Polynomial Preserving Jump-Diffusions on the Unit Interval

Transcription

1 Polynomial Preserving Jump-Diffusions on the Unit Interval Martin Larsson Department of Mathematics, ETH Zürich joint work with Christa Cuchiero and Sara Svaluto-Ferro 7th General AMaMeF and Swissquote Conference Lausanne, 7 10 September 2015

2 Outline Polynomial preserving processes Polynomial preserving processes on [0, 1] 2/20

3 Polynomial preserving processes 3/20

4 Polynomial preserving processes Markov semimartingale X with state space E R d (Extended) generator G given by G f (x) = b(x) f (x) Tr ( a(x) 2 f (x) ) ( ) + f (x + ξ) f (x) ξ f (x) ν(x, dξ) Assumption: ξ 2n ν(x, dξ) K n (1 + x 2n ) for some K n R, all x E, all n N. 4/20

5 Polynomial preserving processes Markov semimartingale X with state space E R d (Extended) generator G given by G f (x) = b(x) f (x) Tr ( a(x) 2 f (x) ) ( ) + f (x + ξ) f (x) ξ f (x) ν(x, dξ) Assumption: ξ 2n ν(x, dξ) K n (1 + x 2n ) for some K n R, all x E, all n N. Definition. G is called polynomial preserving (PP) if G Pol n (E) Pol n (E) for all n N, where Pol n (E) = {polynomials on E of degree n}. In this case, X is called a polynomial preserving process. 4/20

6 Polynomial preserving processes By definition of (PP), G restricts to an operator G Poln(E) on the finite-dimensional vector space Pol n (E) 5/20

7 Polynomial preserving processes By definition of (PP), G restricts to an operator G Poln(E) on the finite-dimensional vector space Pol n (E) Let h 1,..., h N be a basis for Pol n (E), write H(x) = (h 1 (x),..., h N (x)) 5/20

8 Polynomial preserving processes By definition of (PP), G restricts to an operator G Poln(E) on the finite-dimensional vector space Pol n (E) Let h 1,..., h N be a basis for Pol n (E), write Coordinate representations: H(x) = (h 1 (x),..., h N (x)) p(x) = H(x) p G p(x) = H(x) G p p R N G R N N 5/20

9 Polynomial preserving processes By definition of (PP), G restricts to an operator G Poln(E) on the finite-dimensional vector space Pol n (E) Let h 1,..., h N be a basis for Pol n (E), write Coordinate representations: H(x) = (h 1 (x),..., h N (x)) p(x) = H(x) p G p(x) = H(x) G p p R N G R N N Key consequence: E [p(x T ) F t ] = e (T t)g p (X t ) = H(X t ) e (T t)g p (formally) 5/20

10 Polynomial preserving processes By definition of (PP), G restricts to an operator G Poln(E) on the finite-dimensional vector space Pol n (E) Let h 1,..., h N be a basis for Pol n (E), write Coordinate representations: H(x) = (h 1 (x),..., h N (x)) p(x) = H(x) p G p(x) = H(x) G p p R N G R N N Key consequence: E [p(x T ) F t ] = e (T t)g p (X t ) = H(X t ) e (T t)g p (formally) This only involves a matrix exponential as opposed to solving a PDE which leads to tractable pricing models 5/20

11 Polynomial preserving processes Lemma (Cuchiero, Keller-Ressel, Teichmann, 2012): G is (PP) if and only if b i (x) Pol 1 (E) a ij (x) + ξ i ξ j ν(x, dξ) Pol 2 (E) R d ξ k1 1 ξk d d ν(x, dξ) Pol k 1+ +k d (E) if k k d 3 R d Corollary. If X is affine, then it is (PP). 6/20

12 Polynomial preserving processes on [0, 1] 7/20

13 Polynomial preserving processes on [0, 1] Why do we care about [0,1]? Stochastically evolving probabilities (e.g. default probabilities) Stochastically evolving correlations Stochastic recovery rates Electricity modeling Stepping stone toward simplex E = {x R d + : x x d = 1}... and other compact state spaces 8/20

14 Polynomial preserving processes on [0, 1] Let E = [0, 1] and consider an operator of the form G f = 1 2 af + bf + ( f ( + ξ) f ξf ) ν(, dξ) 9/20

15 Polynomial preserving processes on [0, 1] Let E = [0, 1] and consider an operator of the form G f = 1 2 af + bf + ( f ( + ξ) f ξf ) ν(, dξ) Problem: Describe those a, b, ν such that G is the (extended) generator of a polynomial preserving process on [0, 1]. 9/20

16 Polynomial preserving processes on [0, 1] Let E = [0, 1] and consider an operator of the form G f = 1 2 af + bf + ( f ( + ξ) f ξf ) ν(, dξ) Problem: Describe those a, b, ν such that G is the (extended) generator of a polynomial preserving process on [0, 1]. Remarks: Without jumps the solution is the well-known Jacobi process: dx t = (b + BX t )dt + σ X t (1 X t )dw t with b 0, b + B 0. With jumps, much richer behavior is possible. We restrict ν to have simple polynomial jump sizes. 9/20

17 Polynomial jump sizes Definition. We say that ν(x, dξ) has simple polynomial jump sizes if ν(a, dξ) = λ(x) 1 A (γ(x, y))µ(dy) for some measurable λ 0, some measure µ on R N+1 for some N, and γ(x, y) = y 0 + y 1 x + + y N x N Meaning: Jump size is polynomial in the current state, X t = γ(x t, Y ), Y µ, and arrive with intensity λ(x t ) (if µ is a probability measure) Remark: Affine processes do not admit state-dependent jump sizes 10/20

18 Example 1 Jacobi process with constant intensity jumps: dx t = (b + BX t )dt + σ X t (1 X t )dw t N t ( ) + Y i,1 ( X t ) + Y i,2 (1 X t ) i=0 where N t is a Poisson process with constant intensity λ, and {Y i,1, Y i,2 : i = 1, 2,...} iid µ, supp µ [0, 1] [0, 1] with suitable boundary conditions. 11/20

19 Example 1 Jacobi process with constant intensity jumps: dx t = (b + BX t )dt + σ X t (1 X t )dw t N t ( ) + Y i,1 ( X t ) + Y i,2 (1 X t ) i=0 where N t is a Poisson process with constant intensity λ, and {Y i,1, Y i,2 : i = 1, 2,...} iid µ, supp µ [0, 1] [0, 1] with suitable boundary conditions. Coefficients of G : a(x) = σ 2 x(1 x) γ(x, y) = y 1 ( x) + y 2 (1 x) b(x) = b + Bx λ(x) λ 11/20

20 Example 1 Jacobi process with constant intensity jumps: dx t = (b + BX t )dt + σ X t (1 X t )dw t N t ( ) + Y i,1 ( X t ) + Y i,2 (1 X t ) i=0 a(x) 1 λ(x) /20

21 Example 2 Jacobi process with unbounded intensity jumps: a(x) = σ 2 x(1 x) b(x) = b + Bx γ(x, y) = yx λ(x) = c 0 + c 1 x 1 x 0 x y 2 µ(dy) <, supp µ [0, 1], and suitable boundary conditions. 12/20

22 Example 2 Jacobi process with unbounded intensity jumps: a(x) = σ 2 x(1 x) b(x) = b + Bx γ(x, y) = yx λ(x) = c 0 + c 1 x 1 x 0 x y 2 µ(dy) <, supp µ [0, 1], and suitable boundary conditions. a(x) λ(x) /20

23 Example 3 Dunkl-like process: Fix x (0, 1). For x x, a(x) = σ 2 x(1 x) γ(x, y) = y(x x ) b(x) = b + Bx λ(x) = c 0 + c 1 x + c 2 x 2 (x x ) 2 y 2 µ(dy) <, supp µ [0, 1 x 1 1 x ] and suitable boundary conditions. 13/20

24 Example 3 Dunkl-like process: Fix x (0, 1). For x x, a(x) = σ 2 x(1 x) γ(x, y) = y(x x ) b(x) = b + Bx λ(x) = c 0 + c 1 x + c 2 x 2 (x x ) 2 y 2 µ(dy) <, supp µ [0, 1 x 1 1 x ] and suitable boundary conditions. λ(x) a(x) /20

25 Characterization theorem Theorem. Assume ν(x, dξ) has simple polynomial jump sizes. Then G is the (extended) generator of a unique polynomial preserving process on [0, 1] if and only if Affine drift: b(x) = b + Bx for some b, B R Affine jump sizes: Can take µ(dy) on [0, 1] [0, 1] such that y 2 µ(dy) <, and γ(x, y) = y 1 ( x) + y 2 (1 x) Boundary conditions: a(0) = a(1) = 0 b(0) λ(0) y2 µ(dy) 0 b(1) + λ(1) y1 µ(dy) 0 a, λ and supp µ are of one of four types (up to reflection in 1 2 ): 14/20

26 Characterization theorem Case 1 Case 2 a(x) a(x) 0 1 λ(x) 0 1 λ(x) Case 3 a(x) Case 4 a(x) λ(x) λ(x) /20

27 Case 4 Intensity and diffusion are of the form λ(x) = c 0 + c 1 x + c 2 x 2 x 2 + α 2 and a(x) = x(1 x) a 0 + a 1 x + a 2 x 2 x 2 + α 2 for some α C \ R such that ( ) nµ(dy) y 1 ( α) + y 2 (1 α) = 0 for all n 3 where supp µ [0, 1] [0, 1]. We do not know whether such α and µ exist! If none exist, then a(x) is necessarily essentially quadratic. 16/20

28 Extensions Observation: The set { G : G is the generator of a (PP) process on [0, 1] } is a convex cone. 17/20

29 Extensions Observation: The set { G : G is the generator of a (PP) process on [0, 1] } is a convex cone. Thus: Can combine Cases 1 3 to get (PP) generators with (for a.e. x) a(x) = σ 2 x(1 x) b(x) = b+bx ν(x, A) = 1 A (γ(x, y))k(x, dy) where 17/20

30 Extensions Observation: The set { G : G is the generator of a (PP) process on [0, 1] } is a convex cone. Thus: Can combine Cases 1 3 to get (PP) generators with (for a.e. x) a(x) = σ 2 x(1 x) b(x) = b+bx ν(x, A) = 1 A (γ(x, y))k(x, dy) where K(x, dy) = m(dy) + µ(0) 0 + K k=2 µ (k) 0 (dy) + xµ(0) 1 (dy) x + µ(1) 0 (dy) + xµ(k) 1 (dy) + x 2 µ (k) 2 (dy) (x x k ) 2 (dy) + xµ(1) 1 (dy) 1 x with x k (0, 1) and signed measures µ (k) i concentrated on {y [0, 1] [0, 1] : (1 x k )y 2 = x k y 1 } 17/20

31 Extensions This gives a rich class of (PP) processes on [0, 1] but is still not exhaustive or even dense: 18/20

32 Extensions This gives a rich class of (PP) processes on [0, 1] but is still not exhaustive or even dense: Example. A (PP) process is obtained by a(x) 0, b(x) = 1 2x, and ν(x, A) = 1 A (γ(x, y))k(x, dy) where K(x, dy) = 1 x(1 + x) δ (1,0)(dy) + 1 (1 x)(1 + x) δ (0,1/2)(dy) However, neither nor individually gives rise to a (PP) generator. 18/20

33 Conclusion (PP) processes can be used to build flexible and tractable models Beyond the affine case, little is known about (PP) jump-diffusions Characterization of (PP) processes on the unit interval with simple polynomial jump sizes Outlook: What about Case 4? Composite polynomial jump sizes Boundary attainment? (PP) processes on the unit simplex 19/20

34 Thank you! 20/20