Interest Rate Models:

Transcription

1 1/17 Interest Rate Models: from Parametric Statistics to Infinite Dimensional Stochastic Analysis René Carmona Bendheim Center for Finance ORFE & PACM, Princeton University URL: rcarmona/ IPAM / Financial Math January 3-5, 2001

2 Chapter 6 2/17 Infinite Dimensional Ornstein Uhlenbeck Processes

3 One Dimensional O-U Processes 3/17

4 One Dimensional O-U Processes 3/17 Solution of the SDE

5 One Dimensional O-U Processes 3/17 Solution of the SDE dξ t = aξ t dt + bdw t

6 One Dimensional O-U Processes 3/17 Solution of the SDE a R or a > 0, b > 0 dξ t = aξ t dt + bdw t

7 One Dimensional O-U Processes 3/17 Solution of the SDE a R or a > 0, b > 0 dξ t = aξ t dt + bdw t {w t } t 0 one dimensional Wiener process

8 One Dimensional O-U Processes 3/17 Solution of the SDE a R or a > 0, b > 0 dξ t = aξ t dt + bdw t {w t } t 0 one dimensional Wiener process ξ 0 (implicitly) assumed to be independent of {w t } t 0

9 One Dimensional O-U Processes 3/17 Solution of the SDE a R or a > 0, b > 0 dξ t = aξ t dt + bdw t {w t } t 0 one dimensional Wiener process ξ 0 (implicitly) assumed to be independent of {w t } t 0 Explicit solution

10 One Dimensional O-U Processes 3/17 Solution of the SDE a R or a > 0, b > 0 dξ t = aξ t dt + bdw t {w t } t 0 one dimensional Wiener process ξ 0 (implicitly) assumed to be independent of {w t } t 0 Explicit solution ξ t = e ta ξ 0 + b t 0 e (t s)a dw s, 0 t < +

11 One Dimensional O-U Processes 3/17 Solution of the SDE a R or a > 0, b > 0 dξ t = aξ t dt + bdw t {w t } t 0 one dimensional Wiener process ξ 0 (implicitly) assumed to be independent of {w t } t 0 Explicit solution ξ t = e ta ξ 0 + b t 0 e (t s)a dw s, 0 t < +

12 Mean value 4/17

13 Mean value Eξ t = e ta Eξ 0 4/17

14 Mean value Autocovariance function Eξ t = e ta Eξ 0 4/17

15 Mean value Autocovariance function Eξ t = e ta Eξ 0 cov{ξ s, ξ t } = [var{ξ 0 } + b 2a (e2(s t)a 1)]e (s+t)a 4/17

16 Mean value Autocovariance function Eξ t = e ta Eξ 0 cov{ξ s, ξ t } = [var{ξ 0 } + b 2a (e2(s t)a 1)]e (s+t)a {ξ t } is a Gaussian process whenever ξ 0 is Gaussian 4/17

17 Mean value Autocovariance function Eξ t = e ta Eξ 0 cov{ξ s, ξ t } = [var{ξ 0 } + b 2a (e2(s t)a 1)]e (s+t)a {ξ t } is a Gaussian process whenever ξ 0 is Gaussian {ξ t } is a (strong) Markov process. {ξ t } has a unique invariant measure N(0, b/2a) 4/17

18 Mean value Autocovariance function Eξ t = e ta Eξ 0 cov{ξ s, ξ t } = [var{ξ 0 } + b 2a (e2(s t)a 1)]e (s+t)a {ξ t } is a Gaussian process whenever ξ 0 is Gaussian {ξ t } is a (strong) Markov process. {ξ t } has a unique invariant measure N(0, b/2a) if ξ 0 N(0, b/2a), {ξ t } is stationary ergodic Markov (mean zero) Gaussian process. Autocovariance function: Eξ s ξ t = b 2a e a t s. 4/17

19 n-dimensional O-U Processes 5/17

20 n-dimensional O-U Processes 5/17 a > 0 A n n symmetric positive definite matrix

21 n-dimensional O-U Processes 5/17 a > 0 A n n symmetric positive definite matrix b > 0 B n n symmetric positive definite matrix

22 n-dimensional O-U Processes 5/17 a > 0 A n n symmetric positive definite matrix b > 0 B n n symmetric positive definite matrix {w t } t 0 1 dim. Wiener Proc. {W t } t 0 n dim. Wiener Proc.

23 n-dimensional O-U Processes 5/17 a > 0 A n n symmetric positive definite matrix b > 0 B n n symmetric positive definite matrix {w t } t 0 1 dim. Wiener Proc. {W t } t 0 n dim. Wiener Proc. As before, solution of the SDE

24 n-dimensional O-U Processes 5/17 a > 0 A n n symmetric positive definite matrix b > 0 B n n symmetric positive definite matrix {w t } t 0 1 dim. Wiener Proc. {W t } t 0 n dim. Wiener Proc. As before, solution of the SDE dx t = AX t dt + BdW t

25 n-dimensional O-U Processes 5/17 a > 0 A n n symmetric positive definite matrix b > 0 B n n symmetric positive definite matrix {w t } t 0 1 dim. Wiener Proc. {W t } t 0 n dim. Wiener Proc. As before, solution of the SDE dx t = AX t dt + BdW t {W t } t 0 n-dimensional Wiener process

26 n-dimensional O-U Processes 5/17 a > 0 A n n symmetric positive definite matrix b > 0 B n n symmetric positive definite matrix {w t } t 0 1 dim. Wiener Proc. {W t } t 0 n dim. Wiener Proc. As before, solution of the SDE dx t = AX t dt + BdW t {W t } t 0 n-dimensional Wiener process X 0 (implicitly) assumed to be independent of {W t } t 0

27 n-dimensional O-U Processes 5/17 a > 0 A n n symmetric positive definite matrix b > 0 B n n symmetric positive definite matrix {w t } t 0 1 dim. Wiener Proc. {W t } t 0 n dim. Wiener Proc. As before, solution of the SDE dx t = AX t dt + BdW t {W t } t 0 n-dimensional Wiener process X 0 (implicitly) assumed to be independent of {W t } t 0 Explicit solution

28 n-dimensional O-U Processes 5/17 a > 0 A n n symmetric positive definite matrix b > 0 B n n symmetric positive definite matrix {w t } t 0 1 dim. Wiener Proc. {W t } t 0 n dim. Wiener Proc. As before, solution of the SDE dx t = AX t dt + BdW t {W t } t 0 n-dimensional Wiener process X 0 (implicitly) assumed to be independent of {W t } t 0 Explicit solution X t = e ta X 0 + t 0 e (t s)a BdW s, 0 t < +

29 n-dimensional O-U Processes 5/17 a > 0 A n n symmetric positive definite matrix b > 0 B n n symmetric positive definite matrix {w t } t 0 1 dim. Wiener Proc. {W t } t 0 n dim. Wiener Proc. As before, solution of the SDE dx t = AX t dt + BdW t {W t } t 0 n-dimensional Wiener process X 0 (implicitly) assumed to be independent of {W t } t 0 Explicit solution X t = e ta X 0 + t 0 e (t s)a BdW s, 0 t < +

30 Mean value 6/17

31 Mean value EX t = e ta EX 0 6/17

32 Mean value EX t = e ta EX 0 6/17 Autocovariance function

33 Mean value EX t = e ta EX 0 6/17 Autocovariance function cov{x s, X t } = e sa var{x 0 }e ta + s t 0 e (s u)a Be (t u)a du

34 Mean value EX t = e ta EX 0 6/17 Autocovariance function cov{x s, X t } = e sa var{x 0 }e ta + s t 0 e (s u)a Be (t u)a du {X t } is a Gaussian process whenever X 0 is Gaussian

35 Mean value 6/17 EX t = e ta EX 0 Autocovariance function cov{x s, X t } = e sa var{x 0 }e ta + s t 0 e (s u)a Be (t u)a du {X t } is a Gaussian process whenever X 0 is Gaussian {X t } is a (vector valued) Markov process. Could drive the system with a r-dimensional Wiener process: The n n matrix B replaced by a d r matrix C

36 Mean value 6/17 EX t = e ta EX 0 Autocovariance function cov{x s, X t } = e sa var{x 0 }e ta + s t 0 e (s u)a Be (t u)a du {X t } is a Gaussian process whenever X 0 is Gaussian {X t } is a (vector valued) Markov process. Could drive the system with a r-dimensional Wiener process: The n n matrix B replaced by a d r matrix C role of B played by C C

37 Mean value 6/17 EX t = e ta EX 0 Autocovariance function cov{x s, X t } = e sa var{x 0 }e ta + s t 0 e (s u)a Be (t u)a du {X t } is a Gaussian process whenever X 0 is Gaussian {X t } is a (vector valued) Markov process. Could drive the system with a r-dimensional Wiener process: The n n matrix B replaced by a d r matrix C role of B played by C C

38 {X t } has a unique invariant measure, say µ = N(0, Σ) n- dimensional Gaussian measure µ(dx) = 1 Z(Σ) e <Σ 1 x,x>/2 dx, 7/17 Z(Σ) being a normalizing constant Σ = 0 e sa Be sa ds. Notice that the fact that Σ solves: ΣA + AΣ = B is the crucial property of the matrix Σ

39 If X 0 µ, {X t } t 0 is stationary mean zero (Markov) Gaussian process with autocovariance tensor: 8/17 Γ(s, t) = E µ {X s X t } = { Σe (t s)a whenever 0 s t e (s t)a Σ whenever 0 t s.

40 If X 0 µ, {X t } t 0 is stationary mean zero (Markov) Gaussian process with autocovariance tensor: 8/17 Γ(s, t) = E µ {X s X t } = { Σe (t s)a whenever 0 s t e (s t)a Σ whenever 0 t s. Cumbersome formulae when A & B do not commute

41 If X 0 µ, {X t } t 0 is stationary mean zero (Markov) Gaussian process with autocovariance tensor: 8/17 Γ(s, t) = E µ {X s X t } = { Σe (t s)a whenever 0 s t e (s t)a Σ whenever 0 t s. Cumbersome formulae when A & B do not commute

42 If A & B do commute 9/17

43 If A & B do commute Σ = B 0 e 2sA ds = B(2A) 1 9/17

44 If A & B do commute Σ = B 0 e 2sA ds = B(2A) 1 9/17 If x, y R n we have: < x, Γ(s, t)y >= E µ {< x, X s >< y, X t >} =< Σx, Σe t s A y > with the notation: =< x, e t s A y > Σ <, > D =< D, D >=< D, >

45 If A & B do commute Σ = B 0 e 2sA ds = B(2A) 1 9/17 If x, y R n we have: < x, Γ(s, t)y >= E µ {< x, X s >< y, X t >} =< Σx, Σe t s A y > with the notation: =< x, e t s A y > Σ <, > D =< D, D >=< D, >

46 The Markov process {X t } t 0 is symmetric (Fukushima) 10/17

47 The Markov process {X t } t 0 is symmetric (Fukushima) Generated by the Dirichlet form of the measure µ, i.e. to the quadratic form: Q(f, g) = < f(x), g(x) > B µ Σ (dx) R n defined on the subspace Q of L 2 (R n, µ Σ (dx)) comprising the absolutely continuous functions whose first derivatives (in the sense of distributions) are still in the space L 2 (R n, µ Σ (dx)). 10/17

48 Infinite Dimensional O-U Processes 11/17

49 Infinite Dimensional O-U Processes 11/17 Generalizations of the Three Approaches used in the Finite Dimensional Case

50 Infinite Dimensional O-U Processes 11/17 Generalizations of the Three Approaches used in the Finite Dimensional Case The state space R d of the Wiener noise is replaced by a (possibly infinite dimensional) Hilbert space, say H, or a Banach space E

51 Infinite Dimensional O-U Processes 11/17 Generalizations of the Three Approaches used in the Finite Dimensional Case The state space R d of the Wiener noise is replaced by a (possibly infinite dimensional) Hilbert space, say H, or a Banach space E are replaced by a Wiener pro- The Wiener noise terms ξ (j) t cess in H or E

52 Infinite Dimensional O-U Processes 11/17 Generalizations of the Three Approaches used in the Finite Dimensional Case The state space R d of the Wiener noise is replaced by a (possibly infinite dimensional) Hilbert space, say H, or a Banach space E are replaced by a Wiener pro- The Wiener noise terms ξ (j) t cess in H or E The d n dispersion matrix B is replaced by a map σ from H or E into the state space F of f t replacing R n The n n drift coefficient matrix A is replaced by a (possibly unbounded) operator which we will still denote by A

53 Infinite Dimensional O-U Processes 11/17 Generalizations of the Three Approaches used in the Finite Dimensional Case The state space R d of the Wiener noise is replaced by a (possibly infinite dimensional) Hilbert space, say H, or a Banach space E are replaced by a Wiener pro- The Wiener noise terms ξ (j) t cess in H or E The d n dispersion matrix B is replaced by a map σ from H or E into the state space F of f t replacing R n The n n drift coefficient matrix A is replaced by a (possibly unbounded) operator which we will still denote by A

54 The role played by BdW t is now played by a H-valued (cylindrical) Wiener process with covariance given by the operator B. The appropriate mathematical object is a linear function, say W B, from the tensor product L 2 ([0, ), dt) ˆ 2 H B into a Gaussian subspace of L 2 (Ω, F, P) where (Ω, F, P) is the complete probability space we work with. If x H B and t 0, then W B (1 [0,t) ( )x) should play the same role as < x, W t > in the finite dimensional case. 12/17

55 HJM Stochastic PDE 13/17

56 HJM Stochastic PDE 13/17 For the sake of the present discussion:

57 HJM Stochastic PDE 13/17 For the sake of the present discussion: Take α t 0

58 HJM Stochastic PDE 13/17 For the sake of the present discussion: Take α t 0 Choose σ t σ deterministic & independent of t df t dt = Af t + σ dw dt SPDE when A is a partial differential operator

59 HJM Stochastic PDE 13/17 For the sake of the present discussion: Take α t 0 Choose σ t σ deterministic & independent of t df t dt = Af t + σ dw dt SPDE when A is a partial differential operator Integral form f t = f 0 t 0 Af s ds + σw (t)

60 Weak Form: for f H < f, f t >=< f, f 0 > t 0 < A f, f s > ds+ < σ f, W (t) > which makes sense when f belongs to the domain D(A ) of the adjoint operator A and the domain D(σ ) of the adjoint operator σ. Very seldom tractable Evolution Form (variation of constant formula): f t = e ta f 0 + t 0 e (t s)s σdw (s) This requires that the exponentials of operators exist, i.e. that A generates a semigroup of operators. 14/17

61 OU Set Up 15/17

62 OU Set Up 15/17 X = {X t ; t 0} stochastic process in a Banach/Hilbert space F

63 OU Set Up 15/17 X = {X t ; t 0} stochastic process in a Banach/Hilbert space F Probability space (Ω, F, P)

64 OU Set Up 15/17 X = {X t ; t 0} stochastic process in a Banach/Hilbert space F Probability space (Ω, F, P) Filtration (F t ) t 0 (satisfying the usual assumptions)

65 OU Set Up 15/17 X = {X t ; t 0} stochastic process in a Banach/Hilbert space F Probability space (Ω, F, P) Filtration (F t ) t 0 (satisfying the usual assumptions) Drift operator A, possibly unbounded operator on F. A is the infinitesimal generator of a C 0 -semigroup {e ta ; t 0} of bounded operators on F. The domain of A, D(A) = {f F ; lim t 0 e ta f f t is (generally) is a proper subset of F exists.},

66 OU Set Up 15/17 X = {X t ; t 0} stochastic process in a Banach/Hilbert space F Probability space (Ω, F, P) Filtration (F t ) t 0 (satisfying the usual assumptions) Drift operator A, possibly unbounded operator on F. A is the infinitesimal generator of a C 0 -semigroup {e ta ; t 0} of bounded operators on F. The domain of A, D(A) = {f F ; lim t 0 e ta f f t is (generally) is a proper subset of F exists.},

67 Noise W = {W t ; t 0} is a Wiener process in a real separable Banach space E with H W the reproducing kernel Hilbert space associated with the Guassian measure µ (abstract Wiener space) 16/17 E H W H W ( Riesz identification) E

68 Noise W = {W t ; t 0} is a Wiener process in a real separable Banach space E with H W the reproducing kernel Hilbert space associated with the Guassian measure µ (abstract Wiener space) 16/17 E H W H W ( Riesz identification) E The variance/covariance operator B : E F is a bounded linear operator (thus it has a restriction to H W and this restriction is Hilbert-Schmidt when F is a Hilbert space.) Note one could take B from H W only if we start with a cylindrical Brownian motion. Formally the Ornstein Uhlenbeck process satisfies the stochas-

69 tic differential equation or in integral form dx t = AX t dt + BdW t 17/17 X t = X 0 + t 0 AX s ds + t 0 BdW s.