Stochastic control methods: Hedging in a market described by pure jump processes
Let \(\{(X_t, Y_t)\}\) be a pure jump process Markov process. \(\{X_t\}\) is an exogenous process and \(\{Y_t\}\) drives the price process \(S_t = S_0 \exp\{Y_t\}\) of a discounted risky asset. It is assumed that the jump intensity and the jump sizes are bounded. The problem is to hedge a contingent claim of the form \(B_T = B(X_T,Y_T)\) for some measurable function \(B\). Since the market is not complete, there does not exist a perfect hedge. The goal is to minimise an expected utility of the hedging error \(E[u(W_T^\theta - B_T)]\). Here \(u\) is a utility function and \[ W_t^\theta = w_0 + \int_0^t \theta_{r-}\, d S_r \] is the wealth process for the trading strategy \(\{\theta_t\}\). In a first model, exponential utility \(u(x) = 1-e^{-a x}\) for some \(a > 0\) is investigated. In addition, \(B_T = B(X_T)\) does not depend on \(Y_T\). The problem is solved via the dynamic programming approach. An ansatz of the form \(U(t,x,w) = e^{-a w} V(t,x)\) reduces the corresponding Hamilton--Jacobi--Bellman equation to a linear equation with a final condition. It is shown that there is a measurable bounded solution, which gives the expected utility and the optimal strategy. The general case is solved via a contraction method. Since the intensity of the jump process is bounded, the model can be embedded into a marked homogeneous Poisson process. By conditioning on the first event of the marked Poisson process, a contraction operator is obtained. The operator can be compared with a model where investment is allowed only until the \(n\)-th event. In the limit, the optimal utility and the optimal strategy are found.
piecewise deterministic control problems
scientific article; zbMATH DE number 5760429
scientific article