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An Introduction to Stochastic Control An introduction to stochastic control F. J. Silva Dip. di Matematica Guido Castelnuovo March 2012 Some disperse but useful results This section is based on [24, Chapter 1]. Let Ω be a nonempty set, and let F ⊂ 2Ω. We say that F is a π-system if A; B 2 F ) A \ B 2 F. F is a λ-system if (i) Ω 2 F; (ii) A, B 2 F and A ⊆ B imply B n A 2 F; (iii) Ai 2 F, Ai " A implies that A 2 F. Lemma 1. [σ-field of a π-system, or λ-π lemma] If a π system A is contained in a λ system F then σ(A) ⊆ F. Proof: See any standard book on measure theory (e.g. [2]) Example of application: [Uniqueness of the extension of a measure defined on a π-system] Let P and Q be two probability measures on (Ω; F) which coincide on a 1 π-system A, then they coincide on σ(A). In fact, it is enough to define C = fC 2 F ; P (C) = Q(C)g and two verify that it is a λ-system. Corollary 1. [Measurable function w.r.t. the σ-field of a π system] Let A be a π system. Let H be a linear space of functions from Ω to R such that (i)1 2 H;( ii) IA 2 H for all A 2 A (iii) φi 2 H; 0 ≤ φi " φ, φ is bounded ) φ 2 H: Then H contains all σ(A)-measurable functions from Ω to R. Proof: Let φ be σ(A)-measurable. Clearly we have that n + n X −n φ " φ ; with φ (!) := j2 Ifφ+(!)2[j2−n;(j+1)2−n]g (1) j≥0 with an analogous approximation for φ−. Therefore, by (iii), it is enough to show that φn 2 H. But φ is the sum of indicator of elements in σ(A). Therefore it is natural to consider the set F := fA 2 Ω; IA 2 Hg: 2 and to prove that σ(A) ⊆ F. But this follows from lemma 1, since A is a π-system and A ⊂ F, which is easily shown to be a λ-system. Theorem 1. [Dynkin theorem] Let (Ω; F) and (Ω0; F 0) two measurable spaces, and let (U; d) be a Polish space. Let ξ :Ω ! Ω0 and φ :Ω ! U be r.v.'s. Then φ 2 σ(ξ)- measurable, i.e. φ−1(B(U)) ⊆ ξ−1(F 0), iff there exists a measurable η :Ω0 ! U such that φ(!) = η(ξ(!)) for all ! 2 Ω: Proof: Consider the case U = R (the general case can be obtained using an isomorphism theorem, see [19]) and define the set 0 0 H := η(ξ); for some F -measurable map η :Ω ! R : We have to shown that the set of σ(ξ)=B(R) measurable maps belongs to H. This can be done by checking the assumptions of corollary 1 with A = σ(ξ) and proving that H satisfies (i), (ii) and (iii). Exercise: Do the details of the above proof. 3 Lemma 2. [Borel-Cantelli] Let (Ω; F; P) be a probability space and consider a sequence of events Ai 2 F. We have X P(Ai) < 1 ) P(\i [j≥i Aj) = 0: i Proof: Straightforward. Note that P(\i [j≥i Aj) ≤ P([j≥iAj) for all i and use the convergence of the series. Lemma 3. [Chebyshev inequality] Consider a nonnegative r.v. X. Then, for all p 2 (0; 1) and " > 0 we have p E(X ) P(X ≥ ") ≤ : "p Proof: It suffices to note that Z Z p p p X (X ≥ ") = (X ≥ " ) = p p d (!) ≤ d (!): P P IfX ≥" g P p P Ω Ω " 4 Conditional expectation This section is based on [24, Chapter 1]. Consider a probability space (Ω; F; P). Let 1 X 2 L (Ω) and G be a sub-σ-field of F. Define the signed measure µ : G! R as Z µ(A) := X(!)dP(!) for all A 2 G: A 1 By the Radon-Nikodym theorem there exists a unique PjG a.s. r.v. f 2 LG(Ω) (i..e. in particular G-measurable) such that Z Z fdP = XdP for all A 2 G: (2) A A The function f is called the conditional expectation of X given G, and we write E(XjG) := f: 5 Fundamental properties of E(·|G) All the properties below are simple consequences of (2) (Do them as an exercise!) (i) E(·|G) is a linear bounded functional. (ii) For a constant a 2 R, we have E(ajG) = a. 1 (iii) [Mononoticity] For X; Y 2 LF with X ≥ Y we have E(XjG) ≥ E(Y jG). 2 2 (iv) [Take out the measurable part] For X 2 LF and Y 2 LG, we have E(YXjG) = Y E(XjG). (v) [Characterization of independence] X is independent of G iff for every Borel f 1 such that f(X) 2 L (Ω) we have E(f(X)jG) = E(f(X)). (vi) [Tower or \projection" property] If G1 ⊆ G2 ⊆ F, then E(E(XjG1)jG2) = E(E(XjG2)jG1) = E(XjG1): (vii) [Jensen inequality] Let φ convex such that φ(X) 2 L1(Ω), then φ(E(XjG)) ≤ E(φ(X)jG): 6 [Conditioning one r.v. X w.r.t. another r.v. ξ] 1 Let X 2 LF and ξ : (Ω; F) ! (U; B(U)). Note that we can always define −1 E(Xjξ) := E(Xjξ (B(U))) = η(ξ) for some B(U)=B(R)- measurable function η; by Dynkin theorem. Therefore, it is natural to define E(Xjξ = x) := η(x): Another way to define this is by appealing to the Radon-Nikodym. In fact, let us define in B(U) the measure Z ν(B) := X(!)dP(!); ξ−1(B) −1 which is absolutely continuous with respect to Pξ := P ◦ ξ (the image measure of P dν under ξ). Therefore, there exists a Radon -Nikodym derivative (unique "-a.s.) such dP" P that Z dν Z (x)dP"(x) = X(!)dP(!) for all B 2 B(U): B dP" ξ−1(B) 7 We define dν E(Xjξ = x) := (x): dP" It can be checked that −1 dν η(ξ(!)) = E(Xjξ (B(U)))(!) = (ξ(!)) Pjξ−1(B(U)) − a.s. (3) dP" −1 Integrating w.r.t. a set of the form ξ (A) and using the definition of Pξ, we obtain that dν η(x) = (x) P" − a.s. dP" Note that incidentally, this gives another proof of Dynkin theorem. Let us prove (3). We 8 have R dν R dν (ξ(!))d (!) = (x)d "(x) ξ−1(B) dP" P B dP" P = ν(B) R = ξ−1(B) X(!)dP(!) R −1 = ξ−1(B) E(Xjξ (B(U)))(!)dP(!): which yields the result. Now, we define the conditional probability w.r.t. a σ-field G as P(AjG) := E(IAjG): Nota that for each B 2 F we have that P(BjG) is only defined PjG a.s. Thus, it is not sure that we can find some A 2 G with P(A) = 1 such that if we fix any ! 2 A, we can measure all the sets B 2 F . However, we can give a sense to this using the concept of regular conditional probability (see [2] for more on this). 9 [Characterization of E(·|ξ) in terms of fg(ξ); g bounded continuousg] Let us now prove that E(Xjξ) = 0 iff for all bounded continuous g we have E(g(ξ)X) = 0: The \only if" part is direct. To prove the \if part", first note that E(Xjξ) = 0 iff E(φX) = 0 for all φ that are σ(ξ) measurable. Now, consider the set H := fφ : (Ω; F) ! (R; B(R)) ; E(φX) = 0g: We have to show that H contains the σ(ξ)-measurable functions. It is clear that H satisfies the assumptions (i) and (iii) of corollary 1. We only have to construct a π-system A such that IA 2 H for all A 2 A and σ(A) = σ(ξ). Let us take A := fξ−1([a; b]) ; for some a < bg: 10 If A = ξ−1([a; b]) 2 A we have E(IA(ξ)X) = E(I[a;b](ξ)X): n Now, take any sequence of continuous functions g ! I[a;b](ξ), where the convergence is n pointwise. Since E(g (ξ)X) = 0, by passing to the limit we get that E(IA(ξ)X) = 0 and so assumption (ii) of corollary 1 is verified. Note that the same proof yields that if G = σ(ξ1; :::; ξn) we have n E(XjG) = 0 iff for all bounded continuous g : R ! R we have E(g(ξ1; : : : ; ξn)X) = 0: The interesting fact is that the result is also valid when we condition to a countable set of r.v's. More precisely, using the above result and the technique of its proof we get the following result. Proposition 1. [Checking conditions only on a finite set of variables] Consider a 11 sequence of variables ξ1; ξ2;::: and define G := σ(fξi ; i 2 Ng). Then n E(XjG) = 0 iff for all n 2 N and g 2 Cb(R ) we have E(g(ξ1; : : : ; ξn)X) = 0: 12 Stochastic process: Basic definitions Good references for this part are [7, 14]. Let I be a non-empty index set and (Ω; F; P) a probability space.
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