# Continuous-time Stochastic Control and Optimization with by Huyên Pham

By Huyên Pham

Stochastic optimization difficulties come up in decision-making difficulties below uncertainty, and locate numerous functions in economics and finance. nonetheless, difficulties in finance have lately resulted in new advancements within the idea of stochastic control.

This quantity offers a scientific remedy of stochastic optimization difficulties utilized to finance by way of offering the various present equipment: dynamic programming, viscosity strategies, backward stochastic differential equations, and martingale duality tools. the speculation is mentioned within the context of contemporary advancements during this box, with entire and distinctive proofs, and is illustrated through concrete examples from the area of finance: portfolio allocation, choice hedging, genuine suggestions, optimum funding, etc.

This publication is directed in the direction of graduate scholars and researchers in mathematical finance, and also will gain utilized mathematicians attracted to monetary purposes and practitioners wishing to grasp extra concerning the use of stochastic optimization equipment in finance.

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**Sample text**

4. 5, validates the optimality of the candidate solution to the HJB equation. This classical approach to the dynamic programming is called the veriﬁcation step. 6 by solving three examples in ﬁnance. The main drawback of this approach is to suppose the existence of a regular solution to the HJB equation. 7 a simple example inspired by ﬁnance pointing out this feature. 1) where W is a d-dimensional Brownian motion on a ﬁltered probability space (Ω, F, F = (Ft )t≥0 , P ) satisfying the usual conditions.

The new feature is that the constraint carries not directly on α but on the “derivative” of α with respect to the variations dS of the price. 4, the utility of the portfolio and/or option payoﬀ to be optimized, is measured according to the Von Neumann-Morgenstern criterion. The risk of the uncertain payoﬀ −X is valued by the risk measure ρ(−X) written as an expectation −EU (−X) under some ﬁxed probability measure and for a given utility function U . 3) where Q is a set of probability measures. 2) is inspired from the theory of coherent risk measures initiated by Artzner et al.

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, ∞). s. In the sequel, we denote by E Q the expectation operator under Q. When a property is relative to Q, we shall specify the reference to Q. When it is not speciﬁed, the property is implicitly relative to P , the initial probability measure on (Ω, F ). 12 (Bayes formula) Let Q P and Z its martingale density process.