In this paper we study some large deviations principles for conditionally continuous Gaussian processes. Then we find estimates of level crossing probability for such processes. Large deviations theory is concerned with the study of probabilities of very “rare” events. There are events whose probability is very small, however these events are of great importance; they may represent an atypical situation (i.e. a deviation from the average behavior) that may cause disastrous consequences: an insurance company or a bank may bankrupt; a statistical estimator may give a wrong information; a physical or chemical system may show an atypical configuration. The aim of this paper is to extend the theory of large deviations for Gaussian processes to a wider class of random processes – the conditionally Gaussian processes. Such processes were introduced in applications in finance, optimization and control problems. See, for instance, [12, 16, 14] and [1]. More precisely, Doucet et al. in [12] considered modelling the behavior of latent variables in neural networks by Gaussian processes with random parameters; Lototsky in [16] studied stochastic parabolic equations with random coefficients; Gulisashvili in [14] studied large deviations principle for some particular stochastic volatility models where the log-price is, conditionally, a Gaussian process; in [1] probabilities of large extremes of conditionally Gaussian processes were considered, in particular sub-Gaussian processes i.e. Gaussian processes with a random variance. Let $(Y,Z)$ be a random element on the probability space $(\varOmega ,\mathcal{F},\mathbb{P})$, where $Z={({Z_{t}})}_{t\in [0,1]}$ is a process taking values in $\mathbb{R}$ and Y is an arbitrary random element (a process or a random variable). We say that Z is a conditionally Gaussian process if the conditional distribution of the process $Z|Y$ is (almost surely) Gaussian. The theory of large deviations for Gaussian processes and for conditioned Gaussian processes is already well developed. See, for instance, Section 3.4 in [11] (and the references therein) for Gaussian processes, [7] and [13] for particular conditioned Gaussian processes. The extension of this theory is possible thanks to the results obtained by Chaganty in [8].

We consider a family of processes ${({Y^{n}},{Z^{n}})_{n\in \mathbb{N}}}$ on a probability space $(\varOmega ,\mathcal{F},\mathbb{P})$. ${({Y^{n}})_{n\in \mathbb{N}}}$ is a family of processes taking values in a measurable space $({E_{1}},{\mathcal{E}_{1}})$ that satisfies a large deviation principle (LDP for short) and ${({Z^{n}})_{n\in \mathbb{N}}}$ is a family of processes taking values in $({E_{2}},{\mathcal{E}_{2}})$ such that for every $n\in \mathbb{N}$, ${Z^{n}}|{Y^{n}}$ is a Gaussian process ($\mathbb{P}$-a.s.). We want to find a LDP for the family ${({Z^{n}})_{n\in \mathbb{N}}}$.

A possible application of LDPs is computing the estimates of level crossing probability (ruin problem). We will give the asymptotic behavior (in terms of large deviations) of the probability where φ is a suitable function. We will consider the following families of conditionally Gaussian processes.

1) The class of Gaussian processes with random variance and random mean, i.e. the processes of the type ${({Z_{t}})}_{t\in [0,1]}={({Y_{1}}{X_{t}}+{Y_{2}})}_{t\in [0,1]}$, where X is a centered continuous Gaussian process with covariance function k and $Y=({Y_{1}},{Y_{2}})$ is a random element independent of X. Notice that $Z|Y$ is Gaussian with and

2) The class of Ornstein–Uhlenbeck type processes with random diffusion coefficients. More precisely ${({Z_{t}})}_{t\in [0,1]}$ is the solution of the following stochastic differential equation: where $x,{a_{0}},{a_{1}}\in \mathbb{R}$ and Y is a random element independent of the Brownian motion ${({W_{t}})}_{t\in [0,1]}$.

The paper is organized as follows. In Section 2 we recall some basic facts on large deviations theory for continuous Gaussian processes. In Section 3 we introduce the conditionally Gaussian processes and the Chaganty theory. In Section 4 and 5 we study the theoretical problem and we give the main results. Finally in Section 6 we investigate the ruin problem for such processes.

We briefly recall some main facts on large deviations principles and reproducing kernel Hilbert spaces for Gaussian processes we are going to use. For a detailed development of this very wide theory we can refer, for example, to the following classical references: Chapitre II in Azencott [2], Section 3.4 in Deuschel and Strook [11], Chapter 4 (in particular Sections 4.1 and 4.5) in Dembo and Zeitouni [10], for large deviations principles; Chapter 4 (in particular Section 4.3) in [15], Chapter 2 (in particular Sections 2.2 and 2.3) in [5], for reproducing kernel Hilbert space. Without loss of generality, we can consider centered Gaussian processes.

In this section we introduce conditionally Gaussian processes and the Chaganty theorem which allows us to find a LDP for families of such processes. We also recall, for sake of completeness, some results about conditional distributions in Polish spaces. We referred to Section 3.1 in [6] and Section 4.3 in [3].

Let Y and Z two random variables, defined on the same probability space $(\varOmega ,\mathcal{F},\mathbb{P})$, with values in the measurable spaces $({E_{1}},{\mathcal{E}_{1}})$ and $({E_{2}},{\mathcal{E}_{2}})$ respectively, and let us denote by ${\mu _{1}},{\mu _{2}}$ the (marginal) laws of Y and Z respectively and by μ the joint distribution of $(Y,Z)$ on $(E,\mathcal{E})=({E_{1}}\times {E_{2}},{\mathcal{E}_{1}}\times {\mathcal{E}_{2}})$. A family of probabilities $({\mu _{2}}{(dz|y)))_{y\in {E_{1}}}}$ on $({E_{2}},{\mathcal{E}_{2}})$ is a regular version of the conditional law of Z given Y if In this case we have In this section we will use the notation $(E,\mathcal{B})$ to indicate a Polish space (i.e. a complete separable metric space) with the Borel σ-field, and we say that a sequence ${({x_{n}})}_{n\in \mathbb{N}}\subset \mathbb{E}$ converges to $x\in E$, ${x_{n}}\to x$, if ${d_{E}}({x_{n}},x)\to 0$, as $n\to \infty $, where ${d_{E}}$ denotes the metric on E. Regular conditional probabilities do not always exist, but they exist in many cases. The following result, that immediately follows from Corollary 3.2.1 in [6], shows that in Polish spaces the regular version of the conditional probability is well defined.

In what follows we always suppose random variables taking values in a Polish space.

The main tool that we will use to study LDP for a family of conditionally Gaussian processes is provided by Chaganty Theorem (Theorem 2.3 in [8]). Let $({E_{1}},{\mathcal{B}_{1}})$ and $({E_{2}},{\mathcal{B}_{2}})$ be two Polish spaces. We denote by ${({\mu _{n}})}_{n\in \mathbb{N}}$ a sequence of probabilities measures on $(E,\mathcal{B})=({E_{1}}\times {E_{2}},{\mathcal{B}_{1}}\times {\mathcal{B}_{2}})$ (the sequence of joint distributions), by ${({\mu _{1n}})}_{n\in \mathbb{N}}$ the sequence of the marginal distributions on $({E_{1}},{\mathcal{B}_{1}})$ and by ${({\mu _{2n}}(\cdot |{x_{1}}))}_{n\in \mathbb{N}}$ the sequence of conditional distributions on $({E_{2}},{\mathcal{B}_{2}})$ (${x_{1}}\in {E_{1}}$,), given by Proposition 2, i.e.

Let $\alpha >0$ and define ${\mathcal{C}_{\alpha }}([0,1])=\{y\in \mathcal{C}([0,1]):y(t)\ge \alpha ,\hspace{0.1667em}\hspace{0.1667em}t\in [0,1]\}$ (with the uniform norm on compact sets). ${\mathcal{C}_{\alpha }}([0,1])$ is a Polish space. Consider the family of processes ${({Y^{n}},{Z^{n}})_{n\in \mathbb{N}}}$, where ${({Y^{n}})_{n\in \mathbb{N}}}={({Y_{1}^{n}},{Y_{2}^{n}})_{n\in \mathbb{N}}}$ is a family of processes with paths in ${\mathcal{C}_{\alpha }}([0,1])\times \mathcal{C}([0,1])$ and for $n\in \mathbb{N}$, ${Z^{n}}={X^{n}}{Y_{1}^{n}}+{Y_{2}^{n}}$ with ${({Y^{n}})_{n\in \mathbb{N}}}$ independent of ${({X^{n}})_{n\in \mathbb{N}}}$. Suppose ${({({X_{t}^{n}})_{t\in [0,1]}})}_{n\in \mathbb{N}}$ is family of continuous centered Gaussian processes which satisfy the hypotheses of Theorem 2 and suppose that ${({Y^{n}})_{n\in \mathbb{N}}}$ satisfies a LDP with the good rate function ${I_{Y}}$ and the speed $\gamma (n)$. We want to prove a LDP principle for ${({Z^{n}})_{n\in \mathbb{N}}}$.

As far as the LDP is concerned exponentially equivalent measures are indistinguishable. See Theorem 4.2.13 in [10].

Let us denote $J(z|y)={\varLambda _{y}^{\ast }}(z)$, for $z\in \mathcal{C}([0,1])$ and $y\in {\mathcal{C}_{\alpha }}([0,1])\times \mathcal{C}([0,1])$. We want to prove the lower semicontinuity of $J(\cdot |\cdot )$.

Let $\alpha >0$ and let again ${\mathcal{C}_{\alpha }}([0,1])=\{y\in \mathcal{C}([0,1]):y(t)\ge \alpha ,\hspace{0.1667em}\hspace{0.1667em}t\in [0,1]\}$. Consider the family of processes ${({Y^{n}},{Z^{n}})_{n\in \mathbb{N}}}$, where ${({Y^{n}})_{n\in \mathbb{N}}}$ is a family of processes with paths in ${\mathcal{C}_{\alpha }}([0,1])$ and for $n\in \mathbb{N}$, ${Z^{n}}$ is the solution of the following stochastic differential equation. where, $x,\hspace{0.1667em}{a_{0}},\hspace{0.1667em}{a_{1}}\in \mathbb{R}$ and ${({Y^{n}})_{n\in \mathbb{N}}}$ is a family of random processes independent of the Brownian motion ${({W_{t}})}_{t\in [0,1]}$.

Suppose that ${({Y^{n}})_{n\in \mathbb{N}}}$ satisfies a LDP with the good rate function ${I_{Y}}$, and the speed $\gamma (n)=n$. We want to prove a LDP principle for ${({Z^{n}})_{n\in \mathbb{N}}}$.

Let ${Z^{n,y}}$, $y\in {\mathcal{C}_{\alpha }}([0,1])$, be the solution of the following stochastic differential equation, that is $m(t)={e^{{a_{1}}t}}(x+\frac{{a_{0}}}{{a_{1}}}[1-{e^{-{a_{1}}t}}])$. ${({({Z_{t}^{n,y}})_{t\in [0,1]}})}_{n\in \mathbb{N}}$ is a family of Gaussian processes and a family of diffusions. It is well known from the Wentzell–Friedlin theory that ${({({Z_{t}^{n,y}})_{t\in [0,1]}})}_{n\in \mathbb{N}}$ satisfies the LDP in $\mathcal{C}([0,1])$ with the speed $\gamma (n)=n$ and the good rate function where And it is well known from the theory of Gaussian processes that the family ${({({Z_{t}^{n,y}})_{t\in [0,1]}})}_{n\in \mathbb{N}}$ satisfies the LDP in $\mathcal{C}([0,1])$ with the speed $\gamma (n)=n$ and the good rate function where ${\mathcal{H}_{y}}$ is the reproducing kernel Hilbert space associate to the covariance function

The two rate functions (for the unicity of the rate function) are the same rate function. So we can deduce a LDP for the family ${({Z^{n}})_{n\in \mathbb{N}}}$ in two different ways. First let ${({Z_{t}^{n,y}})_{t\in [0,1]}}$ be a family of diffusions.

We now want to prove the lower semicontinuity of $J(\cdot |\cdot )$ on $\mathcal{C}([0,1])\times {\mathcal{C}_{\alpha }}([0,1])$.

Now let ${({Z_{t}^{n,y}})_{t\in [0,1]}}$ be a family of continuous Gaussian processes. We have to prove that ${({Z_{t}^{n,{y^{n}}}})_{t\in [0,1]}}$ satisfies the same LDP as ${({Z_{t}^{n,y}})_{t\in [0,1]}}$ when ${y^{n}}\stackrel{{\mathcal{C}_{\alpha }}([0,1])}{\longrightarrow }y$. Let ${\tilde{Z}_{t}^{n,{y^{n}}}}={Z_{t}^{n,{y^{n}}}}-m(t)$ for every $n\in \mathbb{N}$ and $t\in [0,1]$.

Straightforward calculations show that there exists $L>0$, such that Therefore the family ${({\tilde{Z}^{n,{y^{n}}}})_{n\in \mathbb{N}}}$ is exponentially tight at the speed n. Furthermore, conditions (6) and (7) of Theorem 2 are fullfilled, in fact and where ${k^{y}}(s,t)={e^{{a_{1}}(s+t)}}{\int _{0}^{s\wedge t}}{e^{-2{a_{1}}u}}{y^{2}}(u)\hspace{2.5pt}du$. Therefore ${({\tilde{Z}^{n,{y^{n}}}})_{n\in \mathbb{N}}}$ satisfies a LDP on $\mathcal{C}([0,1])$. Finally, thanks to the contraction principle, the family ${({Z^{n,{y^{n}}}})_{n\in \mathbb{N}}}$ satisfies a LDP on $\mathcal{C}([0,1])$ with the rate function $J(\cdot |y)$ defined in (16).

We have proved that the hypotheses of Theorem 3 are verified, so the LDP for ${({Z^{n}})_{n\in \mathbb{N}}}$ follows.

In this section we will study the probability of level crossing for a family of conditionally Gaussian processes. In particular, we will study the probability as $n\to \infty $, where ${({Z^{n}})_{n\in \mathbb{N}}}$ is a family of conditionally Gaussian process. In this situation the probability ${p_{n}}$ has a large deviation limit The main reference in this section is [4]. We now compute ${\lim \nolimits_{n\to \infty }}\frac{1}{\gamma (n)}\log ({p_{n}})$, for a fixed continuous path $\varphi \in \mathcal{C}([0,1])$. The computation is simple, in fact, since ${({Z^{n}})_{n\in \mathbb{N}}}$ satisfies a LDP with the rate function where ${I_{Y}}(\cdot )$ is the rate function associated to the family of conditioning processes ${({Y^{n}})_{n\in \mathbb{N}}}$, C is the Polish set where ${({Y^{n}})_{n\in \mathbb{N}}}$ takes values, and $J(\cdot |y)$ is the good rate function of the family of Gaussian processes ${({Z^{n,y}})_{n\in \mathbb{N}}}$. If we denote we have that where and It is a simple calculation to show that ${\inf _{w\in \stackrel{\circ }{\mathcal{A}}}}{I_{Z}}(w)={\inf _{w\in \bar{\mathcal{A}}}}{I_{Z}}(w)$. Therefore, For every $t\in [0,1]$ let ${\mathcal{A}_{t}}=\{w\in \mathcal{C}([0,1]):w(t)=1+\varphi (t)\}$, then $\mathcal{A}={\bigcup _{0\le t\le 1}}{\mathcal{A}_{t}}$ and so