Affine Rough Models
aa r X i v : . [ q -f i n . M F ] D ec Affine Rough Models ∗ Martin Keller-Ressel † Martin Larsson ‡ Sergio Pulido § December 20, 2018
Abstract
The goal of this survey article is to explain and elucidate the affine structure ofrecent models appearing in the rough volatility literature, and show how it leads toexponential-affine transform formulas.
Affine stochastic volatility models have a long history in the quantitative finance literature;see e.g. Duffie et al. (2003); Kallsen (2006) and the references listed there. These modelsare generally of the form dS t = S t p V t dB t , (1.1)where S is the asset price, and the (spot) variance V is modeled by an affine process.Arguably, the most prominent example is the Heston (1993) model, where V follows a scalarsquare-root diffusion. The affine property leads to tractable Fourier–Laplace transformsof various quantities of interest. For example, the log-price satisfies the exponential-affinetransform formula E [exp( v log S T ) | F t ] = exp ( v log S t + φ ( T − t ) + ψ ( T − t ) V t ) , (1.2) ∗ The authors would like to thank Eduardo Abi Jaber and Christa Cuchiero for valuable discussions andsuggestions. Martin Keller-Ressel gratefully acknowledges financial support from DFG grants ZUK 64 andKE 1736/1-1. Martin Larsson gratefully acknowledges financial support from SNF Grant 205121 163425.Sergio Pulido gratefully acknowledges financial support from MATH AmSud project SaSMoTiDep 18-MATH-17. † Institute of Mathematical Stochastics, TU Dresden, 01062 Dresden, Germany, [email protected]. ‡ Department of Mathematics, ETH Zurich, R¨amistrasse 101, CH-8092, Zurich, Switzerland, [email protected]. § Laboratoire de Math´ematiques et Mod´elisation d’´Evry (LaMME), Universit´e d’´Evry-Val-d’Essonne,ENSIIE, Universit´e Paris-Saclay, UMR CNRS 8071, IBGBI 23 Boulevard de France, 91037 ´Evry Cedex,France, [email protected]. φ, ψ are the solutions to ordinary differential equations of Riccati type that dependon v . Similar formulas also exist for the spot variance and integrated spot variance.Unfortunately, these models do not produce the rough trajectories of volatility thatseem to occur empirically, see Gatheral et al. (2018), and have trouble capturing the termstructure of implied volatilities and its skew, cf. Fukasawa (2017). Still, it is possible to con-struct stochastic volatility models with these features, and with an “affine structure” thatproduces formulas similar to (1.2). This has recently been done by Guennoun et al. (2018);El Euch and Rosenbaum (2016); Abi Jaber et al. (2017); Gatheral and Keller-Ressel (2018),and related ideas appear already in Comte et al. (2012). The goal of this chapter is to ex-plain and elucidate this “affine structure”, and show how it leads to exponential-affinetransform formulas.We will give four perspectives. The point of departure, in Section 2, is a class of stochastic convolution equations for V with affine coefficients, which contains the roughHeston model of El Euch and Rosenbaum (2016) as an immediate special case. This is thefirst perspective. The second perspective, in Section 3, is to view these models as forwardvariance models , which focus on the forward variance curve ξ t ( T ) = E [ V T | F t ] . (1.3)The third perspective, in Section 4, is to regard a modified forward variance curve as thesolution of a stochastic partial differential equation . Finally, the fourth perspective, inSection 5, is available when the convolution kernel is the Laplace transform of a possiblysigned measure. This leads to a representation as a mixture of mean-reverting processes .Dually, this gives multiple perspectives on the Riccati equations that characterize theassociated Fourier–Laplace functionals.Space constraints prevent us from including rigorous proofs of all results. Still, someproofs and derivations are presented, selected because they are instructive without beingtoo long. We occasionally use the convolution notation ( f ∗ g )( t ) = R t f ( t − s ) g ( s ) ds forfunctions f and g , and similarly ( f ∗ dZ ) t = R t f ( t − s ) dZ s when Z is a semimartingale. Consider the stochastic convolution equation V t = V + Z t K ( t − s ) b ( V s ) ds + Z t K ( t − s ) σ ( V s ) dW s , (2.1)for some real continuous functions b and σ , kernel K ∈ L ( R + ), initial condition V ∈ R ,and Brownian motion W . Solutions to (2.1) are always understood to have continuouspaths. 2 xample 2.1. Taking b ( x ) = λ ( θ − x ) , σ ( x ) = ζ √ x , and the power-law kernel K α -pow ( t ) = t α − / Γ( α ) with α ∈ ( , , we obtain the spot variance process in the rough Heston modelof El Euch and Rosenbaum (2016). With α = 1 , we recover the spot variance process inthe classical Heston model. Our focus is on the case where V is an affine Volterra process , which is when b ( x ) and σ ( x ) are affine in x . This definition naturally generalizes to higher dimension: b ( x ) is thena vector and σ ( x ) a matrix, and one requires b ( x ) and σ ( x ) σ ( x ) ⊤ to be affine in x . In thischapter we focus on the one-dimensional affine case, so that b ( x ) = β − λx and σ ( x ) = α + ax (2.2)for some real parameters β, λ, α, a such that α + aV t ≥ t ≥
0. The latter conditionraises delicate questions of existence of solutions to (2.1) when a = 0, which we do notaddress here in detail. Let us however state the following result, whose proof can be foundin Abi Jaber et al. (2017). Part (ii) of the theorem requires the following assumption onthe kernel: K is strictly positive and completely monotone. There is γ ∈ (0 ,
2] such that R h K ( t ) dt = O ( h γ ) and R T ( K ( t + h ) − K ( t )) dt = O ( h γ ) for every T < ∞ . (2.3)(Recall that a C ∞ function f : (0 , ∞ ) → R is completely monotone if ( − k f ( k ) ≥ k ≥ γ = 2 α − Theorem 2.2.
Consider the equation (2.1) with coefficients b ( x ) and σ ( x ) as in (2.2) andkernel K ∈ L ( R + ) . (i) Assume that α ≥ and a = 0 . Then there exists a pathwise unique strong solution V for any initial condition V ∈ R ; the Volterra Ornstein–Uhlenbeck process . (ii) Assume that α = 0 , a > , β ≥ , and K satisfies (2.3) . Then there exists a uniquein law R + -valued weak solution V for any initial condition V ∈ R + ; the Volterrasquare-root process .In either case, the trajectories of V are H¨older continuous of any order less than γ/ . Remark 2.3.
A solution V is called strong if it is adapted to the filtration generated by theBrownian motion W . This is not required for a weak solution, where one is free to constructthe Brownian motion as needed. Pathwise uniqueness means that any two solutions V and V ′ to (2.1) driven by the same Brownian motion must have identical trajectories (outsidea nullset). It is not known whether pathwise uniqueness holds in (ii) . A Volterra–Heston model is a stochastic volatility model of the form (1.1), where thespot variance V is a Volterra square-root process. Most of this chapter is concerned withsuch models. The process V is generally neither a Markov process nor a semimartingale.This causes difficulties that the alternative perspectives developed in the following sectionshelp to circumvent. 3 Forward variance models
A useful perspective on (2.1) is as a forward variance model , as noted e.g. by B¨uhler (2006);Bergomi and Guyon (2012) and Bayer et al. (2016). Consider a model (1.1) where the spotvariance process V is given by the stochastic convolution equation (2.1) with affine drift b ( x ) = λ ( θ − x ), i.e. V t = V + λ Z t K ( t − s )( θ − V s ) ds + Z t K ( t − s ) σ ( V s ) dW s . (3.1)Our first goal is to derive an SDE for the forward variance ξ t ( T ) defined in (1.3). To thisend, we remark that for any kernel k ∈ L ( R + ) there exists a unique kernel r ∈ L ( R + ),called the resolvent or resolvent of the second kind of k , such that k ( t ) − r ( t ) = Z t r ( t − s ) k ( s ) ds, t ≥ . Example 3.1. If k ( t ) ≡ c is constant, then r ( t ) = ce − ct . If k ( t ) = c t α − / Γ( α ) is pro-portional to the power-law kernel, then r ( t ) = ct α − E α,α ( − ct α ) where E α,α denotes theMittag-Leffler function. The forward variance dynamics can now be described as follows.
Proposition 3.2.
Let R λ be the resolvent of λK . The forward variance ξ t ( T ) associatedto (3.1) satisfies dξ t ( T ) = λ R λ ( T − t ) σ ( V t ) dW t with initial condition ξ ( T ) = V (cid:18) − Z T R λ ( s ) ds (cid:19) + θ Z T R λ ( s ) ds. If λ = 0 , interpret λ − R λ = K , and note that R λ = 0 in this case.Proof. Suppose λ = 0; otherwise, the proof is easier and does not use resolvents. Denoteby the function which takes the constant value 1. The spot variance process V is givenby V = V + λK ∗ ( θ − V ) + K ∗ ( σ ( X ) dW ). Therefore, V − R λ ∗ V = V (1 − R λ ∗ ) + λ ( K − R λ ∗ K ) ∗ ( θ − V ) + ( K − R λ ∗ K ) ∗ ( σ ( V ) dW ) . By definition of the resolvent, K − R λ ∗ K = λ R λ . Plug this in and cancel the − R λ ∗ V terms on both sides to get V = V (1 − R λ ∗ ) + θR λ ∗ + λ R λ ∗ ( σ ( V ) dW ) . (3.2)4he process M u := R u R λ ( T − s ) σ ( X s ) dW s , u ∈ [0 , T ], is a martingale. Therefore, evalu-ating (3.2) at T and taking F t -conditional expectations yields ξ t ( T ) = E [ V T | F t ] = ξ ( T ) + Z T λ R λ ( T − s ) σ ( V s ) dW s , which is the claimed result. Remark 3.3.
We ignored some technical but important points in the proof. First, theassociativity property ( k ∗ k ) ∗ dZ = k ∗ ( k ∗ dZ ) was used for certain kernels k , k and dZ = σ ( X ) dW . This identity can be proved using the stochastic Fubini theorem.Second, we did not verify that M is really a martingale, not just a local martingale. For σ ( x ) = α + ax this can be done by noting that E [ h M i T ] = λ Z T R λ ( T − s ) E [ σ ( X s ) ] ds ≤ C (1 + sup s ≤ T E [ | X s | ]) , where one can take C = λ ( | α | + | a | ) R T R λ ( s ) ds . The right-hand side is finite, so M isactually a square-integrable martingale. Details are given by Abi Jaber et al. (2017). In the affine case (2.2), not only conditional expectations have useful representations, butalso Fourier–Laplace transforms. We now explain how such representations can be derivedonce the Volterra–Heston model (in log-price notation L = log S ) is written in forwardvariance form, ( dL t = − V t dt + p V t dB t dξ t ( T ) = λ R λ ( T − t ) σ p V t dW t . (3.3)Here λ ≥ d h B, W i t = ρdt for ρ ∈ [0 , α = 0 and a = σ for some σ >
0. Define the function Q ( u, z ) = 12 ( u − u ) + σρuz + σ z , u, z ∈ C . (3.4) Theorem 3.4.
Consider the Volterra–Heston model (3.3) . Fix
T > and ( u, v, w ) ∈ C ,and assume that the Riccati–Volterra equation ψ = vK + K ∗ ( Q ( u, ψ ) − λψ + w ) (3.5) has a solution ψ ∈ L (0 , T ) . Then the auxiliary process M t = exp (cid:18) uL t + vξ t ( T ) + w Z T ξ t ( s ) ds + Z Tt ξ t ( s ) Q ( u, ψ ( T − s )) ds (cid:19) (3.6)5 s a local martingale on [0 , T ] , and satisfies dM t M t = u p V t dB t + ψ ( T − t ) σ p V t dW t . (3.7) If M is a true martingale, the joint conditional Fourier–Laplace transform of the triplet ( L T , V T , R T V s ds ) is E [exp( uL T + vV T + w R T V s ds ) | F t ] = M t . The two crucial assumptions are of course that (3.5) has a solution, and that the localmartingale M is really a true martingale. The following theorem gives a sufficient conditionthat guarantees this; the proof can be found in Abi Jaber et al. (2017). Theorem 3.5.
Let K be a kernel satisfying (2.3) and let ( u, v, w ) ∈ C satisfy Re u ∈ [0 , , Re v ≤ , and Re w ≤ . Then the Riccati–Volterra equation (3.5) has a unique globalsolution ψ , and the local martingale M in (3.6) is a true martingale. We now present the proof of Theorem 3.4 in the special case where v = w = 0. Thissimplifies the calculations, and the proof in the general case is similar. Proof of Theorem 3.4 for v = w = 0 . Subtract R λ ∗ ψ from both sides of (3.5), where now v = w = 0, and apply the resolvent equation K − R λ ∗ K = λ R λ to get an equivalent formof the Riccati–Volterra equation, ψ = λ R λ ∗ Q ( u, ψ ) . (3.8)We aim to apply Itˆo’s formula to M t , so we define G t = R Tt Q ( u, ψ ( T − s )) ξ t ( s ) ds . Since ξ t ( s ) = V s for s ≤ t , we can write G t = Z T Q ( u, ψ ( T − s )) ξ t ( s ) ds − Z t Q ( u, ψ ( T − s )) V s ds. Focus on the first term. Using first Proposition 3.2, then the stochastic Fubini theorem(Veraar, 2012, Thm. 2.2), and finally (3.8), we get Z T Q ( u, ψ ( T − s )) ξ t ( s ) ds = Z T Q ( u, ψ ( T − s )) (cid:26) ξ ( s ) + σ Z t ∧ s λ R λ ( s − r ) p V r dW r (cid:27) ds = Z T Q ( u, ψ ( T − s )) ξ ( s ) ds + σ Z t Z Tr Q ( u, ψ ( T − s )) λ R λ ( s − r ) ds p V r dW r = Z T Q ( u, ψ ( T − s )) ξ ( s ) ds + σ Z t ψ ( T − r ) p V r dW r .
6s a result, G t = Z T Q ( u, ψ ( T − s )) ξ ( s ) ds + σ Z t ψ ( T − r ) p V r dW r − Z t Q ( u, ψ ( T − s )) V s ds. This leaves us in a position to apply Itˆo’s formula to M t = exp ( uL t + G t ), giving dM t M t = u dL t + dG t + u d h L i t + u d h L, G i t + 12 d h G i t = (cid:26)
12 ( u − u ) − Q ( u, ψ ( T − t )) + uρσψ ( T − t ) + σ ψ ( T − t ) (cid:27) V t dt + u p V t dB t + ψ ( T − t ) σ p V t dW t . Comparing with (3.4) shows that the dt -term vanishes, so M is indeed the local martingalein (3.7). If M is a true martingale, we conclude that E [exp( uL T ) | F t ] = E [ M T | F t ] = M t ,as claimed. Remark 3.6.
Setting g = Q ( u, ψ ) and applying Q ( u, · ) to both sides of (3.8) yields g = Q ( u, λ R λ ∗ g ) . This is the ‘convolution Riccati equation’ considered by Gatheral and Keller-Ressel (2018),which leads to an equivalent formulation in terms of g instead of ψ . We now discuss a converse to Theorem 3.4 obtained by Gatheral and Keller-Ressel (2018).Consider a general forward variance model of the type dξ t ( T ) = η t ( T ) dW t , where η t ( T ) is decreasing in T and ξ t ( T ) is the forward variance associated to a priceprocess S of the form dS t = S t a ( V t ) dW t . Assume that the conditional cumulant generatingfunction of the log-price L T = log S T is of the form E [exp( uL T ) | F t ] = exp (cid:18) uL t + Z Tt ξ t ( T − s ) g ( s, u ) ds (cid:19) for all u ∈ [0 ,
1] and 0 ≤ t ≤ T , for some continuous function g ≤
0. Under mild integrabilityconditions on η t ( T ), Gatheral and Keller-Ressel (2018) then show that, necessarily, a ( V t ) = a p V t and η t ( T ) = κ ( T − t ) p V t for some constant a ≥ κ . Thus the model is precisely of the form (3.3), and κ is identified with the resolvent λ R λ . 7 .3 Fractional calculus and the rough Heston model Consider the power law kernel K α -pow ( t ) = t α − / Γ( α ) used in the rough Heston model.The Riemann–Liouville fractional integral I α is defined via convolution with this kernel, I α f = K α -pow ∗ f . One then defines the Riemann–Liouville fractional derivative D α as D α f = ddt I − α f , which provides an inverse to the fractional integral in that D α ( I α f ) = I α ( D α f ) = f . It follows that, in the case v = w = 0, (3.5) is equivalent to D α ψ = Q ( u, ψ ) − λψ, which is precisely the fractional Riccati equation derived by El Euch and Rosenbaum (2016).Using Proposition 3.2 and (3.8) we can rewrite the exponent in (3.6), for t = 0, as ξ ∗ Q ( u, ψ ) = V ∗ Q ( u, ψ ) + ( θ − V ) ( ∗ R λ ∗ Q ( u, ψ ))= V ∗ Q ( u, ψ ) + λ ( θ − V )( ∗ ψ )= V I − α ψ + λθ ( ∗ ψ ) . Thus, in the rough Heston model, the unconditional transform formula in Theorem 3.4,with v = w = 0, becomes E [exp( uL T )] = exp (cid:18) uL + λθ Z T ψ ( s ) ds + V I − α ψ ( T ) (cid:19) , which is consistent with El Euch and Rosenbaum (2016). Another perspective on (2.1) via a stochastic partial differential equation arises as follows.Starting with a Volterra process V of the form (2.1), define the process u t ( x ) = E (cid:20) V t + x − Z t + xt K ( t − s + x ) b ( V s ) ds (cid:12)(cid:12)(cid:12) F t (cid:21) . This process is considered by Abi Jaber and El Euch (2018b). We call it the modifiedforward process , because had we not subtracted the time integral, we would have obtainedthe so-called Musiela parameterization ξ t ( t + x ) of the forward process. The only terminside the conditional expectation that is not already F t -measurable is an integral withrespect to W . This gives u t ( x ) = V + Z t K ( t − s + x ) b ( V s ) ds + Z t K ( t − s + x ) σ ( V s ) dW s , (4.1)which can be expressed in terms of the following SPDE.8 roposition 4.1. The process u t ( x ) in (4.1) is a mild solution of the SPDE du t ( x ) = ( ∂ x u t ( x ) + K ( x ) b ( u t (0))) dt + K ( x ) σ ( u t (0)) dW t (4.2) with initial condition u ( x ) = V for all x .Proof. Formally taking the differential in (4.1), using that ∂ t K ( t − s + x ) = ∂ x K ( t − s + x )and that u t (0) = V t , gives (4.2). More rigorously, note that K ( t − s + x ) = T t − s K ( x ), where T t − s is the shift operator that maps any function f to the shifted function f ( t − s + · ). Thederivative ∂ x is the infinitesimal generator of the shift semigroup { T t } t ≥ , so, by definition,(4.1) is actually the mild formulation of the SPDE (4.2); see (Da Prato and Zabczyk, 2014,Section 6.1). The SPDE (4.2) suggests that the process { u t ( · ) } t ≥ is an infinite dimensional Markovprocess. In the affine case (2.2), we therefore expect a Fourier–Laplace transform formulalike E (cid:20) exp (cid:18)Z ∞ h ( x ) u T ( x ) dx (cid:19) (cid:12)(cid:12)(cid:12) F t (cid:21) = exp (cid:18) φ ( T − t ) + Z ∞ Ψ( T − t, x ) u t ( x ) dx (cid:19) , (4.3)where φ ( τ ) and Ψ( τ, x ) are solutions of appropriate Riccati equations. These equationsturn out to be ∂ t φ ( t ) = R φ (cid:0)R ∞ Ψ( t, y ) K ( y ) dy (cid:1) (4.4)Ψ( t, x ) = h ( x − t ) { x ≥ t } + R Ψ (cid:0)R ∞ Ψ( t − x, y ) K ( y ) dy (cid:1) { x At first sight, (4.5) does not look like a differential equation for Ψ( t, x ) .But, along the lines of the proof of Proposition 4.1, (4.5) can actually be viewed as a mildformulation of the formal PDE ∂ t Ψ( t, x ) = − ∂ x Ψ( t, x ) + R Ψ (cid:18)Z ∞ Ψ( t, y ) K ( y ) dx (cid:19) δ ( x ) with initial condition Ψ(0 , x ) = h ( x ) . Let us give a derivation of the Riccati equations (4.4)–(4.5). We assume that V = 0;this does not affect the validity of the Riccati equations, but simplifies the calculations.Suppose that Ψ( t, x ) satisfies (4.5) and define dZ t = b ( V t ) dt + σ ( V t ) dW t , a semimartingale.9sing (4.1), (4.5), and the stochastic Fubini theorem; then a change of variables; and finally(4.5) once again, we get Z ∞ Ψ( T − t, x ) u t ( x ) dx = Z t Z ∞ T − t h ( x − T + t ) K ( t − s + x ) dx dZ s + Z t Z T − t R Ψ (cid:0)R ∞ Ψ( T − t − x, z ) K ( z ) dz (cid:1) K ( t − s + x ) dx dZ s = Z t Z ∞ T − s h ( y − T + s ) K ( y ) dy dZ s + Z t Z T − st − s R Ψ (cid:0)R ∞ Ψ( T − s − y, z ) K ( z ) dz (cid:1) K ( y ) dy dZ s = Z t Z ∞ Ψ( T − s, x ) K ( x ) dx dZ s − Z t Z t − s R Ψ (cid:0)R ∞ Ψ( T − s − y, z ) K ( z ) dz (cid:1) K ( y ) dy dZ s . Combining this with the stochastic Volterra equation (2.1) satisfied by V yields d Z ∞ Ψ( T − t, x ) u t ( x ) dx = Z ∞ Ψ( T − t, x ) K ( x ) dx dZ t − Z t R Ψ (cid:0)R ∞ Ψ( T − t, y ) K ( y ) dy (cid:1) K ( t − s ) dZ s dt = Z ∞ Ψ( T − t, x ) K ( x ) dx dZ t − R Ψ (cid:0)R ∞ Ψ( T − t, y ) K ( y ) dy (cid:1) V t dt. Let M t denote the right-hand side of (4.3). Use the previous equation and (4.4) to get dM t M t = Z ∞ Ψ( T − t, x ) K ( x ) dx σ ( V t ) dW t . (4.7)Thus M is a local martingale, and M T = exp( R ∞ h ( x ) u T ( x ) dx ) since Ψ(0 , x ) = h ( x ). If M is a true martingale we deduce the exponential-affine formula (4.3).This can be used to derive the special case of Theorem 3.4 where u = w = 0 (note that α = 0 and a = σ in that theorem). Formally setting h = vδ with v ∈ C gives E [exp ( vV T ) | F t ] = exp (cid:18) φ ( T − t ) + Z ∞ Ψ( T − t, x ) u t ( x ) dx (cid:19) . There is a connection between the Riccati equation (4.5) and the Riccati–Volterra equation(3.5) in Theorem 3.4. To wit, suppose that Ψ( t, x ) solves (4.5) and define ψ ( t ) = Z ∞ Ψ( t, x ) K ( x ) dx. (4.8)10sing the definition (4.6) of R Ψ , the definition (4.8) of ψ , (4.5), and a change of variables,we get K ∗ (cid:16) − λψ + a ψ (cid:17) ( t ) = Z ∞ K ( x ) R Ψ ( ψ ( t − x )) { x If we are in the classical case K ( t ) = 1 , then µ = δ . In the rough Hestoncase K ( t ) = t α − / Γ( α ) with α ∈ ( , , then µ ( dx ) = x − α Γ( α )Γ(1 − α ) dx . To see how (5.1) leads to a (possibly infinite) mixture of mean-reverting processes, andin order to simplify the presentation, we will assume that V = 0. The general case can bededuced by considering the process e V = V − V ; the reader is invited to work out whathappens in this general case.Substituting (5.1) into (2.1) with V = 0, and interchanging the time- and µ -integrals(justified by the stochastic Fubini theorem) yields the representation V t = Z ∞ u t ( x ) µ ( dx ) , (5.2)11here we define, for all t ≥ u t ( x ) = Z t e − x ( t − s ) b ( V s ) ds + Z t e − x ( t − s ) σ ( V s ) dW s . Crucially, each process { u t ( x ) } t ≥ is a semimartingale, even if V is not. To find its dynamicsmove e − xt outside the time integrals and apply the product rule to get du t ( x ) = ( − xu t ( x ) + b ( V t )) dt + σ ( V t ) dW t . Plugging (2.2) and (5.2) into this expression gives du t ( x ) = (cid:18) β − xu t ( x ) − λ Z ∞ u t ( y ) µ ( dy ) (cid:19) dt + s α + a Z ∞ u t ( y ) µ ( dy ) dW t . (5.3)As x ranges through the support of µ , (5.3) defines a (possibly infinite) coupled systemof mean-reverting processes, and (5.2) expresses V as a mixture of those processes. TheGaussian case a = 0 is covered by results of Carmona et al. (2000); Harms and Stefanovits(2018).Apart from its theoretical interest, this representation can be useful for numerical pur-poses. The idea is to replace µ by an approximation µ n that is supported on finitely manypoints x , . . . , x n . The system (5.3) then becomes an SDE for the n -dimensional Markovprocess { u t ( x ) , . . . , u t ( x n ) } t ≥ . This can be used to approximate the affine Volterra pro-cess V . More details on this construction are given by Abi Jaber and El Euch (2018a) andCuchiero and Teichmann (2018). The drift and squared volatility in (5.3) depend on the curve u t ( · ) in an affine way.This suggests that the process { u t ( · ) } t ≥ is an affine Markov process, possibly infinite-dimensional. In particular, we expect a transform formula similar to (4.3): E (cid:20) exp (cid:18)Z ∞ h ( x ) u T ( x ) µ ( dx ) (cid:19) (cid:12)(cid:12)(cid:12) F t (cid:21) = exp (cid:18) φ ( T − t ) + Z ∞ Ψ( T − t, x ) u t ( x ) µ ( dx ) (cid:19) , (5.4)where φ ( τ ) and Ψ( τ, x ) are solutions of appropriate Riccati equations with initial conditions φ (0) = 0 , Ψ(0 , x ) = h ( x ) . (5.5)In this Markovian situation, one can apply the standard method for deriving the Riccatiequations. Let M t denote the right-hand side of (5.4). Itˆo’s formula and (5.3) give, after12ome computations, dM t M t = h − ∂ t φ ( T − t ) + R φ (cid:0)R ∞ Ψ( T − t, y ) µ ( dy ) (cid:1) + Z ∞ (cid:16) − ∂ t Ψ( T − t, x ) − x Ψ( T − t, x )+ R Ψ (cid:0)R ∞ Ψ( T − t, y ) µ ( dy ) (cid:1)(cid:17) u t ( x ) µ ( dx ) i dt + local martingale , (5.6)with R φ , R Ψ as in (4.6). It is remarkable that the same functions R φ and R Ψ as for theSPDE representation occur also here. This is one manifestation of the underlying abstractpoint of view due to Cuchiero and Teichmann (2018).Suppose that φ and Ψ solve the possibly infinite-dimensional Riccati equations ∂ t φ ( t ) = R φ (cid:0)R ∞ Ψ( t, y ) µ ( dy ) (cid:1) ,∂ t Ψ( t, x ) = − x Ψ( t, x ) + R Ψ (cid:0)R ∞ Ψ( t, y ) µ ( dy ) (cid:1) , (5.7)with initial conditions (5.5). Then, due to (5.6), M is a local martingale with M T =exp( R ∞ h ( x ) u T ( x ) µ ( dx )). If M is actually a true martingale, we obtain the transformformula (5.4), which is nothing but the martingale property E [ M T | F t ] = M t . In particular,if h ( x ) ≡ v is constant, combining (5.2) and (5.4) gives E [exp ( vV T ) | F t ] = exp (cid:18) φ ( T − t ) + Z ∞ Ψ( T − t, x ) u t ( x ) µ ( dx ) (cid:19) . Just as in Section 4, there is a connection between the solution Ψ( t, x ) to the Riccatiequation (5.7), with h ( x ) ≡ v constant, and the solution ψ ( t ) to the Riccati–Volterraequation (3.5), with u = w = 0. 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