Abstract
We introduce a suitable backward stochastic differential equation (BSDE) to represent the value of an optimal control problem with partial observation for a controlled stochastic equation driven by Brownian motion. Our model is general enough to include cases with latent factors in mathematical finance. By a standard reformulation based on the reference probability method, it also includes the classical model where the observation process is affected by a Brownian motion (even in presence of a correlated noise), a case where a BSDE representation of the value was not available so far. This approach based on BSDEs allows for greater generality beyond the Markovian case, in particular our model may include path-dependence in the coefficients (both with respect to the state and the control), and does not require any nondegeneracy condition on the controlled equation. We use a randomization method, previously adopted only for cases of full observation, and consisting in a first step, in replacing the control by an exogenous process independent of the driving noise and in formulating an auxiliary (“randomized”) control problem where optimization is performed over changes of equivalent probability measures affecting the characteristics of the exogenous process. Our first main result is to prove the equivalence between the original partially observed control problem and the randomized problem. In a second step, we prove that the latter can be associated by duality to a BSDE, which then characterizes the value of the original problem as well.
Highlights
The main aim of this paper is to prove a representation formula for the value of a general class of stochastic optimal control problems with partial observation by means of an appropriate backward stochastic differential equation.To motivate our results, let us start with a classical optimal control problem with partial observation, where we consider an Rn-valued controlled process X solution to dXt = b(Xt, αt ) dt + σ 1(Xt, αt ) dVt1 + σ 2(Xt, αt ) dVt2, Received September 2016; revised July 2017. 1Supported by the Italian MIUR-PRIN 2015 “Deterministic and stochastic evolution equations.” MSC2010 subject classifications
Our first main result is to prove the equivalence between the original partially observed control problem and the randomized problem. We prove that the latter can be associated by duality to a backward stochastic differential equation (BSDE), which characterizes the value of the original problem as well
Let us start with a classical optimal control problem with partial observation, where we consider an Rn-valued controlled process X solution to dXt = b(Xt, αt ) dt + σ 1(Xt, αt ) dVt1 + σ 2(Xt, αt ) dVt2, Received September 2016; revised July 2017. 1Supported by the Italian MIUR-PRIN 2015 “Deterministic and stochastic evolution equations.”
Summary
The main aim of this paper is to prove a representation formula for the value of a general class of stochastic optimal control problems with partial observation by means of an appropriate backward stochastic differential equation (backward SDE or BSDE). A standard model, widely used in applications, consists in assuming that W is defined by the formula dWt = h(Xt , αt ) dt + dVt2, W0 = 0 In this problem, b, σ 1, σ 2, f , g, h are given data satisfying appropriate assumptions. One introduces the process of unnormalized conditional distributions defined for every test function φ : Rn → R by the formula ρt (φ) = E φ(Xt )Zt | FtW and proves that ρ is a solution to the so-called controlled Zakai equation:
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