Affine processes on positive semidefinite matrices

Transcription

1 Affine processes on positive semiefinite matrices Josef Teichmann (joint work with C. Cuchiero, D. Filipović an E. Mayerhofer) ETH Zürich Zürich, June 2010 C. Cuchiero, D. Filipović, E. Mayerhofer, J. Teichmann Affine processes on positive semiefinite matrices

2 1 Introuction Overview Motivation: Multivariate stochastic covariance moels Literature 2 Definition of affine processes on S + Affine processes on S + are Feller an regular 3 Main theorem Amissible parameters Ieas of the proof 4 1 / 69

3 Overview Motivation: Multivariate stochastic covariance moels Literature Introuction Affine processes on S +, the cone of positive semiefinite matrices, are stochastically continuous time-homogeneous Markov processes with state space S +, whose Laplace transform have exponential-affine epenence on the initial state, E x [e Tr(uXt)] = e φ(t,u) Tr(ψ(t,u)x). 2 / 69

4 Overview Motivation: Multivariate stochastic covariance moels Literature Introuction Affine processes on S +, the cone of positive semiefinite matrices, are stochastically continuous time-homogeneous Markov processes with state space S +, whose Laplace transform have exponential-affine epenence on the initial state, E x [e Tr(uXt)] = e φ(t,u) Tr(ψ(t,u)x). Aim: Full characterization of this class of processes 3 / 69

5 Overview Motivation: Multivariate stochastic covariance moels Literature Introuction Affine processes on S +, the cone of positive semiefinite matrices, are stochastically continuous time-homogeneous Markov processes with state space S +, whose Laplace transform have exponential-affine epenence on the initial state, E x [e Tr(uXt)] = e φ(t,u) Tr(ψ(t,u)x). Aim: Full characterization of this class of processes Necessary conitions on the parameters of the infinitesimal generator implie by the efinition of an affine process. 4 / 69

6 Overview Motivation: Multivariate stochastic covariance moels Literature Introuction Affine processes on S +, the cone of positive semiefinite matrices, are stochastically continuous time-homogeneous Markov processes with state space S +, whose Laplace transform have exponential-affine epenence on the initial state, E x [e Tr(uXt)] = e φ(t,u) Tr(ψ(t,u)x). Aim: Full characterization of this class of processes Necessary conitions on the parameters of the infinitesimal generator implie by the efinition of an affine process. Sufficient conitions for the existence of affine processes on S +. 5 / 69

7 Overview Motivation: Multivariate stochastic covariance moels Literature Introuction Affine processes on S +, the cone of positive semiefinite matrices, are stochastically continuous time-homogeneous Markov processes with state space S +, whose Laplace transform have exponential-affine epenence on the initial state, E x [e Tr(uXt)] = e φ(t,u) Tr(ψ(t,u)x). Aim: Full characterization of this class of processes Necessary conitions on the parameters of the infinitesimal generator implie by the efinition of an affine process. Sufficient conitions for the existence of affine processes on S +. Probabilistic properties like infinite ivisibility, excursion theory, etc. 6 / 69

8 Overview Motivation: Multivariate stochastic covariance moels Literature Motivation: Affine stochastic covariance moels One-imensional affine stochastic (co)variance moels (Heston [15], Barnorff-Nielsen Shepar moel [1], etc.). 7 / 69

9 Overview Motivation: Multivariate stochastic covariance moels Literature Motivation: Affine stochastic covariance moels One-imensional affine stochastic (co)variance moels (Heston [15], Barnorff-Nielsen Shepar moel [1], etc.). Risk neutral ynamics for the log-price process Y t an the R + -value variance process X t : X t = (b + βx t t + σ X t W t + J t, X 0 = x, ( Y t = r X ) t t + X t B t, Y 0 = y. 2 B, W : correlate Brownian motions, J: pure jump process, r: constant interest rate. 8 / 69

10 Overview Motivation: Multivariate stochastic covariance moels Literature Motivation: Affine stochastic covariance moels One-imensional affine stochastic (co)variance moels (Heston [15], Barnorff-Nielsen Shepar moel [1], etc.). Risk neutral ynamics for the log-price process Y t an the R + -value variance process X t : X t = (b + βx t t + σ X t W t + J t, X 0 = x, ( Y t = r X ) t t + X t B t, Y 0 = y. 2 B, W : correlate Brownian motions, J: pure jump process, r: constant interest rate. Efficient valuation of European options since the moment generating function is explicitly known (up to the solution of an ODE ) an is of the following form E x,y [ e zx t+vy t ] = e Φ(t,z,v)+Ψ(t,z,v)x+vy. 9 / 69

11 Overview Motivation: Multivariate stochastic covariance moels Literature Multivariate affine stochastic covariance moels Extension to multivariate stochastic covariance moels with the aim to / 69

12 Overview Motivation: Multivariate stochastic covariance moels Literature Multivariate affine stochastic covariance moels Extension to multivariate stochastic covariance moels with the aim to......capture the epenence structure between ifferent assets, 11 / 69

13 Overview Motivation: Multivariate stochastic covariance moels Literature Multivariate affine stochastic covariance moels Extension to multivariate stochastic covariance moels with the aim to......capture the epenence structure between ifferent assets,...obtain a consistent pricing framework for multi-asset options such as basket options, 12 / 69

14 Overview Motivation: Multivariate stochastic covariance moels Literature Multivariate affine stochastic covariance moels Extension to multivariate stochastic covariance moels with the aim to......capture the epenence structure between ifferent assets,...obtain a consistent pricing framework for multi-asset options such as basket options,...use them for moel-base ecision-making in the area of portfolio optimization an heging of correlation risk. 13 / 69

15 Overview Motivation: Multivariate stochastic covariance moels Literature Multivariate affine stochastic covariance moels Multivariate stochastic covariance moels consist of a -imensional logarithmic price process with risk-neutral ynamics ( Y t = r1 1 ) 2 X t iag t + X t B t, Y 0 = y, an stochastic covariation process. 0 X = Y, Y. B: -imensional Brownian motion. r: constant interest rate 1: the vector whose entries are all equal to one. X iag : the vector containing the iagonal entries of X. 14 / 69

16 Overview Motivation: Multivariate stochastic covariance moels Literature Multivariate affine stochastic covariance moels Multivariate stochastic covariance moels consist of a -imensional logarithmic price process with risk-neutral ynamics ( Y t = r1 1 ) 2 X t iag t + X t B t, Y 0 = y, an stochastic covariation process. 0 X = Y, Y. B: -imensional Brownian motion. r: constant interest rate 1: the vector whose entries are all equal to one. X iag : the vector containing the iagonal entries of X. In orer to qualify for a covariation process, X must be specifie as a process in S +. An affine ynamics for X is interesting out of several reasons. 15 / 69

17 Overview Motivation: Multivariate stochastic covariance moels Literature Multivariate affine stochastic covariance moels The following affine ynamics for X have been propose in the literature: X t = (b + HX t + X t H )t + X t W t Σ + Σ Wt Xt + J t, X 0 = x S +, b: suitably chosen matrix in S +, H, Σ: invertible matrices, W a stanar -matrix of Brownian motions possibly correlate with B, J a pure jump process whose compensator is an affine function of X. 16 / 69

18 Overview Motivation: Multivariate stochastic covariance moels Literature Multivariate affine stochastic covariance moels Analytic tractability: Uner some (mil) technical conitions, the following affine transform formula hols: [ ] E x,y e Tr(zXt)+v Y t = e Φ(t,z,v)+Tr(Ψ(t,z,v)x)+v y for appropriate arguments z S is an v C. The functions Φ an Ψ solve a system of generalize Riccati ODEs. Option pricing via Fourier methos 17 / 69

19 Overview Motivation: Multivariate stochastic covariance moels Literature Multivariate affine stochastic covariance moels Analytic tractability: Uner some (mil) technical conitions, the following affine transform formula hols: [ ] E x,y e Tr(zXt)+v Y t = e Φ(t,z,v)+Tr(Ψ(t,z,v)x)+v y for appropriate arguments z S is an v C. The functions Φ an Ψ solve a system of generalize Riccati ODEs. Option pricing via Fourier methos v y in the moment generating function correspons to a homogeneity assumption an implies that the covariation process X is a Markov process in its own filtration, which motivates the analysis of affine processes on S / 69

20 Overview Motivation: Multivariate stochastic covariance moels Literature Relate Literature Affine processes on R m + R n : D. Duffie, D. Filipović an W. Schachermayer [11]: Characterization of affine processes on R m + R n. D. Filipović an E. Mayerhofer [12]: Characterization of affine Diffusion processes. M. Keller-Ressel, W. Schachermayer an J. Teichmann [16]: Regularity of affine processes.... many other authors. 19 / 69

21 Overview Motivation: Multivariate stochastic covariance moels Literature Relate Literature Affine processes on R m + R n : D. Duffie, D. Filipović an W. Schachermayer [11]: Characterization of affine processes on R m + R n. D. Filipović an E. Mayerhofer [12]: Characterization of affine Diffusion processes. M. Keller-Ressel, W. Schachermayer an J. Teichmann [16]: Regularity of affine processes.... many other authors. Affine processes on S + - Theory of Wishart processes: M. Bru [3]: Existence an uniqueness of Wishart processes of type X t = (δi )t + X t W t + Wt Xt, X 0 S +, for δ 1. C. Donati-Martin, Y. Doumerc, H. Matsumoto an M. Yor [10]: Properties of Wishart processes. 20 / 69

22 Overview Motivation: Multivariate stochastic covariance moels Literature Relate Literature Affine processes on S + Applications from mathematical Finance: O. Barnorff-Nielsen an R. Stelzer [2]: Matrix-value Lévy riven Ornstein-Uhlenbeck processes of finite variation. B. Buraschi et al. [5, 4]: Correlation risk an portfolio optimization. J. Da Fonseca et al. [6, 7, 8, 9]: Multivariate stochastic covariance an option pricing. C. Gourieroux an R. Sufana [13, 14]: Wishart quaratic term structure moels. M. Leippol an F. Trojani [17]: S + -value affine jump iffusions an financial applications (multivariate option pricing, interest rate moels, etc.) 21 / 69

23 Definition of affine processes on S + Affine processes on S + are Feller an regular Setting an notation S : symmetric -matrices equippe with scalar prouct x, y = Tr(xy). 22 / 69

24 Definition of affine processes on S + Affine processes on S + are Feller an regular Setting an notation S : symmetric -matrices equippe with scalar prouct x, y = Tr(xy). S + : cone of symmetric -positive semiefinite matrices. 23 / 69

25 Definition of affine processes on S + Affine processes on S + are Feller an regular Setting an notation S : symmetric -matrices equippe with scalar prouct x, y = Tr(xy). S + : cone of symmetric -positive semiefinite matrices. S ++ : interior of S + in S, the cone of strictly positive efinite matrices. 24 / 69

26 Definition of affine processes on S + Affine processes on S + are Feller an regular Setting an notation S : symmetric -matrices equippe with scalar prouct x, y = Tr(xy). S + : cone of symmetric -positive semiefinite matrices. S ++ : interior of S + in S, the cone of strictly positive efinite matrices. Partial orer inuce by the cones: x y y x S + x y y x S ++. an 25 / 69

27 Definition of affine processes on S + Affine processes on S + are Feller an regular Setting an notation S : symmetric -matrices equippe with scalar prouct x, y = Tr(xy). S + : cone of symmetric -positive semiefinite matrices. S ++ : interior of S + in S, the cone of strictly positive efinite matrices. Partial orer inuce by the cones: x y y x S + an x y y x S ++. X time-homogeneous Markov process with state space S / 69

28 Definition of affine processes on S + Affine processes on S + are Feller an regular Setting an notation S : symmetric -matrices equippe with scalar prouct x, y = Tr(xy). S + : cone of symmetric -positive semiefinite matrices. S ++ : interior of S + in S, the cone of strictly positive efinite matrices. Partial orer inuce by the cones: x y y x S + an x y y x S ++. X time-homogeneous Markov process with state space S +. (P t ) t 0 : associate semigroup acting on boune measurable functions f : S + R, P t f (x) := E x [f (X t )] = f (ξ)p t (x, ξ), x S +. S + 27 / 69

29 Definition of affine processes on S + Affine processes on S + are Feller an regular Setting an notation S : symmetric -matrices equippe with scalar prouct x, y = Tr(xy). S + : cone of symmetric -positive semiefinite matrices. S ++ : interior of S + in S, the cone of strictly positive efinite matrices. Partial orer inuce by the cones: x y y x S + an x y y x S ++. X time-homogeneous Markov process with state space S +. (P t ) t 0 : associate semigroup acting on boune measurable functions f : S + R, P t f (x) := E x [f (X t )] = f (ξ)p t (x, ξ), x S +. S + X might be not conservative. Stanar extension of p t to S + with cemetery. { } 28 / 69

30 Definition of affine processes on S + Affine processes on S + are Feller an regular Definition Definition The Markov process X is calle affine if 1 it is stochastically continuous, that is, lim s t p s (x, ) = p t (x, ) weakly on S + x S +, an for every t an 2 its Laplace transform has exponential-affine epenence on the initial state: P t e u,x = e u,ξ p t (x, ξ) = e φ(t,u) ψ(t,u),x, S + for all t an u, x S +, for some functions φ : R + S + R + an ψ : R + S + S / 69

31 Definition of affine processes on S + Affine processes on S + are Feller an regular Regularity an Feller property Definition The affine process X is calle regular if the erivatives φ(t, u) ψ(t, u) F (u) =, R(u) = (1) t t t=0+ t=0+ exist an are continuous at u = / 69

32 Definition of affine processes on S + Affine processes on S + are Feller an regular Regularity an Feller property Definition The affine process X is calle regular if the erivatives φ(t, u) ψ(t, u) F (u) =, R(u) = (1) t t t=0+ t=0+ exist an are continuous at u = 0. Theorem Let X be an affine process with state space S +. Then, we have: 1 X is regular. 2 X is a Feller process. 31 / 69

33 Definition of affine processes on S + Affine processes on S + are Feller an regular Remarks on the proof The proof relies on several properties of the functions φ an ψ: Semi-flow property: For all t, s R + φ(t + s, u) = φ(t, u) + φ(s, ψ(t, u)), (2) ψ(t + s, u) = ψ(s, ψ(t, u)). (3) 32 / 69

34 Definition of affine processes on S + Affine processes on S + are Feller an regular Remarks on the proof The proof relies on several properties of the functions φ an ψ: Semi-flow property: For all t, s R + φ(t + s, u) = φ(t, u) + φ(s, ψ(t, u)), (2) ψ(t + s, u) = ψ(s, ψ(t, u)). (3) Orer relations: For all u, v S + with v u an for all t 0, φ(t, v) φ(t, u) an ψ(t, v) ψ(t, u). 33 / 69

35 Definition of affine processes on S + Affine processes on S + are Feller an regular Remarks on the proof The proof relies on several properties of the functions φ an ψ: Semi-flow property: For all t, s R + φ(t + s, u) = φ(t, u) + φ(s, ψ(t, u)), (2) ψ(t + s, u) = ψ(s, ψ(t, u)). (3) Orer relations: For all u, v S + with v u an for all t 0, φ(t, v) φ(t, u) an ψ(t, v) ψ(t, u). Analyticity: u φ(t, u) an u ψ(t, u) are analytic on S / 69

36 Definition of affine processes on S + Affine processes on S + are Feller an regular Remarks on the proof The proof relies on several properties of the functions φ an ψ: Semi-flow property: For all t, s R + φ(t + s, u) = φ(t, u) + φ(s, ψ(t, u)), (2) ψ(t + s, u) = ψ(s, ψ(t, u)). (3) Orer relations: For all u, v S + with v u an for all t 0, φ(t, v) φ(t, u) an ψ(t, v) ψ(t, u). Analyticity: u φ(t, u) an u ψ(t, u) are analytic on S ++. This implies Lemma The function ψ(t, u) S ++ for all u S ++ an t / 69

37 Definition of affine processes on S + Affine processes on S + are Feller an regular Remarks on the proof Remark on the proof of the above theorem: Regularity can be prove by methos on the regularity of (semi-)flows from Montgommery an Zippin. 36 / 69

38 Definition of affine processes on S + Affine processes on S + are Feller an regular Remarks on the proof Remark on the proof of the above theorem: Regularity can be prove by methos on the regularity of (semi-)flows from Montgommery an Zippin. The { Feller property follows from regularity an the fact that e u,x u S ++ } is ense in C0 (S + ) (Stone-Weierstrass). 37 / 69

39 Definition of affine processes on S + Affine processes on S + are Feller an regular Consequences of regularity: From regularity we can conclue local characteristics: Since the following limit exists, Ae u,x = e u,x 1 lim t 0 t E( e u,xt x 1 ), we know that the limit is of Lévy-Khintchine form on the cone S + Rx. Due to the affine property the previous equation is also an equation for the erivatives of φ an ψ. By the Markov property (semigroup property) these equations turn into ODEs for φ an ψ. 38 / 69

40 Definition of affine processes on S + Affine processes on S + are Feller an regular The function φ an ψ are solutions of ODEs: Hence we obtain φ(t, u) = F (ψ(t, u)), t φ(0, u) = 0, (4) ψ(t, u) = R(ψ(t, u)), t ψ(0, u) = u S +, (5) where F an R are efine in (1). Due to the particular form of F an R we shall call them generalize Riccati equations. 39 / 69

41 Definition of affine processes on S + Affine processes on S + are Feller an regular The functions F an R the question of existence of affine processes Theorem Moreover, φ(t, u) an ψ(t, u) solve the ifferential equations (4) an (5), where F an R have the following form F (u) = b, u + c (e u,ξ 1)m(ξ), S + \{0} R(u) = 2uαu + B (u) + γ ( ) µ(ξ) e u,ξ 1 + χ (ξ), u ξ 2 1. S + \{0} Conversely, let (α, b, β ij, c, γ, m, µ) be an amissible parameter set. Then there exists a unique affine process on S + with infinitesimal generator (6). 40 / 69

42 Main theorem Amissible parameters Ieas of the proof Infinitesimal generator Theorem If X is an affine process on S +, then its infinitesimal generator is of affine form: Af (x) = 1 A ijkl (x) 2 f (x) + b + B(x), f (x) (c + γ, x )f (x) 2 x ij x kl i,j,k,l + (f (x + ξ) f (x)) m(ξ) (6) S + \{0} + (f (x + ξ) f (x) χ(ξ), f (x) ) M(x, ξ), S + \{0} for some truncation function χ an amissible parameters ( α, b, B(x) = i,j β ij x ij, c, γ, m(ξ), M(x, ξ) = where A ijkl (x) = x ik α jl + x il α jk + x jk α il + x jl α ik. ) x, µ, ξ / 69

43 Main theorem Amissible parameters Ieas of the proof Relation to semimartingales Corollary Let X be a conservative affine process on S +. Then X is a semimartingale. Furthermore, there exists, possibly on an enlargement of the probability space, a -matrix of stanar Brownian motions W such that X amits the following representation ( t ) X t = x + b + χ(ξ)m(ξ) + B(X s) s, 0 S + \{0} t ( ) + XsWsΣ + Σ W s Xs 0 t ( ) + χ(ξ) µ X (t, ξ) (m(ξ) + M(X t, ξ)) t 0 S + \{0} t + (ξ χ(ξ))µ X (t, ξ), 0 S + \{0} where Σ is a matrix satisfying Σ Σ = α an µ X enotes the ranom measure associate with the jumps of X. 42 / 69

44 Main theorem Amissible parameters Ieas of the proof Amissible parameters linear iffusion coefficient: α S +, 43 / 69

45 Main theorem Amissible parameters Ieas of the proof Amissible parameters linear iffusion coefficient: α S +, linear jump coefficient: -matrix µ = (µ ij ) of finite signe measures on S + \ {0} with µ(e) S + for all E B(S + \ {0}) M(x, ξ) := x,µ(ξ) satisfies ξ 2 1 S + \{0} χ(ξ), u M(x, ξ) < for all x, u S + with x, u = 0, 44 / 69

46 Main theorem Amissible parameters Ieas of the proof Amissible parameters linear iffusion coefficient: α S +, linear jump coefficient: -matrix µ = (µ ij ) of finite signe measures on S + \ {0} with µ(e) S + for all E B(S + \ {0}) M(x, ξ) := x,µ(ξ) satisfies ξ 2 1 S + \{0} χ(ξ), u M(x, ξ) < for all x, u S + with x, u = 0, linear rift coefficient: the linear map B(x) = i,j βij x ij with β ij = β ji S satisfies B(x), u χ(ξ), u M(x, ξ) 0 S + \{0} for all x, u S + with x, u = 0, 45 / 69

47 Main theorem Amissible parameters Ieas of the proof Amissible parameters linear iffusion coefficient: α S +, linear jump coefficient: -matrix µ = (µ ij ) of finite signe measures on S + \ {0} with µ(e) S + for all E B(S + \ {0}) M(x, ξ) := x,µ(ξ) satisfies ξ 2 1 S + \{0} χ(ξ), u M(x, ξ) < for all x, u S + with x, u = 0, linear rift coefficient: the linear map B(x) = i,j βij x ij with β ij = β ji S satisfies B(x), u χ(ξ), u M(x, ξ) 0 S + \{0} for all x, u S + with x, u = 0, linear killing rate coefficient: γ S +, 46 / 69

48 Main theorem Amissible parameters Ieas of the proof Amissible parameters linear iffusion coefficient: α S +, linear jump coefficient: -matrix µ = (µ ij ) of finite signe measures on S + \ {0} with µ(e) S + for all E B(S + \ {0}) M(x, ξ) := x,µ(ξ) satisfies ξ 2 1 S + \{0} χ(ξ), u M(x, ξ) < for all x, u S + with x, u = 0, linear rift coefficient: the linear map B(x) = i,j βij x ij with β ij = β ji S satisfies B(x), u χ(ξ), u M(x, ξ) 0 S + \{0} for all x, u S + with x, u = 0, linear killing rate coefficient: γ S +, constant rift term: b ( 1)α, 47 / 69

49 Main theorem Amissible parameters Ieas of the proof Amissible parameters linear iffusion coefficient: α S +, linear jump coefficient: -matrix µ = (µ ij ) of finite signe measures on S + \ {0} with µ(e) S + for all E B(S + \ {0}) M(x, ξ) := x,µ(ξ) satisfies ξ 2 1 S + \{0} χ(ξ), u M(x, ξ) < for all x, u S + with x, u = 0, linear rift coefficient: the linear map B(x) = i,j βij x ij with β ij = β ji S satisfies B(x), u χ(ξ), u M(x, ξ) 0 S + \{0} for all x, u S + with x, u = 0, linear killing rate coefficient: γ S +, constant rift term: b ( 1)α, constant jump term: Borel measure m on S + ( ξ 1) m(ξ) <, S + \{0} \ {0} with 48 / 69

50 Main theorem Amissible parameters Ieas of the proof Amissible parameters linear iffusion coefficient: α S +, linear jump coefficient: -matrix µ = (µ ij ) of finite signe measures on S + \ {0} with µ(e) S + for all E B(S + \ {0}) M(x, ξ) := x,µ(ξ) satisfies ξ 2 1 S + \{0} χ(ξ), u M(x, ξ) < for all x, u S + with x, u = 0, linear rift coefficient: the linear map B(x) = i,j βij x ij with β ij = β ji S satisfies B(x), u χ(ξ), u M(x, ξ) 0 S + \{0} for all x, u S + with x, u = 0, linear killing rate coefficient: γ S +, constant rift term: b ( 1)α, constant jump term: Borel measure m on S + ( ξ 1) m(ξ) <, S + \{0} constant killing rate term: c R +. \ {0} with 49 / 69

51 Main theorem Amissible parameters Ieas of the proof Remark on the amissible parameters No constant iffusion part, linear part is of very specific form u, A(x)u = 4 x, uαu. 50 / 69

52 Main theorem Amissible parameters Ieas of the proof Remark on the amissible parameters No constant iffusion part, linear part is of very specific form u, A(x)u = 4 x, uαu. Very remarkable constant rift conition: b ( 1)α. For 2 the bounary of S + is curve an implies this relation between linear iffusion coefficient α an rift part b. 51 / 69

53 Main theorem Amissible parameters Ieas of the proof Remark on the amissible parameters No constant iffusion part, linear part is of very specific form u, A(x)u = 4 x, uαu. Very remarkable constant rift conition: b ( 1)α. For 2 the bounary of S + is curve an implies this relation between linear iffusion coefficient α an rift part b. Jumps escribe by m are of finite variation, for the linear jump part we have finite variation for the irections orthogonal to the bounary while parallel to the bounary general jump behavior is allowe. 52 / 69

54 Main theorem Amissible parameters Ieas of the proof Remark on the amissible parameters Parameters of prototype equation are amissible: X t = (b + HX t + X t H )t + X t W t Σ + Σ W t Xt + J t, 53 / 69

55 Main theorem Amissible parameters Ieas of the proof Remark on the amissible parameters Parameters of prototype equation are amissible: X t = (b + HX t + X t H )t + X t W t Σ + Σ W t Xt + J t, Σ R : α = Σ Σ S +, 54 / 69

56 Main theorem Amissible parameters Ieas of the proof Remark on the amissible parameters Parameters of prototype equation are amissible: X t = (b + HX t + X t H )t + X t W t Σ + Σ W t Xt + J t, Σ R : α = Σ Σ S +, b ( 1)Σ Σ, 55 / 69

57 Main theorem Amissible parameters Ieas of the proof Remark on the amissible parameters Parameters of prototype equation are amissible: X t = (b + HX t + X t H )t + X t W t Σ + Σ W t Xt + J t, Σ R : α = Σ Σ S +, b ( 1)Σ Σ, B(x) = Hx + H x: Hx + H x, u = 0 if u, x = 0, 56 / 69

58 Main theorem Amissible parameters Ieas of the proof Remark on the amissible parameters Parameters of prototype equation are amissible: X t = (b + HX t + X t H )t + X t W t Σ + Σ W t Xt + J t, Σ R : α = Σ Σ S +, b ( 1)Σ Σ, B(x) = Hx + H x: Hx + H x, u = 0 if u, x = 0, c = 0, γ = 0, 57 / 69

59 Main theorem Amissible parameters Ieas of the proof Remark on the amissible parameters Parameters of prototype equation are amissible: X t = (b + HX t + X t H )t + X t W t Σ + Σ W t Xt + J t, Σ R : α = Σ Σ S +, b ( 1)Σ Σ, B(x) = Hx + H x: Hx + H x, u = 0 if u, x = 0, c = 0, γ = 0, J a compoun Poisson process with intensity l + λ, x an jump istribution ν supporte on S + : m(ξ) = lν(ξ), M(x, ξ) = λ, x ν(ξ). 58 / 69

60 Main theorem Amissible parameters Ieas of the proof What are the new results Full characterization an exact assumptions uner which affine processes on S + actually exist. 59 / 69

61 Main theorem Amissible parameters Ieas of the proof What are the new results Full characterization an exact assumptions uner which affine processes on S + actually exist. Necessity an sufficiency of the remarkable rift conition b ( 1)α. 60 / 69

62 Main theorem Amissible parameters Ieas of the proof What are the new results Full characterization an exact assumptions uner which affine processes on S + actually exist. Necessity an sufficiency of the remarkable rift conition b ( 1)α. Extension of the moel class. 61 / 69

63 Main theorem Amissible parameters Ieas of the proof What are the new results Full characterization an exact assumptions uner which affine processes on S + actually exist. Necessity an sufficiency of the remarkable rift conition b ( 1)α. Extension of the moel class. General linear rift part B(x) = ij βij x ij. This allows epenency of the volatility of one asset on the other ones which is not possible for B(x) = Hx + xh. Example: = 2 an ( ) x22 x B(x) = 12. x 12 x / 69

64 Main theorem Amissible parameters Ieas of the proof What are the new results Full characterization an exact assumptions uner which affine processes on S + actually exist. Necessity an sufficiency of the remarkable rift conition b ( 1)α. Extension of the moel class. General linear rift part B(x) = ij βij x ij. This allows epenency of the volatility of one asset on the other ones which is not possible for B(x) = Hx + xh. Example: = 2 an ( ) x22 x B(x) = 12. x 12 x 11 Full generality of jumps (quaratic variation jumps parallel to the bounary). 63 / 69

65 Main theorem Amissible parameters Ieas of the proof Ieas an methos applie in the proof Necessary conitions: Since the limit exists Ae u,x = e u,x 1 lim t 0 t E( e u,xt x 1 ), we know that the limit is of Lévy-Khintchine form on the cone S + Rx. The structure of infinitely ivisible ranom variables on cones implies the form of F an R an it etermines the form of the infinitesimal generator since Ae u,x = ( F (u) R(u), x )e u,x. In orer to erive the conition on b we consier et(x t ). Invariance conitions for R + -value process imply then b ( 1)α. 64 / 69

66 Main theorem Amissible parameters Ieas of the proof Ieas an methos applie in the proof Sufficient conitions Problems: Unboune coefficients, x is not Lipschitz continuous at the bounary, infinitely active jumps. Regularization of the coefficients (replace x by x + ɛi ɛi ). Through stochastic invariance, we establish existence of an S + -value solution of the martingale problem for the generator of the regularize process Existence of an S + -value solution of the martingale problem for A. Uniqueness an existence of global solutions φ an ψ of the generalize Riccati equations (4) an (5) yiel uniqueness of the solution of the martingale problem Existence of an affine (Markov) process on S / 69

67 On S +, the laws of X t = δi t + X t W t + Wt Xt are those of non central Wishart istributions an are not infinitely ivisible. As a consequence of the conition on the constant rift, we obtain Corollary X is affine an its one-imensional marginal istributions are infinitely ivisible if an only if α = 0 or = / 69

68 Conclusion an Outlook Conclusion Characterization of S + -value affine processes. Exhaustive moel specification. Characterization of infinitely ivisible affine processes. Outlook Calibration an parameter estimation of multivariate stochastic covariance moels. Affine processes on symmetric cones. Calculation of stochastic quantities beyon marginals in the affine setting. Infinite imensional affine processes. 67 / 69

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