# Probabilistic representation and integration by parts forumulae for some stochastic volatility model with unbounded drift

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## Structure of the market and relative arbitrage

It is possible to describe the complete structure of the market by using the couple (t;t) and introducing the process of total market capitalization, or the size of the market portfolio (t)t0, which could be correlated to the market weights process through the volatility coefficients (;())d =1: dt = b(t)dt + Xd =0 ;(t)dW t ; 0 > 0.
The capitalization of the i-th asset is simply Si t = i t t. By using logarithm representation, the decomposition of growth rate is then obtained: d log Si t = d log i t + d log t = ( i() + ())dt + martingale The growth rate is decomposed into i(), the part associated to the weight of the i-th asset and into () which is the growth of total market capitalization common to all assets.
By taking the total capitalization equal to constant, we obtain the dynamics of the logarithm of market weights of the i-th asset: d log i t = i()dt + Xd =1 (i t)􀀀1i;(t)dW. Recall that the rank of the matrix & is n so ci = Pd =1(&i;)2 > 0. For 2 (0; 1), when the weight i t of the i-th weight is small, the first term after the last equality (i t)2(􀀀1)(1 􀀀 2i t)ci is the dominant one and i(t) ! 1 when i t ! 0, assets with smaller weights tend to have greater volatilities as in volatility-stabilized market models [Pic14], stabilization by volatility is then asymptotic. The smaller the exponent 2 (0; 1), the larger the first term, the higher the effect of volatility stabilization. When 1, the total volatility i(t) is bounded for t 2 n, the asymptotic stabilization by volatility is not observed in general.
In classical stochastic portfolio theory, we define the set of self-financing trading strategies T (S) as Rn-valued processes, which are predictable and integrable with respect to the process of capitalizations. Here by using the numéraire of market portfolio (t)t0, the value of the portfolio (V t )t0, also called wealth process, can be written in terms of market weights (t)t0, with strategy = (t)t0 2 T (). According to [KR17], the set of self-financing trading strategies is the same with respect to market capitalizations and relative weights of the assets, i.e. T () = T (S). For the sake of simplicity, by normalization we can suppose the initial value of the portfolio equal to 1. V t = t t = Xn i=1 i ti t; V  = 1.

### A probabilistic scheme for McKean-Vlasov equations

The second part is dedicated to the numerical approximation of the marginal law of the following McKean-Vlasov differential equation: Xs; t = + Z t s b(Xs; r ; [Xs; ]) dr + Z t s (Xs; r ; [Xs; r ])dWr; [] = 2 P(Rd): (2.2.1) On the probability space ( ;A;P(Rd)), W is a Brownian motion of dimension q, the coefficients b; are defined on Rd and take values in Rd and Rdq respectively. Here [] is the law of a random variable . Under weak regularity assumptions on b and a = , i.e. if they are bounded, Hölder in the space variable and have functional linear derivatives with respect to the measure variable, denoted as b=m and a=m, such that b=m is bounded, (x; y) 7! [a(x;m)=m](y) is Hölder (uniformly with respect to variable m), and finally if the diffusion coefficient a is uniformly elliptic: 9 1; 8(x; u;m) 2 (Rd)2P(Rd), 􀀀1juj2 ha(x;m)u; ui juj2, then the McKean- Vlasov SDE (2.2.1) has a unique weak solution for every initial condition (s; ) 2 R+ P(Rd), see e.g. Chaudru de Raynal and Frikha [RF19].

#### Integration by parts formulae for sensitivities

Delta (respectively Vega) is defined by the derivative with respect to the spot price (respectively the volatility) of the underlying asset. We establish Bismut-Elworthy-Li type formulae for the following quantities which could be seen as Delta and Vega at initial time t = 0, with s0 = exp(X0); y0 = Y0: @s0E[h(XT ; YT )] and @y0E[h(XT ; YT )].
The goal is then to write the derivatives of expectations above as expectations of some functions. First we use the probabilistic representation for the expectation. The next step is to do the “transfer of the derivative » inside the expectation using an integration by parts formula, but due to some integrability issues, we rather apply local IBP formulae to each of the random time intervals [i; i+1], i = 0; ;NT . Finally, the derivatives of expectations, or Delta and Vega, can be written as weighted sums of local IBP formulae, the weights being equal to the lengths of random intervals. Main Result 2.3.2. Under the previous assumptions of regularity and uniform ellipticity, the law of the couple (XT ; YT ) at time T satisfies the following Bismut-Elworthy-Li type formulae. There exists > 0 such that for all h 2 B (R2) and all (s0; y0) 2 R2:

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1 Introduction – Français
1.1 Modélisation des capitalisations relatives des actifs
1.1.1 Processus et distribution des poids du marché
1.1.2 Structure du marché et arbitrage relatif
1.1.3 Applications et perspectives
1.2 Un schéma probabiliste pour les équations de McKean-Vlasov
1.2.1 Itération de Picard et méthode de continuation
1.2.2 Représentation probabiliste de la loi marginale
1.2.3 Analyse de la convergence et de l’erreur et perspectives
1.3 Représentation probabiliste et formules d’intégration par parties pour certains modèles à volatilité stochastique avec drift non-borné
1.3.1 Représentation probabiliste du couple spot-volatilité
1.3.2 Formules d’intégration par parties
1.4 Méthodes numériques pour les EDSPRs issues des jeux à champ moyen .
1.4.1 Jeux à champ moyen et approche probabiliste
1.4.2 Algorithmes d’arbre et de grille pour les EDSPRs
2 Introduction – English
2.1 Modeling of relative capitalizations of assets
2.1.1 Process and distribution of market weights
2.1.2 Structure of the market and relative arbitrage
2.1.3 Applications and perspectives
2.2 A probabilistic scheme for McKean-Vlasov equations
2.2.1 Picard iteration and method of continuation
2.2.2 Probabilistic representation of marginal law
2.2.3 Convergence and error analysis and perspectives
2.3 Probabilistic representation and integration by parts forumulae for some stochastic volatility model with unbounded drift
2.3.1 Probabilistic representation of spot-volatility couple
2.3.2 Integration by parts formulae for sensitivities
2.4 Numerical methods for FBSDEs arising from mean-field games
2.4.1 Mean-field games and probabilistic approach
2.4.2 Tree and grid algorithms for FBSDEs
I Modeling the market by capital distribution
3 Modeling the market by capital distribution
3.1 Introduction
3.2 Equation and process of market weights
3.2.1 Market weights equation and its first properties
3.2.2 Distribution and transition density of market weights
3.2.3 Stationary distribution of market weights
3.3 Structure of market and relative arbitrage
3.3.1 Market portfolio and growth rate
3.3.2 Trading strategy and relative arbitrage
3.3.3 Conditions of arbitrage relative to the market
3.4 Future works and conclusion
3.4.1 Functionally generated portfolio
3.4.2 Estimation of model parameters with application to stock market indices
3.4.3 Long-term portfolio optimization under market weights equation
3.4.4 Conclusion and perspectives
3.5 Appendix
3.5.1 Attainment of boundary for Theorem 3.2.1
3.5.2 Hörmander’s condition and theorem for Proposition 3.2.2
3.5.3 Choice of Lyapunov function V for Theorem 3.2.2
3.5.4 Generalized local martingale problem and candidate measure for
II A Probabilistic Scheme for McKean-Vlasov Equations
4 A Probabilistic Scheme for McKean-Vlasov Equations
4.1 Introduction
4.2 Description of the numerical probabilistic scheme
4.2.1 Assumptions and well-posedness of the McKean-Vlasov SDE
4.2.2 Construction of the Picard iteration scheme
4.2.3 Probabilistic representation of the marginal law of the Picard iteration scheme
4.2.4 Construction of the Monte Carlo estimator
4.3 Convergence Analysis
4.3.1 Decomposition of the global error and complexity of the algorithm
4.3.2 Convergence analysis of the local Picard iteration schemes .
4.3.3 Convergence analysis of the global Picard iteration scheme on [0; T]
4.4 Numerical results
4.4.1 Standard linear model
4.4.2 Kuramoto model
4.4.3 Polynomial drift model
4.5 Appendix
4.5.1 Proof of Theorem 4.2.1
4.5.2 Proof of Proposition 4.3.1
III Probabilistic Representation and Integration by Parts Formulae for some Stochastic Volatility Models
5 Probabilistic Representation and Integration by Parts Formulae for some Stochastic Volatility Models
5.1 Introduction
5.2 Preliminaries: assumptions, definition of the underlying Markov chain and related Malliavin calculus
5.2.1 Assumptions
5.2.2 Choice of the approximation process
5.2.3 Markov chain on random time grid
5.2.4 Tailor-made Malliavin calculus for the Markov chain
5.3 Probabilistic representation for the couple (ST ; YT )
5.4 Integration by parts formulae
5.4.1 The transfer of derivative formula
5.4.2 The integration by parts formulae
5.5 Numerical Results
5.5.1 Black-Scholes Model
5.5.2 A Stein-Stein type model
5.5.3 A model with a periodic diffusion coefficient function
5.6 Proof of Theorem 5.3.1 and Lemma 5.4.1
5.6.1 Proof of Theorem 5.3.1
5.6.2 Proof of Lemma 5.4.1
5.7 Some technical results
5.7.1 Emergence of jumps in the renewal process N
5.7.2 Formulae for the computations of price, Delta and Vega
IV Probabilistic Numerical Methods for Mean-Field Games
6 Probabilistic Numerical Methods for Mean-Field Games
6.1 Introduction
6.2 Overview of Mean Field Games and FBSDEs
6.2.1 N Player Stochastic Differential Games
6.2.2 Mean Field Games
6.2.3 General System
6.3 Algorithms
6.3.1 Global Picard Iteration on a Small Time Interval
6.3.2 Continuation in Time of the Global Method for Arbitrary Interval/ Coupling
6.3.3 Tree Algorithm for the Global Method
6.3.4 Picard Iteration on the Marginal Laws: a Grid Algorithm
6.4 Examples
6.4.1 Linear Example
6.4.2 Trigonometric Drivers Example
6.4.3 Mixed Model
6.4.4 Examples: Linear Quadratic Mean Field Games
6.5 Conclusion
Bibliography

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