By Huyên Pham
Stochastic optimization difficulties come up in decision-making difficulties below uncertainty, and locate a number of functions in economics and finance. however, difficulties in finance have lately resulted in new advancements within the concept of stochastic control.
This quantity offers a scientific therapy of stochastic optimization difficulties utilized to finance by way of featuring the several present equipment: dynamic programming, viscosity suggestions, backward stochastic differential equations, and martingale duality tools. the idea is mentioned within the context of contemporary advancements during this box, with entire and specific proofs, and is illustrated via concrete examples from the area of finance: portfolio allocation, choice hedging, genuine suggestions, optimum funding, etc.
This ebook is directed in the direction of graduate scholars and researchers in mathematical finance, and also will profit utilized mathematicians drawn to monetary functions and practitioners wishing to grasp extra concerning the use of stochastic optimization tools in finance.
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Extra resources for Continuous-time Stochastic Control and Optimization with Financial Applications
We also mention a recent book by Schmidli [Schm08] on stochastic control in insurance. 1 Introduction In this chapter, we use the dynamic programming method for solving stochastic control problems. 2 the framework of controlled diﬀusion and the problem is formulated on ﬁnite or inﬁnite horizon. The basic idea of the approach is to consider a family of control problems by varying the initial state values, and to derive some relations between the associated value functions. 3. 4. 5, validates the optimality of the candidate solution to the HJB equation.
17) 0 0 with the convention that e−βθ(ω) = 0 when θ(ω) = ∞. 3 In the sequel, we shall often use the following equivalent formulation (in the ﬁnite horizon case) of the dynamic programming principle: (i) For all α ∈ A(t, x) and θ ∈ Tt,T : θ f (s, Xst,x , αs )ds + v(θ, Xθt,x ) . 18) (ii) For all ε > 0, there exists α ∈ A(t, x) such that for all θ ∈ Tt,T θ f (s, Xst,x , αs )ds + v(θ, Xθt,x ) . 20) for any stopping time θ ∈ Tt,T . We have a similar remark in the inﬁnite horizon case. e. compute v(θ, Xθt,x ), and then maximizing over controls on [t, θ] the quantity θ E t f (s, Xst,x , αs )ds + v(θ, Xθt,x ) .
12) starting from x at time t. 22). Then v admits the representation T v(t, x) = E e− t for all (t, x) ∈ [0, T ] × Rd . 17). 23) is simply derived by writing that E[MT ] = E[Mt ]. We may also obtain this Feynman-Kac representation under other conditions on v, for example with v satisfying a quadratic growth condition. 5). 6). 22). 24) boundedness conditions on b, σ, and polynomial growth condition on f and g (see Friedman [Fr75] p. 147). 24) but requiring stronger regularity conditions on the coeﬃcients (see Krylov [Kry80] p.