By Huyên Pham

Stochastic optimization difficulties come up in decision-making difficulties lower than uncertainty, and locate a number of purposes in economics and finance. however, difficulties in finance have lately ended in new advancements within the conception of stochastic control.

This quantity presents a scientific therapy of stochastic optimization difficulties utilized to finance by way of featuring the several latest tools: 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 exact proofs, and is illustrated through concrete examples from the realm of finance: portfolio allocation, choice hedging, genuine strategies, 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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Dx ϕ(x) + tr(σ(t, x)σ (t, x)Dx2 ϕ(x)), 2 ϕ ∈ C 2 (Rn ). 12). 12), v(t, x) a (real-valued) function of class C 1,2 on T × Rn and r(t, x) a continuous function on T × Rd , we obtain by Itˆo’s formula Mt := e− Rt 0 t r(s,Xs )ds v(t, Xt ) − e− Rs 0 r(u,Xu )du 0 t = v(0, X0 ) + e − Rs 0 r(u,Xu )du ∂v + Ls v − rv (s, Xs )ds ∂t Dx v(s, Xs ) σ(s, Xs )dWs . 20) 0 The process M is thus a continuous local martingale. 22) n where f (resp. g) is a continuous function from [0, T ] × Rn (resp. Rn ) into R. We also assume that the function r is nonnegative.

The objective is to maximize over control processes the gain function J, and we introduce the associated value function: v(t, x) = sup α∈A(t,x) J(t, x, α). 2 Controlled diﬀusion processes 39 • Given an initial condition (t, x) ∈ [0, T ) × Rn , we say that α ˆ ∈ A(t, x) is an optimal control if v(t, x) = J(t, x, α). ˆ • A control process α in the form αs = a(s, Xst,x ) for some measurable function a from [0, T ] × Rn into A, is called Markovian control. In the sequel, we shall implicitly assume that the value function v is measurable in its arguments.

12) starting at time t. e. Xt = ξ. The uniqueness is pathwise and means that if X and Y are two such strong solutions, we have P [Xs = Ys , ∀t ≤ s ∈ T] = 1. This solution is square integrable: for all T > t, there exists a constant CT such that E sup |Xs |p ≤ CT (1 + E[|ξ|p ]). t≤s≤T This result is standard and one can ﬁnd a proof in the books of Gihman and Skorohod [GS72], Ikeda and Watanabe [IW81], Krylov [Kry80] or Protter [Pro90]. 12) starting from ξ at time t. When t = 0, we simply write X ξ = X 0,ξ .