Ivan G. Avramidi. Heat Kernel Method. and its Applications. July 13, Springer

Ivan G. Avramidi Heat Kernel Method and its Applications July 13, 2015 Springer

To my wife Valentina, my son Grigori, and my parents

Preface I am a mathematical physicist. I have been working in mathematical physics over thirty years. The primary focus of my research, until recently, has been developing advanced methods of geometric analysis and applying them to quantum theory. A financial industry practitioner might ask a natural question: Is there anything useful a mathematical physicist can tell me? Well, I asked myself the same question when I got an email from Michel Crouhy, the head of Research and Development at NATIXIS Corporate and Investment Bank in Paris, inviting me to present a series of lectures for the members of his group. Very soon, with the help of Olivier Croissant, I realized that one of the major problems of quantitative finance, at least in option pricing theory, is the problem of finding the solution of a partial differential equation of parabolic type called a generalized diffusion equation (or heat equation). This is exactly what I have been doing my whole life, and that was exactly the reason why Michel Crouhy asked me to explain to his quants what is the heat kernel and how one can compute it, at least approximately, to price options. This book grew out of these lectures. I believe it might be useful for other quants too as well as for physicists, applied mathematicians and engineers, in fact, anybody who is concerned with the need to solve parabolic partial differential equations. The book consists of four parts: Analysis, Geometry, Perturbations and Applications. In the first part after a short review of some background material I present an introduction to partial differential equations. The second part is devoted to a short introduction to various aspects of differential geometry that will be needed later. The third part is devoted to a systematic development of effective methods for various approximation schemes for parabolic differential equations that make an extensive use of differential geometric and analytical concepts introduced earlier. The heart of the book is the development of a short-time asymptotic expansion for the heat kernel. I explain it in details and give explicit examples of some advanced calculations. We also discuss some advanced methods and extensions, including path integrals, jump diffusion and others. In the forth part I start with a short introduction to vii

viii Preface financial mathematics, in particular, stochastic differential equations and the description of some basic models. I show that all these models, including stochastic volatility models, lead to a valuation equation for the option price, which is nothing but a second order partial differential equation of parabolic type. I demonstrate how the advanced perturbational techniques can be applied to some models of mathematical finance. A remark about the level and the style of the presentation is in order. Since most of the time I start from scratch, the level of the presentation is necessarily uneven. I start from very elementary introductory concepts (that should be boring for a specialist) and go pretty quickly to rather advanced technical methods needed for our purposes. So, it is normal if you are bored at the beginning and lost at the end. Also, I intentionally sacrifice rigor for clarity and accessibility for a wider audience. So, the style is rather informal. Most of the time I do not discuss and state precise conditions (which are, of course, of primary interest for pure mathematicians) under which the statements and results are valid. I provide some references to the original papers and books that I found useful. However, the subject is so huge that it is impossible to give a more or less comprehensive review of the literature. No such attempt has been made. This book is a tutorial for non-specialists rather than a review of the area for the experts. Socorro, July, 2015 Ivan Avramidi

Acknowledgements I would like to express my gratitude to many friends, collaborators and colleagues but I will single out just a few people who had an enormous influence on my education and my career: my PhD advisor Vladislav Khalilov, my postoctoral mentor Julius Wess as well as Thomas Branson, Peter Gilkey, Bryce De Witt, Stuart Dowker and Stephen Fulling from whom I learned a lot of material in this book. I am especially indebted to Michel Crouhy and Olivier Croissant who inspired my interest in quantitative finance and without whom this book certainly would never have been written. Most importantly, I thank my wife, Valentina, for constant support and encouragement. ix

Contents Part I Analysis 1 Background in Analysis................................... 3 1.1 Asymptotic Expansions................................. 3 1.2 Gaussian Integrals...................................... 5 1.3 Laplace Integrals....................................... 8 1.4 Fourier Transform...................................... 12 1.5 Laplace Transform...................................... 14 1.6 Mellin Transform....................................... 15 1.7 Derivative of Determinants.............................. 18 1.8 Hamiltonian Systems................................... 19 1.9 Hilbert Spaces......................................... 21 1.10 Functional Spaces...................................... 22 1.11 Self-Adjoint and Unitary Operators....................... 24 1.12 Integral Operators...................................... 27 1.13 Resolvent and Spectrum................................. 28 1.14 Spectral Resolution..................................... 29 1.15 Functions of Operators.................................. 30 1.16 Spectral Functions...................................... 31 1.17 Heat Semigroups....................................... 32 1.17.1 Definition and Basic Properties.................... 32 1.17.2 Duhamel s Formula and Volterra s Series............ 34 1.17.3 Chronological Exponent........................... 36 1.17.4 Campbell-Hausdorff Formula....................... 38 1.17.5 Heat Semigroup for Time Dependent Operators...... 38 1.18 Notes................................................. 41 2 Introduction to Partial Differential Equations............. 43 2.1 First Order Partial Differential Equations.................. 43 2.2 Second Order Partial Differential Equations................ 44 2.2.1 Elliptic Partial Differential Operators............... 44 xi

xii Contents 2.2.2 Classification of Second-Order Partial Differential Equations....................................... 46 2.2.3 Elliptic Equations................................ 47 2.2.4 Parabolic Equations.............................. 48 2.2.5 Hyperbolic Equations............................. 48 2.2.6 Boundary Conditions............................. 49 2.2.7 Gauss Theorem.................................. 50 2.2.8 Existence and Uniqueness of Solutions.............. 52 2.3 Partial Differential Operators............................ 52 2.3.1 Adjoint Operator................................. 53 2.3.2 Adjoint Boundary Conditions...................... 55 2.3.3 Spectral Theorem................................ 58 2.4 Heat Kernel........................................... 59 2.4.1 Heat Kernel..................................... 59 2.4.2 Heat Kernel of Time-dependent Operators........... 61 2.4.3 Cauchy Problem................................. 63 2.4.4 Boundary Value Problem.......................... 63 2.5 Differential Operators with Constant Coefficients........... 66 2.5.1 Ordinary Differential Equations.................... 66 2.5.2 Indegro-Differential Equations..................... 68 2.5.3 Elliptic Partial Differential Operators............... 72 2.5.4 Parabolic Partial Differential Equations............. 75 2.5.5 Ordinary Differential Equations on Half-line......... 76 2.6 Differential Operators with Linear Coefficients............. 79 2.7 Homogeneous Differential Operators...................... 82 2.8 Notes................................................. 84 Part II Geometry 3 Introduction to Differential Geometry.................... 87 3.1 Differentiable Manifolds................................. 87 3.1.1 Basic Definitions................................. 87 3.1.2 Vector Fields.................................... 88 3.1.3 Covector Fields.................................. 90 3.1.4 Riemannian Metric............................... 90 3.1.5 Arc Length...................................... 92 3.1.6 Riemannian Volume Element...................... 93 3.1.7 Tensor Fields.................................... 94 3.1.8 Permutations of Tensors........................... 94 3.1.9 Einstein Summation Convention.................... 96 3.1.10 Levi-Civita Symbol............................... 97 3.1.11 Lie Derivative.................................... 99 3.2 Connection............................................ 100 3.2.1 Covariant Derivative.............................. 100 3.2.2 Parallel Transport................................ 101

Contents xiii 3.2.3 Geodesics....................................... 104 3.3 Curvature............................................. 105 3.3.1 Riemann Tensor.................................. 105 3.3.2 Properties of Riemann Tensor...................... 105 3.4 Geometry of Two-dimensional Manifolds.................. 108 3.4.1 Gauss Curvature................................. 108 3.4.2 Two-dimensional Constant Curvature Manifolds...... 110 3.5 Killing Vectors......................................... 114 3.6 Synge Function........................................ 115 3.6.1 Definition and Basic Properties.................... 115 3.6.2 Derivatives of Synge Function...................... 118 3.6.3 Van Vleck-Morette Determinant.................... 122 3.7 Operator of Parallel Transport........................... 124 3.7.1 Definition and Basic Properties.................... 124 3.7.2 Derivatives of the Operator of Parallel Transport..... 125 3.7.3 Generalized Operator of Parallel Transport.......... 126 3.8 Covariant Expansions of Two-Point Functions.............. 127 3.8.1 Coincidence Limits of Higher-Order Derivatives...... 127 3.8.2 Covariant Taylor Series........................... 129 3.8.3 Normal Coordinates.............................. 132 3.8.4 Covariant Taylor Series of Two-Point Functions...... 134 3.8.5 Two-point Functions in Symmetric Spaces........... 141 3.9 Parallel Orthonormal Frame............................. 146 3.10 Lie Groups............................................ 147 3.11 Lie Algebras........................................... 149 3.12 Matrix Lie Groups...................................... 152 3.13 Geometry of Lie Groups................................. 154 3.14 Geometry of Symmetric Spaces........................... 158 3.14.1 Algebraic Structure of Curvature Tensor............ 158 3.14.2 Killing Vectors Fields............................. 160 3.14.3 Lie Derivatives................................... 161 3.15 Geometric Interpretation of Partial Differential Operators... 162 3.15.1 Laplace Type Operators........................... 162 3.15.2 Self-adjoint Operators............................. 165 3.16 Notes................................................. 166 Part III Perturbations 4 Singular Perturbations.................................... 169 4.1 Motivation............................................ 169 4.2 Semiclassical Approximation............................. 170 4.2.1 Semi-classical Ansatz............................. 170 4.2.2 Hamilton-Jacobi Equation......................... 172 4.2.3 Hamiltonian System.............................. 173 4.2.4 Transport Equations.............................. 176

xiv Contents 4.3 Singularly Perturbed Heat Equation...................... 178 4.3.1 Asymptotic Ansatz............................... 178 4.3.2 Hamilton-Jacobi Equation and Hamiltonian System.. 180 4.3.3 Action.......................................... 181 4.3.4 Transport Equations.............................. 183 4.3.5 Operators with Constant Coefficients............... 184 4.3.6 Quadratic Hamiltonians........................... 185 4.4 Singular Perturbations of Time-dependent Operators........ 194 4.5 Notes................................................. 196 5 Heat Kernel Asymptotics................................. 197 5.1 Asymptotic Ansatz..................................... 197 5.2 Minackshisundaram-Pleijel Expansion..................... 202 5.3 Recurrence Relations................................... 205 5.4 Green Function........................................ 205 5.5 Non-recursive Solution of Recurrence Relations............. 208 5.6 Matrix Elements....................................... 209 5.7 Diagrammatic Technique................................ 212 5.8 Heat Kernel Coefficients for Constant Curvature............ 215 5.9 Heat Kernel Coefficients in One Dimension................ 219 5.10 Heat Kernel Asymptotics of Time-dependent Operators..... 221 5.11 Boundary Value Problems............................... 227 5.11.1 Geometry of the Boundary........................ 227 5.11.2 Boundary Conditions............................. 230 5.11.3 Interior Heat Kernel.............................. 231 5.11.4 Heat Kernel Near Boundary....................... 233 5.11.5 Method of Geodesics Reflected from the Boundary.... 235 5.12 Notes................................................. 238 6 Advanced Topics......................................... 239 6.1 Various Approximation Schemes.......................... 239 6.2 Leading Derivatives in Heat Kernel Diagonal............... 243 6.3 Fourier Transform Method............................... 248 6.3.1 Non-covariant Fourier Transform................... 248 6.3.2 Covariant Fourier Transform....................... 249 6.4 Long Time Behavior of the Heat Kernel................... 252 6.5 Quantum Operator Method.............................. 254 6.5.1 General Framework............................... 254 6.5.2 Linear Connection................................ 257 6.5.3 Harmonic Oscillator.............................. 259 6.5.4 General Systems with Linear Heisenberg Equations... 261 6.6 Algebraic Methods..................................... 265 6.6.1 Linear Connection in Flat Space................... 268 6.6.2 Linear Connection with Quadratic Potential......... 270 6.7 Heat Kernel on Semi-Simple Lie Groups................... 277

Contents xv 6.7.1 Heat Kernel on H 3 and S 3........................ 278 6.7.2 Heat Kernel on the Hyperbolic Space H n............ 279 6.7.3 Covariantly Constant Fields....................... 282 6.7.4 Heat Semi-group................................. 284 6.7.5 Isometries....................................... 285 6.7.6 Heat Kernel..................................... 288 6.7.7 Hyperbolic Plane H 2............................. 290 6.7.8 Sphere S 2....................................... 292 6.7.9 Duality of H 2 and S 2............................. 294 6.8 Heat Kernel of Non-Selfadjoint Operators................. 295 6.9 Path Integrals.......................................... 296 6.9.1 Discretization.................................... 296 6.9.2 Formal Expression................................ 297 6.9.3 Perturbation Theory.............................. 300 6.9.4 Gaussian Path Integrals........................... 302 6.10 Notes................................................. 304 Part IV Applications 7 Stochastic Processes...................................... 307 7.1 Stochastic Processes.................................... 307 7.1.1 Basic Concepts of Probability...................... 307 7.1.2 Wiener Process.................................. 315 7.1.3 Poisson Process.................................. 317 7.2 Stochastic Calculus..................................... 319 7.2.1 Stochastic Differential Equations................... 319 7.2.2 Change of Variables and Itô s Lemma............... 321 7.2.3 Conditional Probability Density.................... 324 7.3 Notes................................................. 328 8 Applications in Mathematical Finance.................... 329 8.1 Derivatives............................................ 329 8.1.1 Financial Instruments............................. 329 8.1.2 Options......................................... 330 8.2 Models in Mathematical Finance......................... 340 8.2.1 Quantitative Analysis............................. 340 8.2.2 Black-Scholes Model.............................. 341 8.2.3 Higher-Dimensional Black-Scholes Model............ 344 8.2.4 Beyond Black-Scholes............................. 345 8.2.5 Deterministic Volatility Models.................... 346 8.3 Stochastic Volatility Models............................. 346 8.4 Two-dimensional Stochastic Volatility Models.............. 349 8.4.1 Heston Model.................................... 350 8.4.2 Hull-White Model................................ 350 8.4.3 GARCH Model.................................. 351

xvi Contents 8.4.4 Ornstein-Uhlenbeck Model........................ 351 8.4.5 SABR Model.................................... 351 8.4.6 SABR Model with Mean-Reverting Volatility........ 352 8.5 Jump Diffusion Models.................................. 352 8.5.1 Jumps Probability Density........................ 352 8.5.2 Stochastic Volatility Model with Jumps............. 354 8.6 Solution of Two-Dimensional Models...................... 357 8.6.1 Black-Scholes Model.............................. 357 8.6.2 Higher Dimensional Black-Scholes Model............ 359 8.6.3 Two-dimensional Stochastic Volatility Models........ 361 8.6.4 Models on Hyperbolic Plane....................... 366 8.6.5 Heston Model.................................... 370 8.7 Notes................................................. 377 Summary..................................................... 378 References.................................................... 381 Index......................................................... 385

Notation t = t Partial derivative with respect to time Ṡ = t S Dot denotes time derivative i = x Partial derivative with respect to space variables i i Covariant derivative A i = i + A i Generalized covariant derivative X Directional covariant derivative along a vector field X N Normal derivative at the boundary ( Asymptotic equivalence m n) Binomial coefficient p, x Standard pairing between dual vector spaces (f, ϕ) Inner product f = (f, f) Norm f Complex conjugate dx = dx 1 dx n Lebesgue measure on R n dx g 1/2 Riemannian volume element Dx(τ) Path integral measure M Boundary of the manifold M T (i1,...i p) Symmetrization of a tensor T [i1,...i p] Anti-symmetrization of a tensor [f(x, x )] = f(x, x) Coincidence limit of a two-point function A Adjoint operator (or Hermitian conjugate matrix) A T Transposed matrix [X, Y ] Lie bracket (commutator) x + = max (x, 0) Nonnegative maximum function δ(x) Dirac delta function δ(x, x ) Covariant delta function δ ij = δ ij = δ i j Kronecker symbol Δ Laplacian Δ(x, x ) Van Vleck-Morette determinant ε j1...j n 1 Levi-Civita symbol Γ (s) Gamma function xvii

xviii Notation Γ i jk Christoffel symbols θ(x) Heaviside step function ρ(x i, X j ) Correlation matrix σ(x, p) Symbol of a partial differential operator σ(x, x ) Synge function σ(x) Standard deviation χ B (x) Characteristic function of a set B ψ(x) = Γ (x)/γ (x) Digamma function A i Generalized connection C = 0.577... Euler s constant C i jk Structure constants of a Lie group Cov(X i, X j ) Covariance matrix d(x, x ) Geodesic distance between x and x E(X) Expected value E(X A) Conditional expected value of X given A f X (x) Probability density function F X (x) Cumulative distribution function g ij Riemannian metric tensor g = det g ij Determinant of the metric g ij Inverse matrix of the metric ĝ µν Induced Riemannian metric on the boundary ĝ = det ĝ µν Determinant of the induced metric G(λ) Resolvent G(λ; x, x ) Resolvent kernel H n n-dimensional hyperbolic space I Identity operator K Gaussian curvature L X Lie derivative along a vector field X L 2 (M, µ) Hilbert space of square integrable functions on a manifold M with the weight µ N i Inward pointing normal vector to the boundary p X (t, x; t, x ) Conditional probability density function (transitional distribution) P (A) Probability of an event A P (A B) Conditional probability of an event A given B P = (g i j ) Operator of parallel transport along the geodesic P(x, x ) Generalized operator of parallel transport along the geodesic R i jkl Riemann tensor R ij Ricci tensor R Scalar curvature R ij Curvature of the connection A i S n n-dimensional sphere SO(n) Special orthogonal group SO(1, n) Special pseudo-orthogonal group

Notation xix SU(n) U(t) U(t, t ) U(t; x, x ) U(t, x; t, x ) Var(X) x = (x 1,..., x n ) ˆx = (ˆx 1,..., ˆx n 1 ) X t Special unitary group Heat semi-group Heat semi-group of a time-dependent operator Heat kernel Heat kernel of a time-dependent operator Variance Local coordinates on a manifold Local coordinates on the boundary of a manifold Stochastic process