1. Stochastic equations for Markov processes • Filtrations and the Markov property • Ito equations for diffusion processes • Poisson random measures • Ito equations for Markov processes with jumps •First •Prev •Next •Go To •Go Back •Full Screen •Close •Quit 1 Filtrations and the Markov property (Ω; F;P ) a probability space Available information is modeled by a sub-σ-algebra of F Ft information available at time t fFtg is a filtration. t < s implies Ft ⊂ Fs A stochastic process X is adapted to fFtg if X(t) is Ft-measurable for each t ≥ 0. An E-valued stochastic process X adapted to fFtg is fFtg-Markov if E[f(X(t + r))jFt] = E[f(X(t + r))jX(t)]; t; r ≥ 0; f 2 B(E) An R-valued stochastic process M adapted to fFtg is an fFtg-martingale if E[M(t + r)jFt] = M(t); t; r ≥ 0 τ is an fFtg-stopping time if for each t ≥ 0, fτ ≤ tg 2 Ft. For a stopping time τ, Fτ = fA 2 F : fτ ≤ tg \ A 2 Ft; t ≥ 0g •First •Prev •Next •Go To •Go Back •Full Screen •Close •Quit 2 Ito integrals W standard Browian motion (W has independent increments with W (t + s) − W (t) normally distributed with mean zero and variance s.) W is compatible with a filtration fFtg if W is fFtg-adapted and W (t + s) − W (t) is independent of Ft for all s; t ≥ 0. If X is R-valued, cadlag and fFtg-adapted, then Z t X X(s−)dW (s) = lim X(ti)(W (t ^ ti+1) − W (t ^ ti)) 0 max(ti+1−ti)!0 exists, and the integral satisfies Z t Z t E[( X(s−)dW (s))2] = E[ X(s)2ds] 0 0 if the right side is finite. The integral extends to all measurable and adapted processes satisfying Z t X(s)2ds < 1 0 m T W is an R -valued standard Brownian motion if W = (W1;:::;Wm) for W1;:::;Wm independent one-dimensional standard Brownian motions. •First •Prev •Next •Go To •Go Back •Full Screen •Close •Quit 3 Ito equations for diffusion processes σ : Rd ! Md×m and b : Rd ! Rd Consider Z t Z t X(t) = X(0) + σ(X(s))dW (s) + b(X(s))ds 0 0 where X(0) is independent of the standard Brownian motion W . Why is X Markov? Suppose W is compatible with fFtg, X is fFtg-adapted, and X is unique (among fFtg-adapted solutions). Then Z t+r Z t+r X(t + r) = X(t) + σ(X(s))dW (s) + b(X(s))ds t t and X(t + r) = H(t; r; X(t);Wt) where Wt(s) = W (t + s) − W (t). Since Wt is independent of F t, Z E[f(X(t + r))jFt] = f(H(t; r; X(t); w))µW (dw): •First •Prev •Next •Go To •Go Back •Full Screen •Close •Quit 4 Gaussian white noise. (U; dU ) a complete, separable metric space; B(U), the Borel sets µ a (Borel) measure on U A(U) = fA 2 B(U): µ(A) < 1g W (A; t) Mean zero, Gaussian process indexed by A(U) × [0; 1) E[W (A; t)W (B; s)] = t ^ sµ(A \ B), W ('; t) = R '(u)W (du; t) P '(u) = aiIAi (u) P W ('; t) = i aiW (Ai; t) R E[W ('1; t)W ('2; s)] = t ^ s U '1(u)'2(u)µ(du) Define W ('; t) for all ' 2 L2(µ). •First •Prev •Next •Go To •Go Back •Full Screen •Close •Quit 5 Definition of integral P X(t) = i ξi(t)'i process in L2(µ) R P R t IW (X; t) = U×[0;t] X(s; u)W (du × ds) = i 0 ξi(s)dW ('i; s) " Z t Z # 2 X E[IW (X; t) ] = E ξi(s)ξj(s)ds 'i'jdµ i;j 0 U Z t Z = E X(s; u)2µ(du)ds 0 U The integral extends to adapted processes satisfying Z t Z X(s; u)2µ(du)ds < 1 a:s: 0 U so that Z t Z 2 2 (IW (X; t)) − X(s; u) µ(du)ds 0 U is a local martingale. •First •Prev •Next •Go To •Go Back •Full Screen •Close •Quit 6 Poisson random measures ν a σ-finite measure on U,(U; dU ) a complete, separable metric space. ξ a Poisson random measure on U × [0; 1) with mean measure ν × m. For A 2 B(U)×B([0; 1)), ξ(A) has a Poisson distribution with expectation ν ×m(A) and ξ(A) and ξ(B) are independent if A \ B = ;. ξ(A; t) ≡ ξ(A × [0; t]) is a Poisson process with parameter ν(A). ξe(A; t) ≡ ξ(A × [0; t] − ν(A)t is a martingale. ξ is fFtg compatible, if for each A 2 B(U), ξ(A; ·) is fFtg adapted and for all t; s ≥ 0, ξ(A × (t; t + s]) is independent of Ft. •First •Prev •Next •Go To •Go Back •Full Screen •Close •Quit 7 Stochastic integrals for Poisson random measures X cadlag, L1(ν)-valued, fFtg-adapted We define Z Iξ(X; t) = X(u; s−)ξ(du × ds) U×[0;t] in such a way that Z E [jIξ(X; t)j] ≤ E jX(u; s−)jξ(du × ds) U×[0;t] Z = E[jX(u; s)j]ν(du)ds U×[0;t] If the right side is finite, then Z Z E X(u; s−)ξ(du × ds) = E[X(u; s)]ν(du)ds U×[0;t] U×[0;t] •First •Prev •Next •Go To •Go Back •Full Screen •Close •Quit 8 Predictable integrands The integral extends to predictable integrands satisfying Z jX(u; s)j ^ 1ν(du)ds < 1 a:s: U×[0;t] so that Z E[Iξ(X; t ^ τ)] = E X(u; s)ξ(du × ds) U×[0;t^τ] for any stopping time satisfying Z E jX(u; s)jν(du)ds < 1 (1) U×[0;t^τ] If (1) holds for all t, then Z Z X(u; s)ξ(du × ds) − X(u; s)ν(du)ds U×[0;t^τ] U×[0;t^τ] is a martingale. X is predictable if it is the pointwise limit of adapted, left-continuous processes. •First •Prev •Next •Go To •Go Back •Full Screen •Close •Quit 9 Stochastic integrals for centered Poisson random measures X cadlag, L2(ν)-valued, fFtg-adapted We define Z I (X; t) = X(u; s−)ξ(du × ds) ξe e U×[0;t] h i in such a way that E I (X; t)2 = R E[X(u; s)2]ν(du)ds if the right side is finite. ξe U×[0;t] Then I (X; ·) is a square-integrable martingale. ξe The integral extends to predictable integrands satisfying Z jX(u; s)j2 ^ jX(u; s)jν(du)ds < 1 a:s: U×[0;t] so that I (X; t ^ τ) is a martingale for any stopping time satisfying ξe Z E jX(u; s)j2 ^ jX(u; s)jν(du)ds < 1 U×[0;t^τ] •First •Prev •Next •Go To •Go Back •Full Screen •Close •Quit 10 It^oequations for Markov processes with jumps W Gaussian white noise on U0 × [0; 1) with E[W (A; t)W (B; t)] = µ(A \ B)t ξi Poisson random measure on Ui × [0; 1) with mean measure νi. Let ξe1(A; t) = ξ1(A; t) − ν1(A)t Z 2 2 σb (x) = σ (x; u)µ(du) < 1 Z Z 2 α1(x; u) ^ jα1(x; u)jν1(du) < 1; jα2(x; u)j ^ 1ν2(du) < 1 and Z Z t X(t) = X(0) + σ(X(s−); u)W (du × ds) + β(X(s−))ds U0×[0;t] 0 Z + α1(X(s−); u)ξe1(du × ds) U1×[0;t] Z + α2(X(s−); u)ξ2(du × ds) : U2×[0;t] •First •Prev •Next •Go To •Go Back •Full Screen •Close •Quit 11 A martingale inequality Discrete time version: Burkholder (1973). Continuous time version: Lenglart, Lepingle, and Pratelli (1980). Easy proof: Ichikawa (1986). Lemma 1 For 0 < p ≤ 2 there exists a constant Cp such that for any locally square integrable martingale M with Meyer process hMi and any stopping time τ p p=2 E[sup jM(s)j ] ≤ CpE[hMiτ ] s≤τ hMi is the (essentially unique) predictable process such that M 2 − hMi is a local martingale. (A left-continuous, adapted process is predictable.) •First •Prev •Next •Go To •Go Back •Full Screen •Close •Quit 12 Graham's uniqueness theorem Lipschitz condition sZ jσ(x; u) − σ(y; u)j2µ(du) + jβ(x) − β(y)j sZ Z 2 + jα1(x; u) − α1(y; u)j ν1(du) + jα2(x; u) − α2(y; u)jν2(du) ≤ Mjx − yj •First •Prev •Next •Go To •Go Back •Full Screen •Close •Quit 13 Estimate X and Xe solution of SDE. E[sup jX(s) − Xe(s)j] s≤t Z t Z 2 1 ≤ E[jX(0) − Xe(0)j] + C1E[( jσ(X(s−); u) − σ(Xe(s−); u)j µ(du)ds) 2 ] 0 U0 Z t Z 2 1 +C1E[( jα1(X(s−); u) − α1(Xe(s−); u)j ν1(du)ds) 2 ] 0 U1 Z t Z +E[ jα2(X(s−); u) − α2(Xe(s−); u)jν2(du)ds] 0 U2 Z t +E[ jβ(X(s−)) − β(Xe(s−))jds] 0 p ≤ E[jX(0) − Xe(0)j] + D( t + t)E[sup jX(s) − Xe(s)j] s≤t •First •Prev •Next •Go To •Go Back •Full Screen •Close •Quit 14 Boundary conditions X has values in D and satisfies Z t Z t Z t X(t) = X(0) + σ(X(s))dW (s) + b(X(s))ds + η(X(s))dλ(s) 0 0 0 where λ is nondecreasing and increases only when X(t) 2 @D.