By Huyên Pham

Stochastic optimization difficulties come up in decision-making difficulties less than uncertainty, and locate quite a few functions in economics and finance. however, difficulties in finance have lately resulted in new advancements within the thought of stochastic control.

This quantity presents a scientific therapy of stochastic optimization difficulties utilized to finance by way of offering different present tools: dynamic programming, viscosity recommendations, backward stochastic differential equations, and martingale duality tools. the idea is mentioned within the context of modern advancements during this box, with entire and targeted proofs, and is illustrated via concrete examples from the realm of finance: portfolio allocation, choice hedging, genuine strategies, optimum funding, etc.

This e-book is directed in the direction of graduate scholars and researchers in mathematical finance, and also will profit utilized mathematicians drawn to monetary purposes and practitioners wishing to grasp extra in regards to the use of stochastic optimization tools in finance.

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**Extra info for Continuous-time Stochastic Control and Optimization with Financial Applications **

**Example text**

Let Q be a probability measure on (Ω, F) such that Q is absolutely continuous with respect to P , denoted by Q P . P. For all t ∈ T, the restriction of Q to Ft is absolutely continuous with respect to the restriction of P to Ft . Let Zt be the corresponding Radon-Nikodym density, often denoted by Zt = dQ dP . s. ∀ t ∈ T. Up to modiﬁcation, we may suppose that the paths of Z are c`ad-l` ag. Z is also called the martingale density process of Q (with respect to P ). s, which means that Q[τ < ∞] = 0, where τ = inf{t : Zt = 0 or Zt− = 0}: this follows from the fact that the martingale Z vanishes on [τ, ∞).

G) is a continuous function from [0, T ] × Rn (resp. Rn ) into R. We also assume that the function r is nonnegative. We give here a simple version of the FeynmanKac representation theorem. 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.

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. We give here a simple version of the FeynmanKac representation theorem.