Stochastic evolution equations for large portfolios of stochastic volatility models

TL;DR

This paper develops a stochastic partial differential equation framework for modeling large portfolios of defaultable assets with stochastic volatility, analyzing their limiting behavior and density properties.

Contribution

It introduces a measure-valued process limit for large portfolios with correlated stochastic volatility and default risk, using advanced calculus and smoothing techniques.

Findings

Existence of a large portfolio limit as a measure-valued process

Derivation of a stochastic PDE with a density solution

Regularity results for the density, but open problem of uniqueness

Abstract

We consider a large market model of defaultable assets in which the asset price processes are modelled as Heston-type stochastic volatility models with default upon hitting a lower boundary. We assume that both the asset prices and their volatilities are correlated through systemic Brownian motions. We are interested in the loss process that arises in this setting and we prove the existence of a large portfolio limit for the empirical measure process of this system. This limit evolves as a measure valued process and we show that it will have a density given in terms of a solution to a stochastic partial differential equation of filtering type in the two-dimensional half-space, with a Dirichlet boundary condition. We employ Malliavin calculus to establish the existence of a regular density for the volatility component, and an approximation by models of piecewise constant volatilities combined with a kernel smoothing technique to obtain existence and regularity for the full two-dimensional filtering problem. We are able to establish good regularity properties for solutions, however uniqueness remains an open problem.

Equations

$$d{A}_t^i$$

$$A_t^i$$

$$A_0^i$$

$$\begin{array}[]{rcl}dA_{t}^{i}&=&A_{t}^{i}{\mu}_{i}dt+A_{t}^{i}h\left(\sigma_{t}^{i}\right)\left(\sqrt{1-\rho_{1,i}^{2}}dW_{t}^{i}+\rho_{1,i}dW_{t}^{0}\right),\;0\leq t\leq T_{i}\\ d\sigma_{t}^{i}&=&k_{i}(\theta_{i}-\sigma_{t}^{i})dt+\xi_{i}\sqrt{\sigma_{t}^{i}}\left(\sqrt{1-\rho_{2,i}^{2}}dB_{t}^{i}+\rho_{2,i}dB_{t}^{0}\right),\;t\geq 0\\ A_{t}^{i}&=&b^{i},\;t>T_{i}\\ (A_{0}^{i},\,\sigma_{0}^{i})&=&(a^{i},\,\sigma^{i}),\end{array}$$

$$\begin{array}[]{rcl}dX_{t}^{i}&=&\left(r_{i}-\frac{h^{2}(\sigma_{t}^{i})}{2}\right)dt+h(\sigma_{t}^{i})\left(\sqrt{1-\rho_{1,i}^{2}}dW_{t}^{i}+\rho_{1,i}dW_{t}^{0}\right),\;\;0\leq t\leq T_{i}\\ d\sigma_{t}^{i}&=&k_{i}(\theta_{i}-\sigma_{t}^{i})dt+\xi_{i}\sqrt{\sigma_{t}^{i}}\sqrt{1-\rho_{2,i}^{2}}dB_{t}^{i}+\xi\sqrt{\sigma_{t}^{i}}\rho_{2,i}dB_{t}^{0},\;\;t\geq 0\\ X_{t}^{i}&=&0,\;t>T_{i}\\ (X_{0}^{i},\,\sigma_{0}^{i})&=&(x^{i},\sigma^{i}),\end{array}$$

$$v_t^N=\frac{1}{N}\sum _{i=1}^N{\delta }_{X_t^i,{\sigma }_t^i},$$

$$v_{1,t}^N=\frac{1}{N}\sum _{i=1}^N{\delta }_{X_t^i,{\sigma }_t^i}{\mathbb{I}}_{\{T_i>t\}}.$$

$$v_t^N\to v_t=\mathbb{P}\left(\left(X_t^1,{\sigma }_t^1\right)\in \cdot |W_{\cdot }^0,B_{\cdot }^0,\mathcal{G}\right)$$

$$v_{1,t}^N\to v_{1,t}$$

$$u_{t,C_1}(x,y)$$

$$\underset{N\to \infty }{\lim }{v}_t^N(\{0\}\times \mathbb{R})$$

$$v_{3,t,s}^N=\frac{1}{N}\sum _{i=1}^N{\delta }_{X_t^i,{\sigma }_t^i,{\sigma }_s^i}.$$

$$\left(X_t^k,{\sigma }_t^k,{\sigma }_s^k\right)=g\left(t,s,B_{\cdot }^k,W_{\cdot }^k,B_{\cdot }^0,W_{\cdot }^0,C_k,X_0^k,{\sigma }_0^k\right)$$

$$1$$

$${\int }_{{\mathbb{R}}^2}fd{v}_{3,t,s}^N\to \mathbb{E}\left[f\left(X_t^1,{\sigma }_t^1,{\sigma }_s^1\right)|W_{\cdot }^0,B_{\cdot }^0,\mathcal{G}\right]\forall t,s\in {\mathbb{Q}}^{+}$$

$${\int }_{{\mathbb{R}}^2}fd{v}_{3,t,s}=\mathbb{E}\left[f\left(X_t^1,{\sigma }_t^1,{\sigma }_s^1\right)|W_{.}^0,B_{.}^0,\mathcal{G}\right]$$

$$v_{2,t}^N={\delta }_0\sum _{i=1}^N{\delta }_{{\sigma }_t^i}{\mathbb{I}}_{\{T_i<t\}}$$

$$\mathbb{E}\left[f\left(X_t^1,{\sigma }_t^1\right){\mathbb{I}}_{\{T_1<t\}}|W_{.}^0,B_{.}^0,G\right]=\mathbb{E}\left[f\left(0,{\sigma }_t^1\right){\mathbb{I}}_{\{T_1<t\}}|W_{\cdot }^0,B_{\cdot }^0,\mathcal{G}\right],$$

$$\underset{N\to \infty }{\mathrm{liminf}}\frac{1}{N}\sum _{i=1}^{N}f\left({\sigma }_t^i\right){\mathbb{I}}_{\{T_i\le t\}}$$

$$=\underset{N\to \infty }{\mathrm{liminf}}\frac{1}{N}\left[\sum _{i=1}^{N}f\left({\sigma }_{t_1}^i\right){\mathbb{I}}_{\{T_i<t\}}+\sum _{i=1}^N\left(f\left({\sigma }_t^i\right)-f\left({\sigma }_{t_1}^i\right)\right){\mathbb{I}}_{\{T_i<t\}}\right]$$

$$\ge \underset{N\to \infty }{\mathrm{liminf}}\frac{1}{N}\left[\sum _{i=1}^{N}f\left({\sigma }_{t_1}^i\right){\mathbb{I}}_{\{T_i<t\}}-\sum _{i=1}^N\left|f\left({\sigma }_t^i\right)-f\left({\sigma }_{t_1}^i\right)\right|\right]$$

$$\ge \underset{N\to \infty }{\mathrm{liminf}}\frac{1}{N}\sum _{i=1}^{N}f\left({\sigma }_{t_1}^i\right){\mathbb{I}}_{\{T_i<t_1\}}-\underset{N\to \infty }{\mathrm{liminf}}\frac{1}{N}\sum _{i=1}^N\left|f\left({\sigma }_t^i\right)-f\left({\sigma }_{t_1}^i\right)\right|,$$

$$\underset{N\to \infty }{\mathrm{liminf}}\frac{1}{N}\sum _{i=1}^{N}f({\sigma }_t^i){\mathbb{I}}_{\{T_i\le t\}}\ge \mathbb{E}\left[f({\sigma }_t^1){\mathbb{I}}_{\{T_1<t\}}|W_{\cdot }^0,B_{\cdot }^0,\mathcal{G}\right].$$

$$\underset{N\to \infty }{\mathrm{limsup}}\frac{1}{N}\sum _{i=1}^{N}f\left({\sigma }_t^i\right){\mathbb{I}}_{\{T_i\le t\}}$$

$$=\underset{N\to \infty }{\mathrm{limsup}}\frac{1}{N}\left[\sum _{i=1}^{N}f\left({\sigma }_{t_2}^i\right){\mathbb{I}}_{\{T_i<t\}}+\sum _{i=1}^N\left(f\left({\sigma }_t^i\right)-f\left({\sigma }_{t_2}^i\right)\right){\mathbb{I}}_{\{T_i<t\}}\right]$$

$$\le \underset{N\to \infty }{\mathrm{limsup}}\frac{1}{N}\left[\sum _{i=1}^{N}f\left({\sigma }_{t_2}^i\right){\mathbb{I}}_{\{T_i<t\}}+\sum _{i=1}^N\left|f\left({\sigma }_t^i\right)-f\left({\sigma }_{t_2}^i\right)\right|\right]$$

$$\le \underset{N\to \infty }{\mathrm{limsup}}\frac{1}{N}\sum _{i=1}^{N}f\left({\sigma }_{t_2}^i\right){\mathbb{I}}_{\{T_i<t_2\}}+\underset{N\to \infty }{\mathrm{limsup}}\frac{1}{N}\sum _{i=1}^N\left|f\left({\sigma }_t^i\right)-f\left({\sigma }_{t_2}^i\right)\right|$$

$$\underset{N\to \infty }{\mathrm{limsup}}\frac{1}{N}\sum _{i=1}^{N}f\left({\sigma }_t^i\right){\mathbb{I}}_{\{T_i\le t\}}\le \mathbb{E}\left[f\left({\sigma }_t^1\right){\mathbb{I}}_{\{T_1<t\}}|W_{\cdot }^0,B_{\cdot }^0,\mathcal{G}\right].$$

$$\left|{\int }_{{\mathbb{R}}^2}f(y)d{v}_{2,t}^N(x,y)-{\int }_{{\mathbb{R}}^2}f(y)d{v}_{2,t}(x,y)\right|$$

$$\le \left|{\int }_{{\mathbb{R}}^2}(f(y)-p(y))d{v}_{2,t}^N(x,y)\right|+\left|{\int }_{{\mathbb{R}}^2}p(y)d{v}_{2,t}^N(x,y)-{\int }_{{\mathbb{R}}^2}p(y)d{v}_{2,t}(x,y)\right|$$

$$+\left|{\int }_{{\mathbb{R}}^2}(p(y)-f(y))d{v}_{2,t}(x,y)\right|$$

$$\le ||f-p|{|}_{\infty }+\left|{\int }_{{\mathbb{R}}^2}p(y)d{v}_{2,t}^N(x,y)-{\int }_{{\mathbb{R}}^2}p(y)d{v}_{2,t}(x,y)\right|+||f-p|{|}_{\infty }$$

$$\le 3\times \frac{ϵ}{3}=ϵ,$$

$${\int }_{{\mathbb{R}}^2}fd{v}_{1,t}$$

$$v_{t,C_1}\left(\cdot \right)=\mathbb{P}\left[\left(X_t^1,{\sigma }_t^1\right)\in \cdot ,T_1>t|W_{\cdot }^0,B_{\cdot }^0,C_1,\mathcal{G}\right],$$

$$Af\left(x,y\right)$$

Full text

1 Introduction

In the study of large portfolios of assets it is common to model correlation through factor models. In this setting the random drivers of individual asset prices come from two independent sources. Firstly there is an idiosyncratic component that reflects the movements due to the asset’s individual circumstances. Secondly there are systemic components that reflect the impact of macroscopic events at the whole market or sector level. The motivations for this paper come from developing such models firstly for credit derivatives such as CDOs which are functions of large portfolios of credit risky assets, but also for the evolution of large portfolios which have exposure to a significant proportion of the whole market. The financial crisis of 2008 showed that the correlation between credit risky assets was not adequately modelled and in this work we will examine the behaviour of a large market when all the individual assets follow classical stochastic volatility models but are correlated through market factors.

Our starting point is a simple structural model for default in a large portfolio, studied in [BHHJR]. In this setting there is a market of $N$ credit risky assets in which the $i$-th asset price $A^{i}$ for $i=1,\dots,N$ is modelled by a geometric Brownian motion with a single systemic risk factor, in that under a risk neutral measure

[TABLE]

where $T_{i}=\inf\{t\geq 0:\,A_{t}^{i}=b^{i}\}$ for some constant default barrier $b^{i}$ and the parameters $r,\sigma,\rho,a^{i}$ are constants. Here the Brownian motions $W^{0},W^{1},\dots$ are all independent and we see that it is $W^{0}$ which captures the macroscopic effects felt by the whole market. We note that the parameters of the geometric Brownian motions, are the same for each asset, it is just the starting point and idiosyncratic noise which cause the differences in asset prices. By rewriting this in terms of a distance to default process and considering the empirical measure it was shown in [BHHJR] that the limit empirical measure process of the model has a density which is the unique strong solution to an SPDE on the positive half-line. The density takes values in a weighted Sobolev space as the derivatives of the density may not be well behaved at the origin. The exact regularity of the density at the origin was the subject of [Ledger14], where it was shown that the regularity is a function of the parameter $\rho$.

This is a naive model and has the problems that would be expected from such a simple structural default model. The short term credit spreads go to 0 and we see correlation skew when using the model to price the tranches of CDOs. Thus we wish to investigate a model which incorporates more realistic features. In particular we take stochastic volatility models for the underlying assets and allow there to be global volatility factors driving the market volatility as well as idiosyncratic factors for the volatilities of the individual assets. It is also the case that we would like to allow the parameters that describe the volatility and correlation between assets to vary.

In this paper we consider a large portfolio of $N$ credit risky assets, where now stochastic volatility models are used instead of Black-Scholes models to describe the evolution of the asset values. The CIR process is used to model the volatility as it is non-negative and mean reverting. We assume the $i$-th value process $A_{i}$ satisfies the following system of SDEs

[TABLE]

for all $i\in\{1,2,\dots,N\}$, where $T_{i}=\inf\{t\geq 0:\,A_{t}^{i}=b^{i}\}$. Here, $a^{1},\,a^{2},\,...,\,a^{N}$ and $\sigma^{1},\,\sigma^{2},\,...,\,\sigma^{N}$ are the initial values of the asset prices and the volatilities respectively, $b^{i}$ is the constant default barrier for the value of the $i$-th asset, $C_{i}=(k_{i},\,\theta_{i},\,\xi_{i},\,r_{i},\,\rho_{1,i},\,\rho_{2,i})$ for $i\in\{1,\,2,\,...,\,N\}$ are vectors for the various parameters of the model, $h$ is a function with enough regularity, and $W_{t}^{1},\,B_{t}^{1},\,...,\,W_{t}^{N},\,B_{t}^{N}$ are standard Brownian motions. We will assume that $(a^{i},\,\sigma^{i})$ and $C_{i}$ are drawn independently from some distribution and the Brownian motions are independent from each other and from each $a^{i},\,\sigma^{i}$ and $C_{i}$. Finally $\left(W_{t}^{0},B_{t}^{0}\right)$ is a pair of correlated Brownian motions, independent of both $W^{i}$ and $B^{i}$ for all $i=1,\dots,N$ as well as $a^{i},\,\sigma^{i}$ and $C_{i}$, which represents the impact of macroscopic factors on each asset and each volatility respectively.

As is usual in a credit setting we consider the distance to default, or the log asset prices, by setting $X_{t}^{i}=\left(\ln A_{t}^{i}-\ln b^{i}\right)$ in (1.1). Applying Ito’s formula, our model becomes

[TABLE]

for $i\in\{1,\,2,\,...,\,N\}$, where $x^{i}=\left(\ln a^{i}-\ln b^{i}\right)$ and $T_{i}=\inf\{t\geq 0:\,X_{t}^{i}=0\}$, $\forall\,1\leq i\leq N$.

An important output from such large portfolio models is the loss process, which gives the proportion of assets that have defaulted by any time $t$. This can be used to capture some key quantities in risk management, such as the probability of loss from a portfolio and the expected loss given default. In credit derivative pricing, the payoffs of CDO tranches are piecewise linear functions of this loss process.

In our set up the loss process is given by the mass of the two-dimensional empirical measure

[TABLE]

on $\{0\}\times\mathbb{R}$, while the restriction of $v_{t}^{N}$ to $(0,\infty)\times\mathbb{R}$ for $t\geq 0$ is given by

[TABLE]

Section 2 establishes the following convergence result: almost surely and for all positive $t$ we have both

[TABLE]

and

[TABLE]

weakly as $N\rightarrow\infty$, for some $\sigma$-algebra $\mathcal{G}$ containing the initial data, where we denote by $v_{t,\,C_{1}}\left(\cdot\right)$ the measure-valued process $\mathbb{P}\left(\left(X_{t}^{1},\,\sigma_{t}^{1}\right)\in\cdot,T_{1}>t\,|\,W_{\cdot}^{0},\,B_{\cdot}^{0},\,C_{1},\,\mathcal{G}\right)$. In Sections 3 and 4, we prove that $v_{t,C_{1}}$ - depending on the information contained in $(W_{.}^{0}$, $B_{.}^{0})$, $\mathcal{G}$ and the coefficient vector $C_{1}=(k_{1},\theta_{1},\xi_{1},r_{1},\rho_{1,1},\rho_{2,1})$

• has a density $u_{t,C_{1}}$ in a weighted Sobolev-Lebesgue space of the two-dimensional positive half-space, but with no differentiability in the second spatial variable $y$. Moreover, it is shown in these sections that given $C_{1}$, $u_{t,\,C_{1}}$ satisfies an SPDE in that function space, along with a Dirichlet boundary condition at $x=0$. In Section 5 we improve the regularity by obtaining (weak) differentiability also in $y$, along with some good integrability for the derivative. Our SPDE for $u_{t,C_{1}}$ has the form

[TABLE]

where $u_{0}$ is the initial density, $\rho_{3}$ is the correlation coefficient between $W_{t}^{0}$ and $B_{t}^{0}$ (i.e $dW^{0}_{t}\cdot dB^{0}_{t}=\rho_{3}dt$), and the boundary condition $u_{t,C_{1}}(0,y)=0$ is satisfied for all $y\in\mathbb{R}$ and $t\geq 0$. Our result for the case where each parameter vector $C_{i}$ is the same constant vector for all $i$ will lead to a limiting empirical process whose density is precisely the solution to the above initial-boundary value problem.

In order to implement the model we could solve the initial-boundary value problem for the SPDE numerically for samples $C_{i}$ of the parameters. Then, we can approximate the loss process from

[TABLE]

where $\{c_{1},\,c_{2},\,...,\,c_{n}\}$ is a random sample from the distribution of $C_{1}$. As the SPDE satisfied by each $u_{t,c_{i}}$ is driven by the two-dimensional Brownian path $(W_{.}^{0},\,B_{.}^{0})$, we only need to simulate $(W_{.}^{0},\,B_{.}^{0})$ and solve the corresponding SPDEs. This approach is quite efficient when the number of assets $N$ is large, since we do not have to simulate the $2N$ idiosyncratic Brownian paths.

There are other approaches to the modelling of credit risk in large portfolios which lead to stochastic partial differential equations. For example in a reduced form setting, see [GSS, SG, STG, SSG]. However, this is the first structural large portfolio model to incorporate stochastic volatility and also the first to introduce random coefficients in the SDEs describing the evolution of the asset prices. This provides a level of generality which captures many features of asset prices, and by taking a large portfolio limit reduces the complexity of the numerical calculations arising in risk management and in derivatives pricing applications. Of course, a disadvantage of the model is the introduction of a large number of parameters that need to be simulated or estimated in order to implement the model. Moreover, the random coefficients need assumptions on their joint distributions and, when we use Monte Carlo techniques to estimate expectations in (1.6), a very large number of simulations may still be required due to the random parameters. The constant coefficient case is just a special case of the model we have considered, in which the weak limit $v_{1,t}$ of the empirical process coincides with the measure-valued process $v_{t,C_{1}}$ whose density $u_{t,C_{1}}$ satisfies our SPDE. Our main aim in this paper is to establish the theoretical background for the general case.

The calibration of the model for its use in the pricing of CDOs would follow a similar approach to that used in [BHHJR]. In its simplest form we take the version where the parameters of the model are assumed to be constants. The initial condition would be fitted to the CDS prices of the underlying constituents of the portfolio. The parameters of the model are then determined from the market tranche prices of the CDOs with different maturities. This is done by solving the model forward from different parameter settings to find model tranche prices and then minimizing the least squares distance between model and market to locate the best fit parameters.

The approach to solving the model forward must be done numerically. This type of model is more computationally intensive than that considered in [BHHJR] as the SPDE is in two dimensions. The technique is to generate the two dimensional Brownian path and then solve the SPDE using a finite element approach. Speed up could be achieved by extending the work of [GR] where the multilevel Monte Carlo approach was used for the model of [BHHJR]. We will not discuss the numerical analysis for the model as, even in the one-dimensional case, this is challenging.

There are significant mathematical challenges in extending large portfolio models to the stochastic volatility setting. A key point is to estimate the boundary behaviour of the empirical measure and, with a non-constant volatility path, this needs a novel approach. The kernel smoothing technique used by [BHHJR] also needs alteration to cope with this volatility process, to enable us to obtain the best possible regularity for our two-dimensional density.

In Section 2, we assume that the initial data satisfies some reasonable exchangeability conditions in order to obtain the convergence result for the empirical measure process as $N\rightarrow\infty$. This is not just a two-dimensional version of the corresponding result in [BHHJR], since it gives the convergence of the restriction of the empirical measure process to $(0,\infty)\times\mathbb{R}$, while it also gives the form of the limiting measure-valued process. It includes thus a law of large numbers which is particularly important for dealing with this two-dimensional version of the large portfolio analysis problem.

In Section 3 we extend some existing Malliavin calculus results and techniques, in order to obtain a strong norm estimate for the density of a CIR process when a component of the driving Brownian Motion (the market factor) is given. We are only able to do this under a condition on the parameters which is stronger than the Feller condition for the CIR process to not hit 0 at any positive time. This is due to the fact that the CIR process does not have Lipschitz coefficients, which means that standard Malliavin calculus techniques for proving the existence of a density of an Ito process are not directly applicable and approximations with processes having better coefficients are needed. In Section 4 we prove a convergence result for a sequence of stopped Ito processes when the sequence of volatility paths decreases pointwise to a continuous and positive path, in order to extend the results of [BHHJR] to the case when the volatility path is non-constant. Combining this with the results of Section 3 in a divide-and-conquer approach, we obtain the existence of a regular density for the measure-valued process $v_{t,C_{1}}$, for any good enough value of $C_{1}$, and also the SPDE and the boundary condition satisfied by that density.

In Section 5, we extend the kernel smoothing method developed in [BHHJR, KX99, HL16], by proving that the standard heat kernel maintains its smoothing and convergence properties, when it is composed with a square root function, and also in certain weighted $L^{2}$ spaces. This allows us to obtain differentiability of our density in the $y$-direction, and also weighted $L^{2}$ integrability of the derivative. This improved kernel smoothing method does not work in distribution spaces for our SPDE and thus, the regularity results of the previous two sections are crucial. Finally, in Section 6, we discuss the question of uniqueness of the solution.

Remark 1.1*.*

(1) We will not discuss the issue of asymptotic arbitrage which can arise when there is a large portfolio limit of assets (see [KKD1, KKD2]). As we are using the limiting model as an approximation to a large finite model, which will not admit arbitrage, the question is only of theoretical interest.

(2) When calibrating the model for pricing credit derivatives the drift term in the asset’s value process is replaced by a known interest rate. Including more parameters than that used in [BHHJR] should improve the calibration of the model and may allow observed features such as correlation skew in CDOs to be captured. Though we should note that even including jumps in the basic model of [BHHJR] still makes it difficult to capture all the observed features of CDO tranche prices [BR12].

(3) One could view the empirical mean of such a model as a natural model for an index, see [HV15] for the simple case. Here we would produce a stochastic volatility model for the index and this could be used to price volatility dependent derivatives.

(4) It would be natural to develop central limit theorems and a large deviation analysis in further work, potentially by adapting and extending appropriately the ideas of [SSO1, SSO2]. For applications in systemic risk it would also be interesting to add a mean field interaction.

(5) The popular Heston model is just a simple case of the model used to describe the evolution of the asset values in our setting, which is obtained when the function $h$ is just a square root function.

2 Connection to the probabilistic solution of an SPDE

In order to study the asymptotic behaviour of our system of particles, some assumptions have to be made. We assume that $(\Omega,\,\mathcal{F},\,\{\mathcal{F}_{t}\}_{t\geq 0},\,\mathbb{P})$ is a filtered probability space with a complete and right-continuous filtration $\{\mathcal{F}_{t}\}_{t\geq 0}$, $\left\{\left(X_{0}^{1},\,\sigma_{0}^{1}\right),\,\left(X_{0}^{2},\,\sigma_{0}^{2}\right),\,...\right\}$ is an exchangeable sequence of $\mathcal{F}_{0}$-measurable two-dimensional random vectors (see [Aldous] for more on exchangeability), and $C_{i}=\left(k_{i},\,\theta_{i},\,\xi_{i},\,r_{i},\,\rho_{1,i},\,\rho_{2,i}\right)$ for $i\in\mathbb{N}$ are i.i.d $\mathcal{F}_{0}$-measurable random vectors in $\mathbb{R}_{+}^{6}$, independent from each $(X_{0}^{j},\,\sigma_{0}^{j})$, such that $\mathbb{P}$- almost surely we have both $k_{i}\theta_{i}>\frac{3}{4}\xi_{i}^{2}$ and $\rho_{1,i},\,\rho_{2,i}\in(-1,1)$. We note that the condition on $k_{i},\theta_{i},\xi_{i}$ is stronger than the usual Feller condition that ensures that 0 is not reached by a CIR process in finite time. We also consider an infinite sequence $\{W_{t}^{0},\,B_{t}^{0},\,W_{t}^{1},\,B_{t}^{1},\,W_{t}^{2},\,B_{t}^{2},\,...\}$ of $\mathcal{F}_{t}$ - adapted standard Brownian motions, in which only $W_{t}^{0}$ and $B_{t}^{0}$ are correlated and their correlation coefficient is denoted by $\rho_{3}$. Under these assumptions and for each $N\in\mathbb{N}$, we consider the interacting particle system described by equations (1.2) and the corresponding empirical measure processes $v_{t}^{N}$ and $v_{1,t}^{N}$ given by (1.3) and (1.4) respectively. We also define $v_{2,t}^{N}=v_{t}^{N}-v_{1,t}^{N}$, the restriction of $v_{t}^{N}$ to $\{0\}\times\mathbb{R}$, for all $t\geq 0$.

We start with the following convergence theorem, the proof of which is a simple modification of the convergence theorem for the one-dimensional empirical measure process in [BHHJR] and can be found in the Appendix. It is stronger than a convergence result for the empirical measure process $v_{t}^{N}$ but we need it for proving Theorem 2.3, a crucial result for establishing the convergence of the $\{0\}\times\mathbb{R}$ - supported component $v_{2,t}^{N}$ (Theorem 2.5).

Theorem 2.1**.**

For each $N\in\mathbb{N}$ and any $t,\,s\geq 0$, consider the random measure given by

[TABLE]

The sequence $v_{3,t,s}^{N}$ of three-dimensional empirical measures converges weakly to some measure $v_{3,t,s}$ for all $t,\,s\geq 0$, $\mathbb{P}$-almost surely. Moreover, the measure-valued process $\{v_{3,t,s}:\,t,\,s\geq 0\}$ is $\mathbb{P}$-almost surely continuous in both $t$ and $s$ under the weak topology.

The convergence result for $v_{t}^{N}$ is a direct consequence of the above theorem and it is given in the following corollary.

Corollary 2.2**.**

The sequence $v_{t}^{N}$ of two-dimensional empirical measures given by $(1.3)$ converges weakly to some measure $v_{t}$ for all $t\geq 0$, $\mathbb{P}$-almost surely. Moreover, the path $\{v_{t}:\,t\geq 0\}$ is $\mathbb{P}$-almost surely continuous under the weak topology. The measure-valued process $v_{t}$ is the restriction of $v_{3,t,s}$ to the space of functions which are constant in the third variable, for any $t\geq 0$.

Proof.

Since $v_{t}^{N}$ is the restriction of $v_{3,t,s}^{N}$ to the space of functions which are constant in the third variable, the result follows by testing the measure against such functions and by taking $N\rightarrow\infty$. ∎

Next, we prove a theorem which gives us the form of the weak limits of the empirical measures $v_{3,t,s}$.

Theorem 2.3**.**

There exists an $\Omega_{0}\subset\Omega$ with $\mathbb{P}(\Omega_{0})=1$ such that for any $\omega\in\Omega_{0}$, we have $\int_{\mathbb{R}^{2}}fdv_{3,t,s}=\mathbb{E}\left[f\left(X_{t}^{1},\,\sigma_{t}^{1},\,\sigma_{s}^{1}\right)|\,W_{.}^{0},\,B_{.}^{0},\,\mathcal{G}\right]$ for any $t,\,s\geq 0$ and any $f\in C_{b}(\mathbb{R}^{3};\,\mathbb{R})$, where $\mathcal{G}$ is some $\sigma$-algebra contained in $\mathcal{F}_{0}$.

Proof.

By the exchangeability of the initial data, we know that there exists a $\sigma$-algebra $\mathcal{G}$ contained in $\mathcal{F}_{0}$, such that the two-dimensional vectors: $\left(X_{0}^{1},\,\sigma_{0}^{1}\right),\,\left(X_{0}^{2},\,\sigma_{0}^{2}\right),\,...$ are i.i.d given $\mathcal{G}$. Moreover, $\left(B_{\cdot}^{k},\,W_{\cdot}^{k},\,C_{k}\right)$ for $k\in\mathbb{N}$ are i.i.d and since they are also independent from $\left(B_{\cdot}^{0},\,W_{\cdot}^{0},\,\mathcal{G}\right)$, they are also i.i.d. under the probability measure $\mathbb{P}(\,\cdot\,|\,W^{0},\,B^{0},\,\mathcal{G})$. The same holds for the two-dimensional vectors $\left(X_{0}^{1},\,\sigma_{0}^{1}\right),\,\left(X_{0}^{2},\,\sigma_{0}^{2}\right),\,...$, since they are i.i.d given $\mathcal{G}$ and measurable with respect to the bigger $\sigma$-algebra $\mathcal{F}_{0}$, with $\left(W_{\cdot}^{0},\,B_{\cdot}^{0}\right)$ being independent from $\mathcal{F}_{0}$. Thus, noting that there is a function $g$ such that

[TABLE]

it follows that $\left(X_{t}^{k},\sigma_{t}^{k},\sigma_{s}^{k}\right)$ for $k\in\mathbb{N}$ are also i.i.d. random vectors under $\mathbb{P}(\,\cdot\,|W^{0},B^{0},\mathcal{G})$.

Thus, for any $f\in C_{b}(\mathbb{R}^{3};\mathbb{R})$ we have

[TABLE]

where, in the last expectation, by the strong law of large numbers, for each $t,s$ the probability that there is no convergence is zero. Hence, there is an $\Omega^{f}\subset\Omega$ (depending on $f$) with $\mathbb{P}\left(\Omega^{f}\right)=1$, such that

[TABLE]

as $N\rightarrow\infty$, for all $\omega\in\Omega^{f}$.

If we denote by $\Omega_{0}^{f}$ the intersection of $\Omega^{f}$ with the set of events for which the results of Theorem 2.1 hold, we see that $\mathbb{P}\left(\Omega_{0}^{f}\right)=1$ and that for all $\omega\in\Omega_{0}^{f}$ we have

[TABLE]

for any $t,\,s\in\mathbb{Q}^{+}$. Since both quantities in $\eqref{eq:2.1}$ are continuous in $(t,\,s)$ (this follows from Theorem 2.1 for the LHS, and by using the dominated convergence theorem for the RHS) and since they coincide for any $t,\,s\in\mathbb{Q}^{+}$, we conclude that they coincide for all $t,\,s\geq 0$ in $\Omega_{0}^{f}$.

Finally, taking the intersection of all $\Omega_{0}^{p}$ for all $p$ belonging to a countable and dense subset $D$ of $C_{b}\left(\mathbb{R}^{3};\,\mathbb{R}\right)$, we obtain the desired set $\Omega_{0}$. This follows from the fact that both quantities in $\eqref{eq:2.1}$ are bounded functionals in $C_{b}\left(\mathbb{R}^{3};\,\mathbb{R}\right)$ with the supremum norm, where for the LHS this follows by taking limits in the obvious inequality, $\int_{\mathbb{R}^{2}}fdv_{3,t,s}^{n}\leq||f||_{\infty},\>\forall\,n\in\mathbb{N}$. Our proof is now complete. ∎

Corollary 2.4**.**

Let $\{v_{t}:\,t\geq 0\}$ be the measure-valued process defined in Corollary 2.2. There exists an $\Omega_{0}\subset\Omega$ with $\mathbb{P}(\Omega_{0})=1$ such that for any $\omega\in\Omega_{0}$, we have $\int_{\mathbb{R}^{2}}fdv_{t}=\mathbb{E}\left[f\left(X_{t}^{1},\,\sigma_{t}^{1}\right)|\,W_{.}^{0},\,B_{.}^{0},\,\mathcal{G}\right]$ for any $t\geq 0$ and for any $f\in C_{b}\left(\mathbb{R}^{3};\,\mathbb{R}\right)$, where $\mathcal{G}$ is the $\sigma$-algebra defined in Theorem 2.3.

Proof.

This result follows by testing the measure against functions which are constant in the third variable and by recalling Corollary 2.2. ∎

The above corollary completes the convergence result for $v_{t}^{N}$ which was given in Corollary 2.2. However, what we need is a similar result for its restriction to $\{0\}\times\mathbb{R}$, that is $v_{2,t}^{N}$. This is given in the following Theorem, the proof of which is based on the more general convergence result given in Theorem 2.1 and Theorem 2.3.

Theorem 2.5**.**

There exists a measure-valued process $\{v_{2,t}:\,t\geq 0\}$ and an $\Omega_{1}\subset\Omega_{0}$ with $\mathbb{P}(\Omega_{1})=1$, such that for any $\omega\in\Omega_{0}$ we have that $v_{2,t}^{N}\rightarrow v_{2,t}$ weakly for all $t\geq 0$. Moreover, we have $\int_{\mathbb{R}^{2}}fdv_{2,t}=\mathbb{E}\left[f\left(X_{t}^{1},\,\sigma_{t}^{1}\right)\mathbb{I}_{\{T_{1}<t\}}|\,W_{.}^{0},\,B_{.}^{0},\,\mathcal{G}\right]$ for all $t\geq 0$ and for all $f\in C_{b}\left(\mathbb{R}^{2};\,\mathbb{R}\right)$, where $\mathcal{G}$ is the $\sigma$-algebra defined in Theorem 2.3.

Proof.

First, observe that when $\left(W^{0},\,B^{0},\,\mathcal{G}\right)$ is given, $T_{1}$ has a continuous distribution, since it is a stopping time for the Ito process $X_{t}^{1}$. Moreover, observe that

[TABLE]

and that

[TABLE]

for any $t\geq 0$ and $f\in C_{b}\left(\mathbb{R}^{2};\,\mathbb{R}\right)$, which means that we only need to work with functions $f$ which are constant in the first variable.

Let now $v_{2,t}(\cdot)$ be the probability law $\mathbb{P}\left(\left(X_{t}^{1},\,\sigma_{t}^{1}\right)\in\cdot\,;\,T_{1}\leq t\,|\,W^{0},\,B^{0},\,\mathcal{G}\right)$ and fix a function $f$ in $C_{b}\left(\mathbb{R};\,\mathbb{R}\right)$ with positive values. Since $T_{i}$ is adapted to $\left(X_{t}^{i},\,\sigma_{t}^{i}\right)$ for any $i\in\mathbb{N}$, by the independence obtained in the first paragraph of the proof of Theorem 2.3 and a Law of Large Numbers argument similar to the one in that proof, we have that the desired convergence holds for the chosen $f$, for all $t\in\mathbb{Q}^{+}$ and for all $\omega$ in some $\Omega_{1,f}\subset\Omega$ with $\mathbb{P}\left(\Omega_{1,f}\right)=1$. By intersecting with the full-probability set $\Omega_{0}$ given in Theorem 2.3, we can take $\Omega_{1,f}\subset\Omega_{0}$. Now for a $t\geq 0$ and an $\omega\in\Omega_{1,f}$, we pick any two rational numbers $t_{1},\,t_{2}\geq 0$ such that $t_{1}\leq t\leq t_{2}$. Then we have

[TABLE]

where the first term equals $\mathbb{E}\left[f\left(\sigma_{t_{1}}^{1}\right)\mathbb{I}_{\{T_{1}<t_{1}\}}\,|\,W_{.}^{0},\,B_{.}^{0},\,\mathcal{G}\right]$ for each rational time $t_{1}$. Next, by recalling Theorem 2.3 for $f(x,\,y,\,z)=|f(y)-f(z)|$, and $s=t_{1}$ we find that the second term equals $\mathbb{E}\left[|f\left(\sigma_{t}^{1}\right)-f\left(\sigma_{t_{1}}^{1}\right)|\,|\,W_{.}^{0},\,B_{.}^{0},\,\mathcal{G}\right]$. Now taking $t_{1}\rightarrow t$ and using the Dominated Convergence Theorem and the fact that the random variable $T_{1}$ has a continuous distribution, we obtain

[TABLE]

Similarly, we have

[TABLE]

and by the same argument for the rational number $t_{2}\rightarrow t$, we find

[TABLE]

Hence, by (2.2) and (2.3), the desired convergence holds in $\Omega_{1,f}\subset\Omega_{0}$ for all $t\geq 0$ and any $f$ in $C_{b}(\mathbb{R};\,\mathbb{R})$ with positive values. By linearity, and since every continuous function can be decomposed into its continuous positive part and its continuous negative part, we can have our convergence result for any $f\in C_{b}(\mathbb{R};\,\mathbb{R})$. Let now $S=\{p_{i}:\,i\in\mathbb{N}\}$ be a countable basis of $C_{b}(\mathbb{R};\,\mathbb{R})$. Then, by linearity, the desired convergence holds for all $t\geq 0$, all $p\in\left[S\right]$ and all $\omega\in\Omega_{1}=\cap_{i\in\mathbb{N}}\Omega_{1,p_{i}}$, with $\mathbb{P}(\Omega_{1})=1$. Now for any $f\in C_{b}(\mathbb{R};\,\mathbb{R})$ and $\epsilon>0$, we can pick $p\in\left[S\right]$ such that $||f-p||_{\infty}<\frac{\epsilon}{3}$, so we have

[TABLE]

for all $N$ sufficiently large. Thus, we have our convergence result for all $t\geq 0$ and all $\omega\in\Omega_{1}=\cap_{i\in\mathbb{N}}\Omega_{1,p_{i}}$ with $\mathbb{P}(\Omega_{1})=1$, so we are done. ∎

Next, by Corollary 2.2 and Theorem 2.5, we have that $v_{1,t}^{N}=v_{t}^{N}-v_{2,t}^{N}\rightarrow v_{t}-v_{2,t}=:v_{1,t}$ weakly, for all $t\geq 0$, $\mathbb{P}$-almost surely. Also, it follows from Corollary 2.4 and Theorem 2.5 that

[TABLE]

for any $f\in C_{b}\left(\mathbb{R}^{2};\,\mathbb{R}\right)$ and $t\geq 0$. It is therefore reasonable to study the behaviour of the process of measures $v_{t,C_{1}}(\cdot)$ defined as

[TABLE]

for a given value of $C_{1}=\left(k_{1},\,\theta_{1},\,\xi_{1},\,r_{1},\,\rho_{1,1},\,\rho_{2,1}\right)$. The behaviour of this process of measures is given in the following Theorem.

Theorem 2.6**.**

Let $A$ be the two-dimensional differential operator mapping any smooth function $f:\mathbb{R}^{+}\times\mathbb{R}\rightarrow\mathbb{R}$ to

[TABLE]

for all $\left(x,\,y\right)\in\mathbb{R}^{+}\times\mathbb{R}$. Then, the measure-valued stochastic process $v_{t,C_{1}}$ satisfies the following weak form SPDE

[TABLE]

for all $t\geq 0$ and any $f\in C_{0}^{test}=\left\{g\in C_{b}^{2}\left(\mathbb{R}^{+}\times\mathbb{R}\right):\,g\left(0,\,y\right)=0,\>\forall\,y\in\mathbb{R}\right\}$.

Proof.

By using Ito’s formula for the stopped two-dimensional process $\left\{\left(X_{t}^{1},\,\sigma_{t}^{1}\right):\,t\geq 0\right\}$ given by (1.2) and by recalling that $f\left(0,\,y\right)=0$ for all $y$, we obtain

[TABLE]

and the desired result follows by taking conditional expectations given $\left(W_{\cdot}^{0},\,B_{\cdot}^{0}\right)$, $C_{1}$ and $\mathcal{G}$, by noticing that Ito integrals with respect to $B_{\cdot}^{1}$ and $W_{\cdot}^{1}$ vanish due to the pairwise independence of the Brownian Motions, and by taking the given coefficients out of the conditional expectations. ∎

3 Volatility Analysis - A Malliavin Calculus approach

Now that we have connected our problem to the study of the probabilistic solution of an SPDE, we need to establish the best possible regularity result for that solution. Before showing that the measure-valued process $\mathbb{E}\left[\cdot(X_{t}^{1},\,\sigma_{t}^{1})\mathbb{I}_{T_{1}>t}\,|\,B_{\cdot}^{0},\,W_{\cdot}^{0},\,C_{1},\,\mathcal{G}\right]$ does indeed have a density for almost all paths of $\left(B_{\cdot}^{0},\,W_{\cdot}^{0}\right)$ with some good regularity, it is natural (and important as we will see) to ask whether the same holds for the 1-dimensional measure-valued process describing the evolution of $\mathbb{E}\left[f(\sigma_{t})\,|\,B_{\cdot}^{0},\,\mathcal{G}\right]$, for suitable $f$, where $\sigma$ is a CIR process driven by a combination of $B_{\cdot}^{0}$ and $B_{\cdot}^{1}$, that is a process satisfying

[TABLE]

We assume that the coefficients of equation (3.1) satisfy: $k\theta>\frac{3}{4}\xi^{2}$, which is stronger than the standard Feller boundary condition for a CIR process, and also $\rho_{2}\in(-1,\,1)$. Then, the answer to our question is given in the next theorem.

Theorem 3.1**.**

Assume that $\sigma_{0}$ is a random variable in $L^{p}\left(\Omega,\,\mathcal{F}_{0},\,\mathbb{P}\right)$ for all $p>p_{0}=1-\frac{2k\theta}{\xi^{2}}$, such that given $\mathcal{G}$, $\sigma_{0}$ has a continuous density $p_{0}(\cdot\,|\,\mathcal{G})$ which is supported in $[0,\infty)$ and which satisfies

[TABLE]

for all $\gamma\in\left[-\frac{2k\theta}{\xi^{2}},\,1\right]$. Then, for every path of $B_{\cdot}^{0}$ and $t\geq 0$, the conditional probability measure $\mathbb{P}(\sigma_{t}\in A\,|\,B_{\cdot}^{0},\,\mathcal{G})$ posseses a continuous probability density $p_{t}(\cdot\,|\,B_{\cdot}^{0},\,\mathcal{G})$ which is supported in $[0,\infty)$. Moreover, for any $T>0$, any $1<q<\frac{4k\theta}{3\xi^{2}}$ and any $\alpha\geq 0$, we have

[TABLE]

To prove this Theorem we need a few lemmas. First, we will need the following finiteness result for the moments of the supremum of a CIR / Ornstein-Uhlenbeck process up to some finite time. The proof of this technical lemma can be found in the Appendix.

Lemma 3.2**.**

Under the assumptions of Theorem 3.1, for any $p\geq 0$ and $T>0$ we have

[TABLE]

Moreover, if $\left\{u_{t}:\,t\geq 0\right\}$ is the Ornstein-Uhlenbeck process which solves the SDE

[TABLE]

under the initial condition $u_{0}=\sqrt{\sigma_{0}}$, then we have also

[TABLE]

Next, we need a few results that involve the notion of Malliavin differentiability. The Malliavin derivative of a random variable adapted to a Brownian path is a stochastic process measuring, in some sense, the rate of change of the random variable when the Brownian path changes at any time $t$. Extending this to random variables taking values in Banach spaces, we can define the $k$-th Malliavin derivative as a random function of $k$ time variables (provided that it exists). The existence and behaviour of these derivatives are inextricably connected to the existence of a regular density for the random variable. We refer to [Nualart] for the basics of Malliavin calculus and, as in [Nualart], we denote by $\mathbb{D}^{n,p}(V)$ the space of $n$-times Malliavin differentiable random variables taking values in the Banach space $V$, whose $k$-th Malliavin derivative has an $L^{2}$ norm (as a function in $k$ time variables taking values in $V$) of a finite $L^{p}$ norm as a random variable, for all $0\leq k\leq n$.

In [AE1] and [AE2] it is proven that the CIR process has a Malliavin derivative under the probability measure $\mathbb{P}$, which is given by a quite similar formula. In [AE1] it is also proven that under our strong assumption $k\theta>\frac{3}{4}{\xi}^{2}$, a second Malliaving derivative with some regularity also exists. The next two lemmas extend these results to the case where the path of the market noise $B_{\cdot}$ is given. This is exactly what we need in order to prove Theorem 3.1. The proofs of these extensions are more or less based on the same ideas as the corresponding initial results (except that Lemma 3.2 is also needed at some points) and can be found in the Appendix.

Lemma 3.3**.**

There exists an $\Omega_{1}\subset\Omega$ with $\mathbb{P}\left(\Omega_{1}\right)=1$, such that for all $\omega\in\Omega_{1}$ the random probability measure $\mathbb{P}(\cdot\,|\,B_{\cdot}^{0},\,\mathcal{G})$ has the following property: Under $\mathbb{P}(\cdot\,|\,B_{\cdot}^{0},\,\mathcal{G})$, the process $\{\sigma_{t}:\,t\geq 0\}$ has a Malliavin derivative with respect to the Brownian Motion $B_{\cdot}^{1}$ which is given by

[TABLE]

for all $t>0$ and $0\leq t^{\prime}\leq t$. This is a process in $t^{\prime}$ which belongs to $L_{B_{\cdot}^{0},\,\mathcal{G}}^{2}\left([0,\,t]\times\Omega\right)$ for any fixed $t\geq 0$, where the notation $L_{B_{\cdot}^{0},\,\mathcal{G}}^{q}$ is used for any $L^{q}$ space when the probability measure $\mathbb{P}$ is replaced by $\mathbb{P}(\cdot\,|\,B_{\cdot}^{0},\,\mathcal{G})$.

Lemma 3.4**.**

For any $1\leq q^{\prime}<\frac{4k\theta}{3\xi^{2}}$ and $T>0$, there exists an $\Omega_{2}\subset\Omega$ with $\mathbb{P}\left(\Omega_{2}\right)=1$, such that for all $\omega\in\Omega_{2}$ the random probability measure $\mathbb{P}(\cdot\,|\,B_{\cdot}^{0},\,\mathcal{G})$ has the following property: Under $\mathbb{P}(\cdot\,|\,B_{\cdot}^{0},\,\mathcal{G})$, the process $\{\sigma_{t}:\,0<t\leq T\}$ belongs to the space $\mathbb{D}^{2,q^{\prime}}$ with respect to the Brownian Motion $B_{\cdot}^{1}$, and the second order Malliavin derivative is given by

[TABLE]

for all $0<t\leq T$ and $0\leq t^{\prime},\,t^{\prime\prime}\leq t$, where the first order derivatives are given by Lemma 3.3. Furthermore we have

[TABLE]

The same holds for the process $v_{t}=\sqrt{\sigma_{t}}$, but this time the second Malliavin Derivative is given by

[TABLE]

Finally, we need a lemma that connects the existence of a regular density to the existence of a regular non-vanishing Malliavin Derivative. There are many results of this kind in the literature (many of them can be found in [Nualart]), but in our case we need the following.

Lemma 3.5**.**

Let $F$ be a random variable in the space $\mathbb{D}^{1,2}\cap L^{p}\left(\Omega\right)$ for all $p>1$. Assume that for some process $\{u_{t}:\,0\leq t\leq T\}$ of $L^{2}$-integrable paths, we have $\left\langle u_{.},\,D_{.}F\right\rangle_{L^{2}}\neq 0$ almost surely and also $\frac{u_{t}}{\left\langle u_{.},\,D_{.}F\right\rangle_{L^{2}}}\in\mathbb{D}^{1,q^{\prime}}\left(L^{2}\left(\left[0,\,T\right]\right)\right)$ for some $q^{\prime}>1$. Then $F$ has a continuous density $p(\cdot)$ for which it holds

[TABLE]

for some $C>0$ and for all $\alpha\geq 0$ for which $\mathbb{E}\left[F^{\frac{\alpha q^{\prime}}{q^{\prime}-1}}\right]$ is finite.

The proof of the above lemma has also been put in the Appendix, since it is almost identical to that of Proposition 2.1.1 in [Nualart] (page 78), except that in the end we need to recall Meyer’s inequality in order to obtain the estimate for the supremum. We are now ready to prove the main result of this section.

Proof of Theorem 3.1.

Lemma 3.3 implies that for almost all $\omega\in\Omega$, $\sigma_{t}\in\mathbb{D}^{1,2}$ with respect to $B_{\cdot}^{1}$ under the probability measure $\mathbb{P}\left(\cdot\,|\,B_{\cdot}^{0},\,\mathcal{G}\right)$. We would like to apply Lemma 3.5 on $\sigma_{t}$ for an appropriate process $\{u_{t^{\prime}}:\,0\leq t^{\prime}\leq t\}$. Let $u_{.}$ be the unique pathwise solution to the linear integral equation

[TABLE]

Then, $u_{t^{\prime}}=u_{0}e^{\int_{0}^{t^{\prime}}e^{-\int_{s}^{t}\left[\left(\frac{k\theta}{2}-\frac{\xi^{2}}{8}\right)\frac{1}{\sigma_{s^{\prime}}}+\frac{k}{2}\right]ds^{\prime}}\sqrt{\sigma_{t}}ds}$ for any $t^{\prime}\leq t$, which is almost surely a differentiable and strictly increasing function on $[0,\,t],$ always bounded by $u_{0}e^{t\sqrt{\sigma_{t}}}>0$. Then it is easy to check that

[TABLE]

for any $t^{\prime}<t$. Thus, we have that

[TABLE]

Next, we want to show that $U_{t^{\prime}}$ is Malliavin differentiable and compute its derivative for any $0<t^{\prime}<t$. By Lemma 3.3 we have that $D_{.}\sigma_{t}$ lies in the space

[TABLE]

for any $t^{\prime}<t$ and any $1<q^{\prime}\leq\frac{4k\theta}{3\xi^{2}}$, which implies that

[TABLE]

for all $0\leq t^{\prime}\leq t$ and any $1<q^{\prime}\leq\frac{4k\theta}{3\xi^{2}}$. Since the LHS of the above is positive (follows easily from Lemma 3.4 and our assumptions for the coefficients), the above implies also that $\int_{t^{\prime}}^{t}D_{s}\sigma_{t}ds\in\mathbb{D}^{1,q^{\prime}}$ for any $t^{\prime}<t$ and that

[TABLE]

Consider now a smooth function $F$ satisfying

[TABLE]

and which has bounded derivatives. Then we can easily check that $U_{t^{\prime}}=F\left(\int_{t^{\prime}}^{t}D_{s}\sigma_{t}ds\right)$, and by the standard Malliavin Chain rule we obtain $U_{t^{\prime}}\in\mathbb{D}^{1,q^{\prime}}$ for any $1<q^{\prime}<\frac{4k\theta}{3\xi^{2}}$, with

[TABLE]

for all $0\leq t^{\prime}\leq t$. Finally, from (3.7) and (3.8) we have that $D_{.}U_{.}$ belongs to the space $L^{q^{\prime}}\left(\Omega;\,L^{2}\left(\left[0,\,t\right]^{2}\right)\right)$ and thus, by (3.6) we deduce that $U_{.}\in\mathbb{D}^{1,q^{\prime}}\left(L^{2}\left[0,\,t\right]\right)$.

Recall now that $\sigma_{t}^{p}$ has a finite expectation under $\mathbb{P}\left(\cdot\,|\,B_{\cdot}^{0},\,\mathcal{G}\right)$, for any exponent $p>0$ and any $t\leq T$, for all $\omega\in\Omega_{2}$ with $\mathbb{P}\left(\Omega_{2}\right)=1$ (this follows easily from Lemma 3.2 and the law of total expectation). Thus, by Lemma 3.5, $\sigma_{t}$ possesses for all $\omega\in\Omega_{2}$ a continuous density $p_{t}(\cdot\,|\,B_{\cdot}^{0},\,\mathcal{G})$ under $\mathbb{P}\left(\cdot\,|\,B_{\cdot}^{0},\,\mathcal{G}\right)$, such that for all $t\geq 0$

[TABLE]

for any $\alpha\geq 0$.

It is not hard to see that the constant $C^{\prime}>0$ does not depend on the fixed path $B_{\cdot}^{0}$ of the Market factor, since it depends on the maximum of the derivative of $F$ and the universal constant of Proposition 1.5.4 in [Nualart] (changing the measure here is the same as changing the process $\sigma_{t}$ by changing its Market factor, under the law of the idiosyncratic factor). Therefore, for any $1<q<q^{\prime}$, by Holder’s inequality, the estimate (3.9) and the law of total expectation, we have

[TABLE]

Since $\frac{(q^{\prime}-1)q}{q^{\prime}-q}>1$, by applying Holder’s inequality and the law of total expectation once more, we find that the first factor on the RHS of (LABEL:eq:3.26) is bounded by

[TABLE]

which is finite by Lemma 3.2. On the other hand, the second factor on the RHS of (LABEL:eq:3.26) is bounded by

[TABLE]

which is finite by Lemma 3.4 for any $T>0$, so the desired result follows. ∎

4 Existence of a regular two-dimensional density

In this section, we combine the results we have obtained in the previous section for the volatility process with the regularity results we have for the constant volatility model of [BHHJR], in order to obtain a regular density for the probabilistic solution of the SPDE obtained in Section 2, when the value of $C_{1}$ is given. First, for any Hilbert space $H$, we denote by $L^{2}\left(\Omega\times\left[0,\,T\right];\,H\right)$ the space of $H$- valued stochastic processes, which are adapted to the Brownian path $(W_{\cdot}^{0},\,B_{\cdot}^{0})$. For our purpose we will need the following useful Theorem which extends the results of [BHHJR] to the case where the volatility path is non-constant.

Theorem 4.1**.**

Let $\{X_{t}:\,t\geq 0\}$ satisfy the stopped SDE

[TABLE]

for $\tau=\inf\{t\geq 0:\,X_{t}=0\}$, under the initial condition $X_{0}=x_{0}$, where $x_{0}$ is a continuous random variable with an $L^{2}$ density $u_{0}=u_{0}(\cdot\,|\,\mathcal{G})$ given $\mathcal{G}\subset\sigma(x_{0})$, $W_{\cdot}^{0}$ and $W_{\cdot}^{1}$ are pairwise independent standard Brownian motions which are also independent of $x_{0}$, $r>0$ and $\rho_{1}\in(-1,\,1)$ are given constants, and $\{\sigma_{t}:\,t\geq 0\}$ is just a deterministic path which is continuous and positive. Let $\{V_{t}:\,t\geq 0\}$ be the measure-valued process given by

[TABLE]

for any Borel set $A\subset\left(0,\,\infty\right)$. Then almost surely, the following are true for all $T>0$;

$V_{t}$* possesses a density $u(t,\,\cdot)=u\left(t,\,\cdot\,;\,W_{\cdot}^{0},\,\mathcal{G}\right)$ for all $0\leq t\leq T$, which is the unique solution to the SPDE*

[TABLE]

in $L^{2}\left(\Omega\times\left[0,\,T\right];\,H_{0}^{1}\left(\mathbb{R}^{+}\right)\right)$ under the initial condition $u(0,\cdot)=u_{0}$, where $u_{0}$ is the density of $x_{0}$ given $\mathcal{G}$. 2. 2.

For all $0\leq t\leq T$, the following identity holds

[TABLE]

To prove the above theorem, we need the following convergence result for a sequence of stopped Ito processes, when the corresponding sequence of volatility paths decreases pointwise to some continuous and positive path.

Lemma 4.2**.**

Let $\{\sigma_{t}^{0}:\,0\leq t\leq T\}$ be a continuous and strictly positive path, which is approximated from above by a pointwise decreasing sequence $\{\{\sigma_{t}^{m}:\,0\leq t\leq T\}\}_{m\in\mathbb{N}}$ of positive and bounded paths. For any $m\in\mathbb{N}\cup\{0\}$, denote by $X_{.}^{m}$ the stopped Ito process given by

[TABLE]

where $\tau^{m}=\inf\{t\geq 0:\,X_{t}^{m}=0\}$ and $W_{\cdot}$ is a standard Brownian Motion, with the initial condition

[TABLE]

for $x_{0}\geq 0$ and $l_{m}=\left(\left\|\sqrt{\sigma_{.}^{m}}-\sqrt{\sigma_{.}^{0}}\right\|_{L^{2}\left[0,\,T\right]}\right)^{1/2}$. Then, for a sequence $\{m_{k}:\,k\in\mathbb{N}\}\subset\mathbb{N}$, we have almost surely: $X_{t}^{m_{k}}\rightarrow X_{t}^{0}$ uniformly on any compact interval $\left[0,\,T\right]$.

The proof of the above lemma is quite technical and can be found in the Appendix

Proof of Theorem 4.1.

Without loss of generality we can assume $\rho_{1}\in[0,1)$. We will prove first that 2. holds for the unique solution $u$ of (4.2) in the space $L^{2}\left(\Omega\times\left[0,\,T\right];\,H_{0}^{1}\left(\mathbb{R}^{+}\right)\right)$. Since the existence and uniqueness of this solution follows from the main results of [KR81] (pages 18-20), for any fixed volatility path and any square integrable initial density, we do not need to use 1. The estimate (4.3) for that $u$ is also obtained without using 1., which means that we can use 2. for proving 1. next. Indeed, applying Ito’s formula for the $L^{2}$ norm (Theorem 3.2 in [KR81] for the triple $H_{0}^{1}\subset L^{2}\subset H^{-1}$) on (4.2) and observing that $\int_{0}^{\infty}u_{xx}u=-\int_{0}^{\infty}u_{x}^{2}$ (by the definition of the distributional second derivative, since $u\in H_{0}^{1}$ can be approximated by smooth and compactly supported functions in that space), we obtain

[TABLE]

for all $0\leq t\leq T$, for all $\omega\in\Omega^{T}$ with $\mathbb{P}\left(\Omega^{T}\right)=1$. The desired result follows since $\mathbb{P}\left({\displaystyle\Omega^{\infty}}\right)=1$ for $\Omega^{\infty}={\displaystyle\cap_{N\in\mathbb{N}}\Omega^{N}}$.

We proceed now to the proof of 1. which will be divided into 3 steps. In what follows, we will be working on the set ${\displaystyle\Omega^{\infty}}$ which is defined above and which is a set of full probability.

1. The constant volatility case:

We assume first that the path $\sigma_{t}$ is constant in $t\geq 0$, i.e $\sigma_{t}=\sigma^{2}\;\forall\,t\geq 0$. In that case, $V_{t}$ is the limit empirical process studied in [BHHJR] and [HL16] (without the compactly supported initial data restriction), scaled by $\sigma>0$, so it does have a density $u(t,\,\cdot)=u(t,\,\cdot,\,W_{\cdot}^{0})$ which is the unique solution of the SPDE

[TABLE]

in $L^{2}\left(\Omega\times\left[0,\,T\right];\,H_{0}^{1}\left(\mathbb{R}^{+}\right)\right)$, under the initial condition $u(0,\cdot)=u_{0}$, which is actually (4.2). It holds also that $xu_{xx}(t,\,x)$ is square integrable.