Диссертация (1137347), страница 5
Текст из файла (страница 5)
Namelyp(s, t, x, y) ≤ c1 exp((1 ∨ T 1/2 )c1 [(t − s)1/2 ])pc (t − s, y − x).17(2.11)The upper bound enjoy the semigroup property, i.e. ∀0 ≤ s < u < t ≤ T,Zpc (u − t, z − x)pc (s − u, y − z)dz = pc (t − s, y − x),Rdwhich allows to propagate the upper bound (2.11) from small times to arbitrary but finite time.2.4Markov ChainsOne of the main advantages of the formal expansion in (2.9) is that it has a direct discrete counterpartin the Markov chain setting. Indeed, denote by (Yttji ,x )j≥i the Markov chain starting from x at timeti with dynamics:√Ytk+1 = Ytk + b(tk , Ytk )h + σ(tk , Ytk ) hξk+1 , Y0 = x,(2.12)where h > 0 is a given time step, for which we denote for all k ≥ 0, tk := kh and the (ξk )k≥1 arecentered i.i.d.
random variables satisfying some integrability conditions. Observe first that if theinnovations (ξk )k≥1 have a density then so does the chain at time tk .Let us now introduce its generator at time ti , i.e. for all ϕ ∈ C02 (Rd , R), x ∈ Rd :i ,xLhti ϕ(x) := h−1 E[ϕ(Ytti+1) − ϕ(x)].In order to give a representation of the density of ph (ti , tj , x, y) of Yttji ,x at point y for j > i, weintroduce similarly to the continuous case, the Markov chain (or inhomogeneous random walk) withcoefficients frozen in space at y. For given (ti , x) ∈ [0, T ] × Rd , tj ≥ ti we set:Ỹttji ,x,y := x + h1/2j−1Xσ(tk , y)ξk+1 ,k=iand denote its density p̃h,y (ti , tj , x, .).
Its generator at time ti writes for all ϕ ∈ C02 (Rd , R), x ∈ Rd :−1i ,x,yL̃h,yE[ϕ(Ỹtti+1) − ϕ(x)].ti ϕ(x) = hUsing the notation p̃h (ti , tj , x, y) := p̃h,y (ti , tj , x, y), we introduce now for 0 ≤ i < j ≤ N theparametrix kernel:hH h (ti , tj , x, y) := (Lhti − L̃h,yti )p̃ (ti + h, tj , x, y).Analogously to Lemma 3.6 in [KM00], which follows from a direct algebraic manipulation, wederive the following representation for the density. Assuming boundness and Lipschitz continuity forcoefficients b and σ with the uniform ellipticity for σ, one can get for 0 ≤ ti < tj ≤ T thathp (ti , tj , x, y) =j−iXp̃h ⊗h H h,(r) (ti , tj , x, y),(2.13)r=0where the discrete time convolution type operator ⊗h is defined byf ⊗h g(ti , tj , x, y) =j−i−1Xk=0Zhf (ti , ti+k , x, z)g(ti+k , tj , z, y)dz.RdAlso g⊗h H h,(0) = g and for all r ≥ 1, H h,(r) := H h ⊗h H h,(r−1) denotes the r-fold discrete convolutionof the kernel H h .18The key point to prove is the direct definition of the discrete kernel function.
SinceZhH (ti , tj , x, y) =h−1 [ph (ti , ti+1 , x, z) − p̃h (ti , ti+1 , x, z)]p̃h (ti+1 , tj , z, y)dz.RdUsing the Markov property we get the following identity:ph (ti , tj , x, y) − p̃h (ti , tj , x, y) =j−1 ZXhk=iZ×ph (ti , tk , x, z)Rdh−1 [ph (tk , tk+1 , z, z 0 ) − p̃h (tk , tk+1 , z, z 0 )]p̃h (tk+1 , tj , z 0 , y)dz 0 dzRd=j−1 ZXhk=iph (ti , tk , x, z)H h (tk , tj , z, y)dzRd= (ph ⊗h H h )(ti , tj , x, y).The expansion (2.13) follows by iterative application of this identity.19Chapter 3Stability of diffusion transitiondensities3.1Stability resultsFor a fixed given deterministic final horizon T > 0, let us consider, as in the Chapter 2, the followingmultidimensional SDE:dXt = b(t, Xt )dt + σ(t, Xt )dWt , t ∈ [0, T ],(3.1)where b : [0, T ] × Rd → Rd , σ : [0, T ] × Rd → Rd ⊗ Rd are bounded coefficients, measurable in timeand Hölder continuous in space (this last condition will be relaxed for the drift term b) and W is aRd Brownian motion on some filtered probability space (Ω, F, (Ft )t≥0 , P).
Also, a(t, x) := σσ ∗ (t, x)is assumed to be uniformly elliptic. In particular those assumptions guarantee that (3.1) admits aunique weak solution, see e.g. Bass and Perkins [BP09], [SV79], [Men11] from which the uniquenessto the martingale problem for the associated generator can be derived under the current assumptions.We now introduce, for a given parameter ε > 0, a perturbed version of (3.1) with dynamics:(ε)dXt(ε)(ε)= bε (t, Xt )dt + σε (t, Xt )dWt , t ∈ [0, T ],(3.2)where bε : [0, T ] × Rd → Rd , σε : [0, T ] × Rd → Rd ⊗ Rd satisfy at least the same assumptions as b, σand are in some sense meant to be close to b, σ when ε is small.(ε)It is known that, under the previous assumptions, densities of processes (Xt )t≥0 , (Xt )t≥0 exist and satisfy some Gaussian bounds, see e.g Aronson [Aro59] or [DM10] for extensions to somedegenerate cases.The goal of this Chapter is to investigate how the closeness of (bε , σε ) and (b, σ) is reflected onthe respective densities of the associated processes.
Important applications can for instance be foundin mathematical finance.In the framework of parameter estimation it can be useful, having at hand estimators (bε , σε ) ofthe true parameters (b, σ) and some controls for the differences |b − bε |, |σ − σε | in a suitable sense,to quantify the difference pε − p of the densities corresponding respectively to the dynamics with theestimated parameters and the one of the model.Another important application includes the case of mollification by spatial convolution.
Thisspecific kind of perturbation is useful to investigate the error between the densities of a non-degeneratediffusion of type (3.1) with Hölder coefficients (or with piecewise smooth bounded drift) and its Eulerscheme. In this framework, some explicit convergence results can be found in [KM17].20More generally, this situation can appear in every applicative field for which the diffusion coefficientmight be misspecified.Our stability results will also apply to two Markov chains with respective dynamics:√Ytk+1 = Ytk + b(tk , Ytk )h + σ(tk , Ytk ) hξk+1 , Y0 = x,√(ε)(ε)(ε)(ε)(ε)Ytk+1 = Ytk + bε (tk , Ytk )h + σε (tk , Ytk ) hξk+1 , Y0 = x,(3.3)where h > 0 is a given time step, for which we denote for all k ≥ 0, tk := kh and the (ξk )k≥1 arecentred i.i.d.
random variables satisfying some integrability conditions. Again, the key tool will bethe parametrix representation for the densities of the chains and the Gaussian local limit theorem.3.1.1Assumptions and Main Results.For better readability let us now repeat assumptions, introduced in Chapter 1, which we use throughout this Section. Below, the parameter ε > 0 is fixed and the constants appearing in the assumptionsdo not depend on ε.(A1) (Boundedness of the coefficients). The components of the vector-valued functions b(t, x), bε (t, x)and the matrix-functions σ(t, x), σε (t, x) are bounded measurable. Specifically, there exist constantsK1 , K2 > 0 s.t.|b(t, x)| +sup(t,x)∈[0,T ]×Rdsup|σ(t, x)| +sup|bε (t, x)| ≤ K1 ,(t,x)∈[0,T ]×Rd(t,x)∈[0,T ]×Rdsup|σε (t, x)| ≤ K2 .(t,x)∈[0,T ]×Rd(A2) (Uniform Ellipticity).
The matrices a := σσ ∗ , aε := σε σε∗ are uniformly elliptic, i.e. thereexists Λ ≥ 1, such that ∀(t, x, ξ) ∈ [0, T ] × (Rd )2 ,Λ−1 |ξ|2 ≤ ha(t, x)ξ, ξi ≤ Λ|ξ|2 , Λ−1 |ξ|2 ≤ haε (t, x)ξ, ξi ≤ Λ|ξ|2 .(A3) (Hölder continuity in space). For some γ ∈ (0, 1] , κ < ∞, for all t ∈ [0, T ],|σ(t, x) − σ(t, y)| + |σε (t, x) − σε (t, y)|γ≤ κ |x − y| .Observe that the last condition also readily gives, thanks to the boundedness of σ, σε that a, aε arealso uniformly γ-Hölder continuous.For a given ε > 0, we say that assumption (A) holds when conditions (A1)-(A3) are in force.Let us now introduce, under (A), the quantities that will bound the difference of the densities in ourmain results below.
Set for ε > 0:∆ε,b,∞:={|b(t, x) − bε (t, x)|}, ∀q ∈ (1, +∞),sup(t,x)∈[0,T ]×Rd∆ε,b,q:=sup kb(t, .) − bε (t, .)kLq (Rd ) .t∈[0,T ]Since σ, σε are both γ-Hölder continuous, see (A3) we also define∆ε,σ,γ := sup |σ(u, .) − σε (u, .)|γ ,u∈[0,T ]21where for γ ∈ (0, 1], |.|γ stands for the usual Hölder norm in space on Cbγ (Rd , Rd ⊗ Rd ) (space ofHölder continuous bounded functions, see e.g. Krylov [Kry96]) i.e. :|f |γ := sup |f (x)| + [f ]γ , [f ]γ :=x∈Rdsupx6=y,(x,y)∈(Rd )2|f (x) − f (y)|.|x − y|γThe previous control in particular implies for all (u, x, y) ∈ [0, T ] × (Rd )2 :|a(u, x) − a(u, y) − aε (u, x) + aε (u, y)| ≤ 2(K2 + κ)∆ε,σ,γ |x − y|γ .We eventually set for q ∈ (1, +∞],∆ε,γ,q := ∆ε,σ,γ + ∆ε,b,q ,(3.4)which will be the key quantity governing the error in our results.We will denote, from now on, by C a constant depending on the parameters appearing in (A)and T .
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
