\mathrm{Bor}(E,\mathbb R)\) is the set of Borel functions from \(E\) to \(\mathbb R\).
It is shown that such BSDEs have a solution if a sequence of Brownian BSDEs of the form \[ Y^{k}_{t} = \xi^{k} + \int_{(t,T]}f^{k}(s,Y^{k}_{s},Z^{k}_{s},Y^{k+1}_{s} - Y^{k}_{s})\, d s - \int_{(t,T]}Z^{k}_{s}\, dW_s \] all have solutions satisfying some boundedness conditions; \(f^{k},Y^{k},Z^{k}\) denote the canonical decomposition of \(f,Y,Z\) as \[ X_{t} = X^{0}_{t} \cdot I_{t \leq \tau_{1}} + X^{n}_{t}((\tau_{i})_{i\leq n},(\zeta_{i})_{i\leq n}) \cdot I_{\tau_{n} \leq t} + \sum_{k=1}^{n-1}X^{k}_{t}((\tau_{i})_{i\leq k},(\zeta_{i})_{i\leq k}) \cdot I_{\tau_{k} \leq t \leq \tau_{k+1}}. \] For quadratic BSDEs with jumps, the existence of solutions is obtained from this existence theorem under mild assumptions. Based on a comparison theorem, a general result establishing the uniqueness of solutions is also provided, which takes a particularly simple form for quadratic BSDEs.
The authors present two applications of this theory: they derive explicit formulas for the price and hedging strategy of a European option on an underlier whose price may jump; and they solve for a self-financing strategy which maximizes an exponential utility on trading an asset subject to counterparty credit risk (the default of the counterparty induces jumps in the asset price). The trading strategy is obtained as the solution to a minimization problem.