By Paul Roman

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**Example text**

S>t Recall that a process (Mt )t∈R+ in L1 (Ω) is called an Ft -martingale if IE[Mt |Fs ] = Ms , 0 ≤ s ≤ t. For example, if (Xt )t∈[0,T ] is a (non homogeneous) Markov process with semigroup (Ps,t )0≤s≤t≤T satisfying Ps,t f (Xs ) = IE[f (Xt ) | Xs ] = IE[f (Xt ) | Fs ], 0 ≤ s ≤ t ≤ T, on Cb2 (Rn ) functions, with Ps,t ◦ Pt,u = Ps,u , 0 ≤ s ≤ t ≤ u ≤ T, then (Pt,T f (Xt ))t∈[0,T ] is an Ft -martingale: IE[Pt,T f (Xt ) | Fs ] = IE[IE[f (XT ) | Ft ] | Fs ] = IE[f (XT ) | Fs ] 0 ≤ s ≤ t ≤ T. = Ps,T f (Xs ), 32 N.

Xsn ) | Xt ], 0 ≤ s1 < · · · < sn < t < t1 < · · · < tn . In discrete time, a sequence (Xn )n∈N of random variables is said to be a Markov chain if for all n ∈ N, the σ-algebras Fn = σ({Xk : k ≤ n}) and Fn+ = σ({Xk : k ≥ n}) are independent conditionally to Xn . In particular, for every Fn+ -measurable bounded random variable F we have IE[F | Fn ] = IE[F | Xn ], n ∈ N. 2 Transition Kernels and Semigroups A transition kernel is a mapping P (x, dy) such that i) for every x ∈ E, A → P (x, A) is a probability measure, and ii) for every A ∈ B(E), the mapping x → P (x, A) is a measurable function.

3. e. IE[Mt2 − t|Fs ] = Ms2 − s, 0 ≤ s < t. Proof. This follows from the equalities IE[(Mt − Ms )2 |Fs ] − (t − s) = IE[Mt2 − Ms2 − 2(Mt − Ms )Ms |Fs ] − (t − s) = IE[Mt2 − Ms2 |Fs ] − 2Ms IE[Mt − Ms |Fs ] − (t − s) = IE[Mt2 |Fs ] − t − (IE[Ms2 |Fs ] − s). Throughout the remainder of this chapter, (Mt )t∈R+ will be a normal martingale. We now turn to the Brownian motion and the compensated Poisson process as the fundamental examples of normal martingales. e. centered and with unit variance) Gaussian random variables under γN , constructed as the canonical projections from (RN , BRN , γN ) into R.