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National Research University Higher School of EconomicsInternational Laboratory of Stochastic Analysis and Its ApplicationsAnna KozhinaParametrix Method and its Applications in ProbabilityTheorySummary of the PhD thesisfor the purpose of obtainingPhilosophy Doctor in Mathematics HSEAcademic supervisor:Valentin Konakov Doctor ofScience, professorMoscow – 2018Modelling of many natural phenomena is still a challenging task.

Data and observations which we receive from the real word usually contain a lot of inaccuraciesand noisy factors. Using deterministic models only often makes predictions inefficient and imprecise. Thus, researchers in many fields are forced to apply modelswith additional randomness inside.A possible way to model the uncertainty is to describe dynamics of the processin terms of Stochastic Differential Equations (SDEs further). We are interested instudying Brownian SDEs of the following formZ tZ tZt = z +b(s, Zs )ds +σ(s, Zs )dWs ,(1)00kwhere (Ws≥0 ) is an R -valued Brownian motion on some filtered probability space(Ω, F, (Ft )t≥0 , P), Zt is Rm valued, with m ∈ N possibly different of k. Thecoefficients b, σ are Rm and Rm ⊗ Rk valued respectively and such that a uniqueweak solution to (1) exists.Equation (1) appears in many applied fields varied from physics to finance.

Letus mention Hamiltonian mechanics [Tal02], financial mathematics [JYC10] andbiologic simple epidemic model ([Bai17]; [BY89]).Except from some very specific cases, the SDE (1) cannot be solved explicitlyand it therefore seems natural to investigate some related approximation procedures. The Euler - Maruyama method (usually simply called the Euler method),introduced in the current SDE framework in [Mar55], is still one of the simplesteffective computational methods. Let us fix a finite time horizon T > 0. For agiven integer N , representing the number of time steps to be considered along thetime interval [0, T ], introducing the time step h = T /N and for all t ∈ [0, T ]:ZthZ=z+thb(φ(s), Zφ(s))dsZ+0thσ(φ(s), Zφ(s))dWs ,(2)0Studying the accuracy of the approximation of the scheme proposed in (2) forthe initial SDE (1) two main types of errors are usually considered.

The first oneto be investigated (see e.g. [Mar55], Gikhman and Skorokhod [GS67], [GS82]) wasthe so-called strong error. Namely, for all p ∈ [1, +∞), with the usual Markoviannotations for the processes Zsh , Zs , started from z at the moment 0 it holds:1/pES (T, z, h, p) := Ez [ sup |Zsh,0,z − Zs0,z |p ].(3)s∈[0,T ]When the coefficients in (1) are Lipschitz continuous in space and at least 1/2Hölder continuous in time, it is easily seen from usual stochastic analysis techniques, namely Itô’ s formula Burkholder-Davis-Gundy inequalities and the Gronwall, Lemma that:∃Cp (T, b, σ), ES (T, z, h, p) ≤ Cp (T, b, σ)h1/2 .1On the other hand, in many applications, such as the pricing and hedging offinancial derivatives, only so called weak error, introduced in (1) and (2), is ofinterest.

For a suitable test function f (we remain here a bit vague about thefunction space to which f belongs to), one introduces:EW (T, z, h, f ) := Ez [f (ZTh,0,z )] − Ez [f (ZT0,z )].(4)There are two sets of assumptions which guarantee that the convergence rate forEW (T, z, h, f ) is actually of order h. Namely, if(i) b, σ, f are smooth and without any specific non-degeneracy assumptionsor(ii) b, σ enjoy some structure property (i.e.

the generator associated with (1) iselliptic or hypoelliptic) and some smoothness, and for f that enjoys suitable growthconditions (and that can even be a Dirac mass)then|EW (T, z, h, f )| = |Ez [f (ZTh )] − Ez [f (ZT )]| ≤ C(T, f, σ, b)h.(5)In the both cases the main tool for the analysis is the correspondence betweenEz [f (ZTh )] and the solution of a second order parabolic PDE.

This correspondenceis provided by the Feynman-Kac representation formula. Precisely, under the aboveassumptions we have that, with the usual Markovian notations, v(t, z) := E[f (ZTt,z )]solves((∂t + Lt )v(t, z) = 0, (t, z) ∈ [0, T [×Rm ,(6)v(T, z) = f (z), z ∈ Rm ,where1 Lt v(t, z) = hb(t, z), ∇z v(t, z)i + Tr a(t, z)Dz2 v(t, z) , a(t, z) := σσ ∗ (t, z),2is the generator associated with (1).

Assuming some smoothness on v, one can2writeEW (T, z, h, f )= E[f (ZTh,0,z )] − E[f (ZT0,z )] =N−1XE[v(ti+1 , Zth,0,z) − v(ti , Zth,0,z)]i+1i(7)i=0=N−1XhZEti+1tii=0n∂s v(s, Zsh,0,z ) + ∇z v(s, Zsh,0,z )b(ti , Zth,0,z)i−1 h Z ti+1 no i NXoi1h,0,zTr(Dz2 v(s, Zsh,0,z )a(ti , Zth,0,z))ds=E∂v+Lv(Z)dssssi2tii=0h Z ti+1 n+ E∇z v(s, Zsh,0,z ) · (b(ti , Zth,0,z) − b(s, Zsh,0,z ))i+tio i1h,0,zTr(Dz2 v(s, Zsh,0,z )(a(ti , Zth,0,z)−a(s,Z)))ds+si2N−1 h Z ti+1 nXE∇z vε (s, Zsh,0,z ) · (b(ti , Zth,0,z) − b(s, Zsh,0,z ))=ii=0+tio i1Tr(Dz2 v(s, Zsh,0,z )(a(ti , Zth,0,z) − a(s, Zsh,0,z ))) ds ,i2exploiting the PDE satisfied by v for the last equality and Itô formula for the thirdequality. For a functionf in Cb2+β (Rk , R), ∀β ∈ (0, 1] the spatial derivatives of vup to order two are globally bounded on [0, T ].

Through Taylor like expansions,whenever (i) or (ii) holds, one can control (8), deriving that each contribution in(8) has the order h2 . This leads to the error of order h achieved after summingfrom 0 to N − 1.In case (i), which was considered among the others in the seminal paper byTalay and Tubaro [TT90], the smoothness of v is simply derived via stochastic flowtechniques. In case (ii) let us mention that in the hypoelliptic setting (see Section4.1.1 for additional details on hypoellipticity), Bally and Talay [BT96a], [BT96b]established (5) for bounded Borel functions f and Dirac masses.

It respectivelybases on the controls of Kusuoka and Stroock [KS84], [KS85] for the derivatives ofthe density of the diffusion process. We carefully mention that, for this method,which anyhow allows to consider a broad class of potential degeneracies, to apply,the coefficients are assumed to be smooth. The estimates on the tangent processesand Malliavin matrices in the works by Kusuoka and Stroock indeed require sucha smoothness. In the uniformly elliptic case yet another approach has been developed by Konakov and Mammen [KM00], [KM02] which is based on parametrixexpansions.Parametrix expansions, which roughly consists in approximating the density ofa process with variable coefficients by the density of the corresponding dynamicswith constant coefficients, have been a successful tool in many fields.

In particular,when a good proxy is available (which is, for instance, the case the coefficients bad σ in (1) are non-degenerate and bounded), parametrix allowed to derive the3(8)controls required for the analysis of the weak error under rather mild assumptions. We can mention the work of Il’in et al. [IKO62], who derived Gaussianheat kernel for the density of (1) for bounded Hölder coefficients when σσ ∗ isnon-degenerate. Similar bounds have been successfully exploited by Konakov andMenozzi [KM17] to derive, in the non-degenerate Hölder continuous setting, thatfor b, σ ∈ C γ/2,γ ([0, T ], Rk ), γ ∈ (0, 1] and f ∈ C β (Rk , R), β ∈ (0, 1]:|EW (T, z, h, f )| = |Ez [f (Zsh )] − Ez [f (Zs )]| ≤ C(T, f, σ, b)hγ/2 ,(9)improving the previous result by Mikulevičius and Platen [MP91] who also obtainedthe bound (9) for a function f ∈ C 2+γ (Rk , R).

This additional smoothness wasdue to the fact that they based their analysis on the associated Schauder estimates(which could already be found in [IKO62]). Going directly to the heat-kernel allowsto notably alleviate the smoothness assumptions on the final condition, which mightbe useful for applications.Intuitively, the above convergence rate can be explained by the fact that, in thelow regularity setting, the terms of order greater than one in the telescopic sum(7) cannot be expanded much further. Namely, we can only exploit the γ-Höldercontinuity of the coefficients which leads to an error controlled by the incrementshhE[|b(s, Zsh ) − b(φ(s), Zφ(s))|] + E[|a(s, Zsh ) − a(φ(s), Zφ(s))|] ≤ C(b, σ)hγ/2 .In other words, the convergence rate is closer to the one associated with the strongerror in (3).For many applications, e.g.

for neuro-sciences or diffusions in random media,it is far important to handle rougher coefficients, for instance piecewise smoothdrifts in (1). In that case, the previously mentioned heat-kernel bounds do nothold. Motivated by the investigation of the related weak error for Dirac massestest functions, we have developed, with V. Konakov and S. Menozzi, a sensitivityanalysis of the density of (1) (when suitable good Gaussian bounds exist) with respect to a perturbation of the coefficients. This is the first main result of the Thesiswhich led to the publication [KKM17] and is thoroughly developed in Chapter 3.Namely, let us introduce the SDE of the form:dXt = b(t, Xt )dt + σ(t, Xt )dWt , t ∈ [0, T ],dddd(10)dwhere b : [0, T ] × R → R , σ : [0, T ] × R → R ⊗ R are bounded coefficientsthat are measurable in time and Hölder continuous in space (this last conditionwill be relaxed for the drift term b).

Also, a(t, x) := σσ ∗ (t, x) is assumed to beuniformly elliptic. In particular, those assumptions guarantee that (10) admits aunique weak solution, see e.g. Bass and Perkins [BP09], [Men11], from which theuniqueness to the martingale problem for the associated generator can be derivedunder the current assumptions.We now introduce, for a given parameter ε > 0, a perturbed version of (10)with dynamics:(ε)dXt(ε)(ε)= bε (t, Xt )dt + σε (t, Xt )dWt , t ∈ [0, T ],4(11)where bε : [0, T ] × Rd → Rd , σε : [0, T ] × Rd → Rd ⊗ Rd satisfy at least the sameassumptions as b, σ and being meant to be close to b, σ when ε is small.It is known that, under the previous assumptions, densities of processes (Xt )t≥0 ,(ε)(Xt )t≥0 exist and satisfy some Gaussian bounds, see e.g Aronson [Aro59] or[DM10] for extensions to some degenerate cases.In the Chapter 3 we investigate, applying the parametrix technique, how thecloseness of (bε , σε ) and (b, σ) is reflected on the respective densities of the associated processes.

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