Risk-sensitive control for a class of nonlinear systems with multiplicative noise

Abstract

In this paper, we consider the problem of optimal control for a class of nonlinear stochastic systems with multiplicative noise. The nonlinearity consists of quadratic terms in the state and control variables. The optimality criteria are of a risk-sensitive and generalised risk-sensitive type. The optimal control is found in an explicit closed-form by the completion of squares and the change of measure methods. As applications, we outline two special cases of our results. We show that a subset of the class of models which we consider leads to a generalized quadratic affine term structure model (QATSM) for interest rates. We also demonstrate how our results lead to generalisation of exponential utility as a criterion in optimal investment.

Robust risk‐sensitive control

“…If conditions of Theorem 3 hold for all T ∈ (0, ∞) in the current case of constant coefficients and Nash equilibrium (u * ∞ , v * ∞ ), then the requirements (33) and (35) are satisfied for all T ∈ (0, ∞). In remains to find (sufficient) conditions under which the stability requirements (34) and (36) are satisfied under the pair (u * ∞ , v * ∞ ). Note first that by the Girsanov theorem processes Wu and Wv defined as:…”

“…Let Σ u (t) > 0, Σ v (t) > 0, H u (t) > 0, and H v (t) > 0, for all t > 0. The stability conditions (34) and (36)…”

Backward stochastic differential equations with unbounded generators

“…Moreover, this also implies that { √ α 1 y n } n≥1 is a Cauchy sequence in M 2 1 (0, T ; R d ). Hence, the limiting processes y * = lim n→∞ y n and z * = lim n→∞ z n are the solution pair of (11). In addition, when such a pair of processes is substituted in (11), then (11) becomes an example of (8) with ψ(•) = z * (•).…”

“…Hence, the limiting processes y * = lim n→∞ y n and z * = lim n→∞ z n are the solution pair of (11). In addition, when such a pair of processes is substituted in (11), then (11) becomes an example of (8) with ψ(•) = z * (•). Therefore, Lemma 2.1 applies, and we have that y * (•) ∈ H 2 1 (0, T ; R d×k ).…”

“…The interest in these equations is not only theoretical, but is also motivated by applications in mathematical finance. Indeed, some very important interest rate models are given by stochastic differential equations (see, for example, [44], [11], [4]). The problem of market completeness (and thus of pricing and hedging of derivatives) in such models gives rise to BSDEs with possibly unbounded coefficients (see [43] for details).…”