An Anticipative Stochastic Minimum Principle under Enlarged Filtrations

Optimal Equivalent Probability Measures under Enlarged Filtrations

“…This time, instead of (10), we study the minimum relative entropy problem where the relative entropy is still given by ℑ(ℚ|ℙ) = 𝔼 ℚ ln X T but yet with state process X = X h satisfying (23). In (24), the minimum relative entropy measure is denoted by Q ∶= ℚ ĥ , while the set A now contains all integrable, càdlàg and F tadapted control processes h(t, ⋅) . Note that (24) in particular corresponds to problem (10) subject to (22).…”

“…In (24), the minimum relative entropy measure is denoted by Q ∶= ℚ ĥ , while the set A now contains all integrable, càdlàg and F tadapted control processes h(t, ⋅) . Note that (24) in particular corresponds to problem (10) subject to (22). Hence, we are presently dealing with the problem of finding the minimum relative entropy measure under a constraint.…”

“…Proposition 2.2 Suppose that the relative entropy is given by ℑ(ℚ|ℙ) = 𝔼 ℚ ln X T with state process X = X h satisfying (23). Then for all t ∈ [0, T] , the optimal control process ĥ(t, ⋅) solving the minimum relative entropy problem (24) is implicitly determined by the integral equation With this optimal control ĥ(t, ⋅) at hand, for all t ∈ [0, T] the corresponding optimal process Ĝt is given by Proof By putting the solution of the geometric SDE ( 23) into (8), we get (23)…”

“…In order to solve problem (24), we maximize the functional point-wise with respect to h(t, ⋅) for every fixed t ∈ [0, T] . To this end, for fixed t we compute the derivatives We further denote by ĥ ∈ A the critical value related to problem (24).…”

“…In order to solve problem (24), we maximize the functional point-wise with respect to h(t, ⋅) for every fixed t ∈ [0, T] . To this end, for fixed t we compute the derivatives We further denote by ĥ ∈ A the critical value related to problem (24). Then, the necessary optimality condition � ĥ(t, ⋅) = 0 is equivalent to (25) and we observe which is strictly positive for all t ∈ [0, T] , since the Lévy measure is positive by assumption.…”