A discrete-time model that weakly converges to a continuous-time geometric Brownian motion with Markov switching drift rate

This research is devoted to studying a geometric Brownian motion with drift switching driven by a 2 × 2 Markov chain. A discrete-time multiplicative approximation scheme was developed, and its convergence in Skorokhod topology to the continuous-time geometric Brownian motion with switching has been proved. Furthermore, in a financial market where the discounted asset price follows a geometric Brownian motion with drift switching, market incompleteness was established, and multiple equivalent martingale measures were constructed.

1 Introduction

In this article, we study a geometric Brownian motion with Markov switching in the drift coefficient. Assume that (_Xt)t≥0 follows a linear stochastic differential equation

$\begin{array}{ll}d{X}_t=({\delta }_0{Y}_t+{\delta }_1(1-Y_t))X_{t}dt+\sigma X_{t}d{W}_t,X_0=x_0,& (1)\end{array}$

where x₀ > 0 is non-random, δ₀, δ₁ ∈ ℝ, (_Wt)t≥0 is a Brownian motion, and (_Yt)t≥0 is an independent of W continuous-time Markov jump process with the values in the set {0, 1}, with the initial value Y₀ = 0 and with an infinitesimal matrix

$\begin{array}{ll}\text{𝔸}=\left(\begin{array}{cc}-{\lambda }_0& {\lambda }_0\\ {\lambda }_1& -{\lambda }_1\end{array}\right),& (2)\end{array}$

for some positive λ₀ and λ₁. Moreover, let the processes (_Yt)t≥0 and (_Wt)t≥0 be defined on a stochastic basis with filtration $(\text{Ω},\text{F},{({\text{F}}_t)}_{t\ge 0},\mathbb{P})$, where ${\text{F}}_t=\sigma \left\{W_s,Y_s,0\le s\le t\right\}$. It is well known that the strong solution of Equation (1) can be represented as an exponent of the form

$\begin{array}{ll}{X}_t=X_{0}exp\left({\displaystyle {\int }_0^t}({\delta }_0{Y}_s+{\delta }_1(1-Y_s))ds+\sigma W_t-\frac{{\sigma }^2}{2}t\right).& (3)\end{array}$

Drift-switching models have been applied in finance and economics for several decades. Early applications of drift switching in the context of time-series econometrics can be found in Quandt [1] or Quandt and Goldfeld [2]. Hamilton [3] used drift switching to model the business cycle, where the expected growth rates of a national product switch according to a Markov chain. In finance, geometric Brownian motion with a Markov chain-modulated drift rate has become popular for modeling asset price dynamics. For instance, Ang and Timmermann [4] and Sotomayor and Cdenillas [5] studied regime-switching models in finance, while Dai et al. [6] and Dai et al. [7] investigated optimal trend-following trading strategies for an asset price modeled by a stochastic differential (Equation 1). In this context, the switching drift rates correspond to bull and bear market conditions. Maheu et al. [8] focus on the identification and estimation aspects of such models. In a similar setting, Décamps et al. [9] and Klein [10] examine optimal investment timing in a risky project with a sunk cost. The study by Aingworth, Das and Motwani [11] was devoted to pricing equity options with Markov switching. Elliott et al. [12] also studied option pricing in models with Markov switching. Bae et al. [13] investigate the problem of asset allocation under regime switching and Ekström and Lu [14] study an optimal irreversible sale of an asset, while Ekström and Lindberg [15] analyze optimal closing strategies for momentum trades. Henderson et al. [16] study exercise patterns of American call executive stock options written on a stock whose drift parameter falls to a lower value at an exponentially distributed random time.

This study focuses on discretizing a geometric Brownian motion with a Markov switching drift rate, as described by Equation (1). Since explicit solutions to models with switching drift rates are rare, rigorous discretization and an understanding of its properties are essential for implementing numerical methods such as binomial and multinomial trees, PDE solvers, or Monte Carlo simulations for these models. Furthermore, in time-series econometrics, a discrete-time version of Equation (1) is typically used from the outset, albeit with only a vague connection to the continuous-time model. Our analysis rigorously connects the continuous- and discrete-time models and provides their convergence properties.

Note that a wide class of theorems on diffusion approximation of additive schemes were proved in the book of Liptser and Shiryaev [17] and generalized to multiplicative schemes in the book of Mishura and Ralchenko [18]. The present study is, in a context, a modification of the functional limit theorems obtained in Chapter 1 of the book [18]. However, to the best of our knowledge, multiplicative Markov switching schemes and their corresponding functional limit theorems have not been previously established.

In addition to the problem of the approximation (in the context of functional limit theorems) of a market with switching, we also investigated the question of the incompleteness of such a market. Intuitively, this incompleteness is obvious, since we have one risky asset with two independent sources of randomness. At the same time, it is easy to construct the so-called minimum martingale measure. It is more difficult to construct a class of equivalent martingale measures other than the minimal one. We managed to construct a fairly wide class of such measures, although it is obvious that all equivalent martingale measures are not exhausted by such a construction.

This study is organized as follows: In Sections 2 and 3, we develop a discretization for the switching component of the process (Equation 1) and prove the weak convergence of the respective probability measures, generated by the prelimit and limit processes, respectively. Section 4 is devoted to the weak convergence of the measures corresponding to the component responsible for volatility. Then, due to the independence of these processes and, consequently, of respective probability measures, we get the weak convergence of the products of these measures, or that is, of the sequence of probability measures generated by the prelimit sequence of probability measures, to the measure corresponding to the limit process. Note also the following: while prelimit and limit Markov processes (chains) are discontinuous, we can establish their weak convergence in Skorokhod topology. However, their integral sums and also the components that are responsible for the weak convergence to geometric Brownian motion converge even in the uniform topology. So, finally, our processes converge in the uniform topology. Finally, Section 5 is devoted to the construction of a wide class of equivalent martingale measures for the market, where Equation (1) represents the discounted price of a risky asset.

2 Discrete-time multiplicative approximation of the diffusion model with Markov switching

The main goal of this study is to construct a sequence of discrete-time versions of X, the geometric Brownian motion with Markov modulated drift given by Equation (1) and Equation (3), such that these discretized versions weakly converge in Skorokhod topology (in fact, convergence will be even in the uniform topology) to the process X on the fixed time interval [0, T].

So, following this direction, we consider the limit process (_{Xt)t∈[0, T]} on the fixed time interval [0, T], where T > 0 is a maturity date, and create a series of discrete-time models numbered by N ∈ ℕ. Our Nth discrete-time market corresponds to the partition of the interval [0, T] into N subintervals of the form $\left[\frac{(k-1)T}{N},\frac{kT}{N}\right]$, 1 ≤ k ≤ N. Let $X_0^{(N)}=x_0$, and $X_k^{(N)}$ be a strictly positive discounted price of the asset at a time $\frac{kT}{N}$ of Nth discrete-time market, 1 ≤ k ≤ N.

Taking into account the multiplicative nature of the limit model, together with the assumption of independence of Y and W on [0, T], we can assume that the ratio $\frac{X_k^{(N)}}{X_{k-1}^{(N)}}$, 1 ≤ k ≤ N can be represented as a product

$\begin{array}{ll}\frac{X_k^{(N)}}{X_{k-1}^{(N)}}=(1+R_k^{(1,N)})(1+R_k^{(2,N)}),& (4)\end{array}$

where random variables $R_k^{(i,N)}$, i ∈ {1, 2}, 1 ≤ k ≤ N are independent and $R_k^{(i,N)}>-1$ almost surely (a.s.) Taking logarithms in Equation (4), we can write

$\begin{array}{l}{U}_k^{(N)}=log(X_k^{(N)})=log(X_0)+{\displaystyle \sum _{j=1}^k}log(1+R_j^{(1,N)})\\ \hspace{1em}\hspace{1em}+{\displaystyle \sum _{j=1}^k}log(1+R_j^{(2,N)}),& (5)\end{array}$

where 1 ≤ k ≤ N and $U_0^{(N)}=log(x_0)$. We assume that the process X^(N) is defined on the stochastic basis $({\text{Ω}}^{(N)},{\text{F}}^{(N)},R_k^{(1,N)}{({\text{F}}_t^{(N)})}_{t\in \left[0,T\right]},{\mathbb{P}}^{(N)})$, where filtration is generated by the respective random variables $R_k^{(i,N)},i=1,2$ so that $X_k^{(N)}$ is ${\text{F}}_{\frac{kT}{N}}^{(N)}$-measurable. In this model, random variables $R_k^{(1,N)}$ represent non-volatile net profit rates generated by the price process on the time intervals $\left[\frac{(k-1)T}{N},\frac{kT}{N}\right]$, 1 ≤ k ≤ N in a model with switching. Recall that we consider (_Ys)s≥0, which is the jump Markov process with values in the set {0, 1} and an infinitesimal matrix (Equation 1). This process governs the switching in a continuous-time model. Recall also that state 0 generates income with intensity δ₀ and state 1 generates income with intensity δ₁. Once we consider a discrete-time model, we have to introduce a discrete-time switching process (note that in such a model, the switching of the interest rate may only occur at times $\frac{kT}{N}$). Let ${(Y_k^{(N)})}_{k\ge 0}$ be a discrete-time, 2 × 2 Markov chain defined on the same probability space as R^{(1, N)}, R^{(2, N)}, and U^(N) which is defined in Equation (5). It is independent of the processes R^{(2, N)} and U^{(2, N)}. The chain takes values in the set {0, 1} and has initial values $Y_0^{(N)}=0$ and $Y_k^{(N)}=0$, implying that the intensity of the interest on the kth interval of the Nth discrete-time market equals δ₀. Similarly, $Y_k^{(N)}=1$ means that such intensity equals δ₁.

The definition of the transition probabilities matrix for the process Y^(N) follows from the requirement for occupation times of Y^(N) to be close to those of (_Ys)s≥0. This leads to the following definition of the transition probabilities of the chain Y^(N) for i ∈ {0, 1}:

$\begin{array}{l}{\mathbb{P}}^{(N)}\left(Y_{k+1}^{(N)}=i|Y_k^{(N)}=i\right)\\ =\mathbb{P}\left(Y_s=i,\frac{Tk}{N}\le s\le \frac{T(k+1)}{N}|Y_{\frac{Tk}{N}}=i\right)\\ =\mathbb{P}(Y_s=i,0\le s\le T/N|Y_0=i)=e^{-\frac{{\lambda }_{i}T}{N}},\end{array}$

where we used the Markov property of the process (_Ys)s≥0. Such probabilities define a one-step transition probability matrix

$\begin{array}{ll}\left(\begin{array}{cc}{e}^{-{\lambda }_0\frac{T}{N}}& 1-e^{-{\lambda }_0\frac{T}{N}}\\ 1-e^{-{\lambda }_1\frac{T}{N}}& e^{-{\lambda }_1\frac{T}{N}}\end{array}\right).& (6)\end{array}$

Using the switching process Y^(N), we can define random variables $R_k^{(1,N)},0\le k\le N$, as follows:

$\begin{array}{ll}{R}_k^{(1,N)}=\frac{{\delta }_{0}T}{N}{Y}_k^{(N)}+\frac{{\delta }_{1}T}{N}(1-Y_k^{(N)}).& (7)\end{array}$

Definition 7 has the following financial interpretation: Since $R_k^{(1,N)},1\le k\le N$ is a profit rate generated by the risky asset on the kth time interval, the accrual on this interval equals to

$\begin{array}{ll}1+R_k^{(1,N)}=exp\left(\frac{{\delta }_{0}T}{N}\right)Y_k^{(N)}+exp\left(\frac{{\delta }_{1}T}{N}\right)(1-Y_k^{(N)}).& (8)\end{array}$

Equation (8) can be written as:

$\begin{array}{ll}{R}_k^{(1,N)}=\left(exp\left(\frac{{\delta }_{0}T}{N}\right)-1\right)Y_k^{(N)}+\left(exp\left(\frac{{\delta }_{1}T}{N}\right)-1\right)\left(1-Y_k^{(N)}\right).& (9)\end{array}$

Using the Taylor formula, we can write $R_k^{(1,N)}$ as follows:

$\begin{array}{l}{R}_k^{(1,N)}=\left(\frac{{\delta }_{0}T}{N}+o\left(\frac{{\delta }_{0}T}{N}\right)\right)Y_k^{(N)}+\left(\frac{{\delta }_{1}T}{N}+o\left(\frac{{\delta }_{1}T}{N}\right)\right)(1-Y_k^{(N)}).\end{array}$

By neglecting asymptotically small terms $o\left(\frac{{\delta }_{0}T}{N}\right)$ and $o\left(\frac{{\delta }_{1}T}{N}\right)$, we arrive at the definition (Equation 8).

Now, we turn our attention to $R_k^{(2,N)}$. This random variable represents the pure volatility in the model. In our discrete-time markets, the sums ${\displaystyle \sum _{j=1}^k}log(1+R_j^{(2,N)})$, roughly speaking, will approximate the process $\sigma W_t-\frac{{\sigma }^2}{2}t$.

Now, as we defined discrete-time markets and prelimit processes $(U_k^{(N)},0\le k\le N)$, we can give a mathematical formulation for the main goal of this study, which is the convergence of discrete-time markets to the market described by Equation (1). By “convergence of discrete-time markets,” we mean weak convergence of probability measures associated with stochastic processes that drive such markets, or convergence of random processes in Skorokhod or uniform topology. It will be specified explicitly in any theorem.

Next, we define the logarithm of the limit price process by

$U_t=log(X_t)=log(X_0)+{\displaystyle {\int }_0^t}({\delta }_0{Y}_s+{\delta }_1(1-Y_s))ds+\sigma W_t-\frac{{\sigma }^{2}t}{2},$

t ∈ [0, T]. It is convenient to separate the components of U_t and $U_k^{(N)}$ as follows:

$\begin{array}{c}{U}_t=log{X}_0+U_t^{(1)}+U_t^{(2)},\\ U_k^{(N)}=log{X}_0+U_k^{(1,N)}+U_k^{(2,N)},\end{array}$

where

$\begin{array}{c}{U}_t^{(1)}={\displaystyle {\int }_0^t}({\delta }_0{Y}_s+{\delta }_1(1-Y_s))ds,U_t^{(2)}=\sigma W_t-\frac{{\sigma }^{2}t}{2},\\ U_k^{(i,N)}={\displaystyle \sum _{j=1}^k}log(1+R_j^{(i,N)}),i\in \left\{1,2\right\},1\le k\le N,U_0^{(i,N)}=0.\end{array}$

Let us define for $t\in [\frac{(k-1)T}{N},\frac{kT}{N})$,

$\begin{array}{c}{U}_t^{(N)}=U_{k-1}^{(N)},U_T^{(N)}=U_N^{(N)},\\ U_t^{(i,N)}=U_{k-1}^{(i,N)},U_T^{(i,N)}=U_N^{(i,N)},\\ R_t^{(N)}=R_{k-1}^{(N)},R_T^{(N)}=R_N^{(N)},\\ Y_t^{(N)}=Y_{k-1}^{(N)},Y_T^{(N)}=Y_N^{(N)},1\le k\le N,& (10)\end{array}$

i ∈ {1, 2}, 1 ≤ k ≤ N. So, we consider step-wise discrete-time approximations of the limit process U. Thus, our goal is to prove the weak convergence of the sequence of stochastic processes ${(U_t^{(N)})}_{t\in \left[0,T\right]}$ to the process (_{Ut)t∈[0, T]}. To this end, we will establish the convergence of ${(Y_t^{(N)})}_{t\in \left[0,T\right]}$ to (_{Yt)t∈[0, T]} (Theorem 3), then the convergence of ${(U_t^{(i,N)})}_{t\in \left[0,T\right]}$ to ${(U_t^{(i)})}_{t\in \left[0,T\right]}$, i ∈ {1, 2} (Theorems 4 and 5), and the desired result then follows because of the independence of probability measures respective to Markov chains and the components that converge to the geometric Brownian motion. Therefore, the respective products of the probability measures weakly converge to the product of probability measures corresponding to the limit Markov chain and the limit geometric Brownian motion, respectively.

3 Weak convergence of discrete-time Markov chains to the limit Markov process

In this section, we prove that the sequence of processes ${(Y^{(N)})}_{t\in \left[0,T\right]}$ introduced in Section 2 converges in Skorokhod topology to the process (_{Yt)t∈[0, T]}. As a consequence, we will obtain the convergence of the processes ${(U^{(1,N)})}_{t\in \left[0,T\right]}$ to ${(U^{(1)})}_{t\in \left[0,T\right]}$, however, even in the uniform topology.

Let N_t be the number of jumps of a process (Y_s, s ≥ 0) on a time interval [0, t]. Let us introduce the occupation times

$\begin{array}{l}{\theta }_0=inf\left\{t>0|Y_t\ne Y_0\right\},{\theta }_n=inf\left\{t>{\theta }_{n-1}:Y_t\ne Y_{{\theta }_{n-1}}\right\}\\ -{\theta }_{n-1},n\ge 1,\end{array}$

and jump times

${\tau }_k={\displaystyle \sum _{j=0}^k}{\theta }_k,k\ge 0.$

Recall that the Markov chain $(Y_k^{(N)},k\ge 0)$, introduced in Section 2, has an initial value of $Y_0^{(N)}=0$ and the transition probability matrix (Equation 6). For this chain, let us define the total number of jumps on the time interval [0, T]

${\nu }_N={\displaystyle \sum _{j=0}^{N-1}}|Y_{j+1}^{(N)}-Y_j^{(N)}|,$

occupation times

$\begin{array}{l}{\theta }_0^{(N)}=inf\left\{k>0|{\displaystyle Y_k^{(N)}}\ne Y_0^{(N)}\right\},\\ {\theta }_n^{(N)}=inf\left\{k>{\displaystyle {\theta }_{n-1}^{(N)}}:Y_k^{(N)}\ne Y_{{\theta }_{n-1}^{(N)}}^{(N)}\right\},n\ge 1,\end{array}$

and jump times

${\tau }_k^{(N)}={\displaystyle \sum _{k=0}^k}{\theta }_j^{(N)},k\ge 0.$

For a given t ∈ [0, T] and integer N, define k_{t, N} ∈ {0, …, N} in the following way: k_{T, N} = N, and for t ∈ [0, T), we have $t\in [\frac{k_{t,N}T}{N},\frac{(k_{t,N}+1)T}{N}).$ We will also use the notation $t^{(N)}=\frac{k_{t,N}T}{N}$.

Lemma 1. For all k ≥ 1, the following inequality holds:

$\begin{array}{ll}\mathbb{P}(N_T=k)\le d{C}^{k}exp\left(-\frac{|{\lambda }_1-{\lambda }_0|T}{2}k\right),& (11)\end{array}$

where

$\begin{array}{ll}d=max\left\{{\left(\frac{{\lambda }_0}{{\lambda }_1}\right)}^{\frac{1}{2}},1\right\}max\left\{e^{-{\lambda }_{0}T},e^{-\frac{|3{\lambda }_1-{\lambda }_0|}{2}T},e^{-\frac{|{\lambda }_0-{\lambda }_1|}{2}T}\right\},& (12)\end{array}$

and

$\begin{array}{ll}C=\frac{e^{|{\lambda }_1-{\lambda }_0|}-1}{|{\lambda }_1-{\lambda }_0|}{({\lambda }_0{\lambda }_1)}^{\frac{1}{2}}.& (13)\end{array}$

Proof. We have the following relations:

$\begin{array}{l}\mathbb{P}(N_T=2m)\\ \hspace{1em}={\displaystyle {\int }_0^T}{\displaystyle {\int }_{t_0}^T}\dots {\displaystyle {\int }_{t_{2m-1}}^T}{\lambda }_0{e}^{-{\lambda }_0{t}_0}{\lambda }_1{e}^{-{\lambda }_1(t_1-t_0)}{\lambda }_0{e}^{-{\lambda }_0(t_2-t_1)}\\ \hspace{1em}\dots {\lambda }_1{e}^{-{\lambda }_1(t_{2m-1}-t_{2m-2})}{e}^{-{\lambda }_0(T-t_{2m-1})}d{t}_0\dots d{t}_{2m-1}\\ \hspace{1em}\le {\displaystyle {\int }_{{\left[0,T\right]}^{2m}}}{\lambda }_0{e}^{-{\lambda }_0{t}_0}{\lambda }_1{e}^{-{\lambda }_1(t_1-t_0)}\dots e^{-{\lambda }_0(T-t_{2m-1})}d{t}_0\dots d{t}_{2m-1}\\ \hspace{1em}={({\lambda }_0{\lambda }_1)}^m{e}^{-{\lambda }_{0}T}{\displaystyle {\int }_{{\left[0,T\right]}^{2m}}}\\ \hspace{1em}exp(({\lambda }_1-{\lambda }_0)(t_0-t_1+t_2\dots +t_{2m-1}))d{t}_0\dots d{t}_{2m-1}\\ \hspace{1em}={({\lambda }_0{\lambda }_1)}^m{e}^{-{\lambda }_{0}T}{\displaystyle \prod _{j=0}^{2m-1}}{\displaystyle {\int }_0^T}{e}^{{(-1)}^j({\lambda }_1-{\lambda }_0)t_j}d{t}_j\\ \hspace{1em}={({\lambda }_0{\lambda }_1)}^m{e}^{-{\lambda }_{0}T}{\displaystyle \prod _{j=0}^{2m-1}}\frac{{(-1)}^j(e^{{(-1)}^j({\lambda }_1-{\lambda }_0)T}-1)}{{\lambda }_1-{\lambda }_0}\\ \hspace{1em}={({\lambda }_0{\lambda }_1)}^m{e}^{-{\lambda }_{0}T}\frac{{(e^{({\lambda }_1-{\lambda }_0)T}-1)}^m{(1-e^{-({\lambda }_1-{\lambda }_0)T})}^m}{{({\lambda }_1-{\lambda }_0)}^{2m}}\\ \hspace{1em}={({\lambda }_0{\lambda }_1)}^m{e}^{-{\lambda }_{0}T}{\left(\frac{e^{({\lambda }_1-{\lambda }_0)T}-1}{{\lambda }_1-{\lambda }_0}\right)}^{2m}{e}^{-({\lambda }_1-{\lambda }_0)Tm}.\end{array}$

In the case when λ₁ > λ₀, from these relations, we immediately get inequality (Equation 11) for k = 2m. In the case when λ₀ > λ₁, we can rewrite previous estimates as

$\begin{array}{l}\mathbb{P}(N_T=2m)\\ \hspace{1em}\le {({\lambda }_0{\lambda }_1)}^m{e}^{-{\lambda }_{0}T}{\left(\frac{e^{({\lambda }_1-{\lambda }_0)T}-1}{{\lambda }_1-{\lambda }_0}\right)}^{2m}{e}^{-({\lambda }_1-{\lambda }_0)Tm}\\ \hspace{1em}={({\lambda }_0{\lambda }_1)}^m{e}^{-{\lambda }_{0}T}{\left(\frac{1-e^{({\lambda }_1-{\lambda }_0)T}}{{\lambda }_0-{\lambda }_1}\right)}^{2m}{e}^{-({\lambda }_1-{\lambda }_0)Tm}\\ \hspace{1em}={({\lambda }_0{\lambda }_1)}^m{e}^{-{\lambda }_{0}T}{\left(\frac{e^{({\lambda }_0-{\lambda }_1)T}-1}{{\lambda }_0-{\lambda }_1}\right)}^{2m}{e}^{-({\lambda }_1-{\lambda }_0)Tm+2({\lambda }_0-{\lambda }_1)Tm}\\ \hspace{1em}={({\lambda }_0{\lambda }_1)}^m{e}^{-{\lambda }_{0}T}{\left(\frac{e^{({\lambda }_0-{\lambda }_1)T}-1}{{\lambda }_0-{\lambda }_1}\right)}^{2m}{e}^{-({\lambda }_0-{\lambda }_1)Tm},\end{array}$

and also get the inequality (Equation 11).

Let us now switch to the case k = 2m−1. Following the same process as before, we obtain the inequality

$\begin{array}{l}\mathbb{P}(N_T=2m-1)\\ \le {\lambda }_0{({\lambda }_0{\lambda }_1)}^{m-1}{e}^{-{\lambda }_{1}T}{\left(\frac{e^{({\lambda }_1-{\lambda }_0)T}-1}{{\lambda }_1-{\lambda }_0}\right)}^{2m-1}{e}^{-({\lambda }_1-{\lambda }_0)Tm}.\end{array}$

If λ₁ > λ₀, then we can write

$\begin{array}{l}\mathbb{P}(N_T=2m-1)\le {\left(\frac{{\lambda }_0}{{\lambda }_1}\right)}^{\frac{1}{2}}{C}^{2m-1}exp(-({\lambda }_1-{\lambda }_0)Tm-{\lambda }_{1}T)\\ \le C^{2m-1}exp\left(-({\lambda }_1-{\lambda }_0)T\frac{2m-1}{2}-\frac{3{\lambda }_1-{\lambda }_0}{2}T\right),\end{array}$

so that Equation (11) holds true in this case.

If λ₀ > λ₁, then

$\begin{array}{l}\mathbb{P}(N_T=2m-1)\\ \le {\lambda }_0{({\lambda }_0{\lambda }_1)}^{m-1}{e}^{-{\lambda }_{1}T}{\left(\frac{e^{({\lambda }_0-{\lambda }_1)T}-1}{{\lambda }_0-{\lambda }_1}\right)}^{2m-1}{e}^{-({\lambda }_0-{\lambda }_1)Tm-({\lambda }_0-{\lambda }_1)T}\\ ={\left(\frac{{\lambda }_0}{{\lambda }_1}\right)}^{\frac{1}{2}}{C}^{2m-1}{e}^{-({\lambda }_0-{\lambda }_1)Tm}\\ ={\left(\frac{{\lambda }_0}{{\lambda }_1}\right)}^{\frac{1}{2}}{C}^{2m-1}exp\left(-({\lambda }_0-{\lambda }_1)T\frac{2m-1}{2}-\frac{{\lambda }_0-{\lambda }_1}{2}T\right),\end{array}$

and Equation (11) holds true.    □

Corollary 1.

$\begin{array}{ll}𝔼exp\left(\frac{|{\lambda }_1-{\lambda }_0|T}{4}{N}_T\right)<\infty .& (14)\end{array}$

Proof. Let us define Λ = |λ₁ − λ₀| and let constants C and d be as in Equations (13) and (12), respectively. From Lemma 1, we see that

$\begin{array}{ll}\mathbb{P}(N_t=k)\le d{C}^k{e}^{-\frac{\text{Λ}T}{2}k},& (15)\end{array}$

Inequality (14) is a direct consequence of Inequality (15), indeed, we can put $\alpha :=\frac{\text{Λ}T}{4}>0$ and get that

$\text{𝔼}{e}^{\alpha N_T}\le d{\displaystyle \sum _{k=0}^{\infty }}{C}^k{e}^{-\frac{\text{Λ}T}{4}k}<\infty .$

                □

Theorem 1. Denote by f_m(t₀, …, t_m) a conditional density of (τ₀, …, τ_m) given N_T = m and put

$g_m(t_0,\dots ,t_m)=f_m(t_0,\dots ,t_m)\mathbb{P}(N_T=m).$

Then, for any ε > 0, there exists an integer N(m) such that for all N ≥ N(m) and all 0 ≤ t₀ < … < t_m ≤ T, we have

$|g_m(t_0,\dots ,t_m)-{\left(\frac{N}{T}\right)}^{m+1}{\widehat{p}}_N(k_{t_0,N},\dots ,k_{t_m,N})|<\epsilon ,$

where ${\widehat{p}}_N(k_0,\dots ,k_m)={\mathbb{P}}^{(N)}\{{\tau }_0^{(N)}=k_0$, ${\tau }_1^{(N)}=k_1,\dots ,$ ${\tau }_m^{(N)}=k_m$, ${\tau }_{m+1}^{(N)}>N\}$.

Proof. We prove the statement for even m (so that we will write 2m in the following theorem). The proof for the odd m is the same.

Let ${\tilde{f}}_{2m}(t_0,\dots ,t_{2m})$ be a conditional density of (θ₀, …, θ_2m) given N_T = 2m and ${\tilde{g}}_{2m}(t_0,\dots ,t_{2m})={\tilde{f}}_{2m}(t_0,\dots ,t_{2m})\mathbb{P}(N_T=2m)$. Since {θ_j, 0 ≤ j ≤ 2m} are independent random variables with alternating exponential distributions, we can write

$\begin{array}{l}{\tilde{g}}_{2m}(t_0,\dots ,t_{2m})\\ ={\displaystyle \underset{h\to 0}{lim}}\frac{1}{{(2h)}^{2m+1}}\mathbb{P}(|{\theta }_j-t_j|<h,{\tau }_{2m+1}>T,0\le j\le 2m)\\ =\mathbb{P}({\theta }_{2m+1}>T-(t_0+\dots +t_{2m}))\\ {\displaystyle \underset{h\to 0}{lim}}{\displaystyle \prod _{j=0}^{2m}}\left(\frac{1}{2h}\mathbb{P}(|{\theta }_j-t_j|<h)\right)\\ ={\lambda }_0{e}^{-{\lambda }_0{t}_0}{\lambda }_1{e}^{-{\lambda }_1{t}_1}\dots {\lambda }_0{e}^{-{\lambda }_0{t}_{2m}}{e}^{-{\lambda }_1(T-(t_0+\dots +t_{2m}))},\end{array}$

for all t_j ≥ 0, 0 ≤ j ≤ 2m, such that t₀+…+t_2m ≤ T. Recall that θ₀ = τ₀ and θ_j = τ_j − τ_j−1, 1 ≤ j ≤ 2m. So we have for all 0 ≤ t₀ < … < t_2m ≤ T

$\begin{array}{l}{g}_{2m}(t_0,\dots ,t_{2m})\\ ={\tilde{g}}_{2m}(t_0,t_1-t_0,\dots ,t_{2m}-t_{2m-1})\\ ={\lambda }_0{e}^{-{\lambda }_0{t}_0}{\lambda }_1{e}^{-{\lambda }_1(t_1-t_0)}\dots {\lambda }_0{e}^{-{\lambda }_0(t_{2m}-t_{2m-1})}{e}^{-{\lambda }_1(T-t_{2m})}\\ ={\lambda }_0{({\lambda }_0{\lambda }_1)}^m{e}^{-{\lambda }_{1}T}exp\left(({\lambda }_1-{\lambda }_0){\displaystyle \sum _{j=0}^{2m}}{(-1)}^j{t}_j\right).\end{array}$

To simplify the further derivations, let us omit indices in k_{ti, N} and simply write k_i. Then we can rewrite ${\widehat{p}}_N(k_0,\dots ,k_m)$ as

$\begin{array}{c}{\widehat{p}}_N(k_0,\dots ,k_{2m})=e^{-{\lambda }_0\frac{k_{0}T}{N}}\left(1-e^{\frac{-{\lambda }_{0}T}{N}}\right)e^{-{\lambda }_1\frac{(k_1-k_0)T}{N}}\left(1-e^{\frac{-{\lambda }_{1}T}{N}}\right)\times \dots \times \\ \times e^{-{\lambda }_0\frac{k_{0}T}{N}}\left(1-e^{\frac{-{\lambda }_{0}T}{N}}\right)e^{\frac{-{\lambda }_1(N-k_{2m})T}{N}}={\left(1-e^{\frac{-{\lambda }_{0}T}{N}}\right)}^{m+1}{\left(1-e^{\frac{-{\lambda }_{1}T}{N}}\right)}^m\times \\ \times exp(-\frac{T}{N}({\lambda }_0{k}_0+{\lambda }_1(k_1-k_0)+{\lambda }_0(k_2-k_1)\dots \\ +{\lambda }_0(k_{2m}-k_{2m-1})+{\lambda }_1(N-k_{2m})))\\ ={\left(1-e^{\frac{-{\lambda }_{0}T}{N}}\right)}^{m+1}{\left(1-e^{\frac{-{\lambda }_{1}T}{N}}\right)}^m{e}^{-{\lambda }_{1}T}exp\left(({\lambda }_1-{\lambda }_0){\displaystyle \sum _{j=0}^{2m}}{(-1)}^j\frac{k_{j}T}{N}\right)\\ ={\left(1-e^{\frac{-{\lambda }_{0}T}{N}}\right)}^{m+1}{\left(1-e^{\frac{-{\lambda }_{1}T}{N}}\right)}^m{e}^{-{\lambda }_{1}T}exp\left(({\lambda }_1-{\lambda }_0){\displaystyle \sum _{j=0}^{2m}}{(-1)}^j{t}_j^{(N)}\right).\end{array}$

Furthermore, the following limit holds:

$\begin{array}{c}{\left(\frac{T}{N}\right)}^{-(2m+1)}{\left(1-e^{-{\lambda }_0\frac{T}{N}}\right)}^{m+1}{\left(1-e^{-{\lambda }_1\frac{T}{N}}\right)}^m\\ ={\left(\frac{1-e^{-{\lambda }_0\frac{T}{N}}}{(T/N)}\right)}^{m+1}{\left(\frac{1-e^{-{\lambda }_1\frac{T}{N}}}{(T/N)}\right)}^m\\ \to {\lambda }_0^{m+1}{\lambda }_1^m,\end{array}$

as N → ∞. For any ε₁ > 0, we can now find an integer N(m, ε₁) such that for all N ≥ N(m, ε₁), we have

$|{\left(\frac{T}{N}\right)}^{-2m-1}{\left(1-e^{\frac{-{\lambda }_{0}T}{N}}\right)}^{m+1}{\left(1-e^{\frac{-{\lambda }_{1}T}{N}}\right)}^m-{\lambda }_0^{m+1}{\lambda }_1^m|<{\epsilon }_1.$

Put

${\epsilon }_2={\lambda }_0^{m+1}{\lambda }_1^m{\epsilon }_1,\text{and }B=exp(|{\lambda }_1-{\lambda }_0|T).$

Note that

$exp\left(({\lambda }_1-{\lambda }_0){\displaystyle \sum _{j=0}^n}{(-1)}^j{s}_j\right)\le B,$

for all integer n > 0 and all 0 ≤ s₀ < s₁ < … < s_n ≤ T. We can now write

$\begin{array}{l}\frac{e^{{\lambda }_{1}T}}{{\lambda }_0^{m+1}{\lambda }_1^m}|g_{2m}(t_0,\dots ,t_{2m})-{\left(\frac{N}{T}\right)}^{2m+1}{\widehat{p}}_N(k_{t_0,N},\dots ,k_{t_{2m},N})|\le \\ \le |exp\left(({\lambda }_1-{\lambda }_0){\displaystyle \sum _{j=0}^{2m}}{(-1)}^j{t}_j\right)\\ -exp\left(({\lambda }_1-{\lambda }_0){\displaystyle \sum _{j=0}^{2m}}{(-1)}^j{t}_j^{(N)}\right)|+\\ +{\epsilon }_{2}exp\left(({\lambda }_1-{\lambda }_0){\displaystyle \sum _{j=0}^{2m}}{(-1)}^j{t}_j^{(N)}\right)=\\ =B\left(|exp\left(({\lambda }_1-{\lambda }_0){\displaystyle \sum _{j=0}^{2m}}{(-1)}^j(t_j-t_j^{(N)})\right)-1|+{\epsilon }_2\right)\\ \le B\left(|B^{\frac{2m+1}{N}}-1|+{\epsilon }_2\right).\end{array}$

Clearly, we can now choose an integer N₀ = N(m, ε) such that for all N ≥ N₀,

$|g_{2m}(t_0,\dots ,t_{2m})-{\left(\frac{N}{T}\right)}^{2m+1}{\widehat{p}}_N(k_{t_0,N},\dots ,k_{t_{2m},N})|<\epsilon .$

The theorem is proved.        □

Theorem 2. Let 0 ≤ t₀<t₁ < … < t_n ≤ T be fixed. Then, for any ε > 0 there exists an integer N(n, ε) such that for all N ≥ N(n, ε), we have

$\begin{array}{ll}|\mathbb{P}(Y_{t_i}=x_i,0\le i\le n)-{\mathbb{P}}^{(N)}\left(Y_{t_i^{(N)}}^{(N)}=x_i,0\le i\le n\right)|<\epsilon ,& (16)\end{array}$

where x_i ∈ {0, 1}, 0 ≤ i ≤ m.

Proof. This result follows from Theorem 1 and Lemma 1. Indeed, for every fixed ε > 0, we can find an integer m such that $\mathbb{P}(N_T>m)+{\mathbb{P}}^{(N)}({\nu }_N>m)<\epsilon /2$ for all N > 0, so that (Equation 16) is reduced to

$\begin{array}{r}|\mathbb{P}(Y_{t_i}=x_i,0\le i\le n,N_t\le m)\\ -{\mathbb{P}}^{(N)}\left(Y_{t_k^{(N)}}^{(N)}=x_i,0\le i\le n,{\nu }_N\le m\right)|<\epsilon /2.& (17)\end{array}$

Let us introduce the random variables r_j of the form

$\begin{array}{ll}{r}_j=inf\left\{k\ge 0:{\tau }_k\le t_j<{\tau }_{k+1}\right\}.& (18)\end{array}$

In fact, r_j is the index number of the occupation interval that covers the fixed point t_j. Note that r_j is defined on the same probability space as (_Ys)s≥0, and for ω ∈ {N_t ≤ m}, each r_k(ω) takes value in the set {0, 1, …, m}. Put $A_n^m=\left\{(r_0(\omega ),\dots ,r_n(\omega )),\omega \in \text{Ω}\right\}\subset {ℝ}^{n+1}$. It is clear that $A_n^m$ is a finite set, and

$\left|A_n^m\right|\le m^{n+1}.$

Then using formula (18), we get an equality

$\begin{array}{c}\mathbb{P}(Y_{t_i}=x_i,0\le i\le n,N_t\le m)\\ ={\displaystyle \sum _{(r_0,\dots ,r_n)\in A_n^m}}\mathbb{P}({\tau }_{r_j}\le t_j<{\tau }_{r_j+1},0\le j\le n,N_t\le m).\end{array}$

By Theorem 1, we can find an integer N(ε, n) such that for all N ≥ N(ε, n)

$\begin{array}{c}|\mathbb{P}({\tau }_{r_j}\le t_j<{\tau }_{r_j+1},0\le j\le n,N_t\le m)\\ -{\mathbb{P}}^{(N)}({\tau }_{r_j}^{(N)}\le t_j<{\tau }_{r_j+1}^{(N)},0\le j\le n,N_t\le m)|\\ \hspace{1em}\\ \le \frac{\epsilon }{2m^{n+1}},\end{array}$

which proves Equation 17 and hence the statement of the theorem follows.            □

Theorem 3. Processes ${(Y_t^{(N)})}_{t\in \left[0,T\right]}$ converge to (Y_t)_{t∈[0, T]}, N → ∞ in Skorokhod topology.

Proof. In Theorem 2, we already proved the convergence of finite-dimensional distributions. Therefore, by Theorem 4, Section VI.5 from Gikhman and Skokohod [19], we have to verify that for all ε > 0

$\begin{array}{ll}{\displaystyle \underset{h\to 0}{lim}}{\displaystyle \underset{N\to \infty }{lim sup}}{\displaystyle \underset{x\in \left\{0,1\right\},0\le s-t\le h}{sup}}{\mathbb{P}}^{(N)}\left(|Y_s^{(N)}-Y_t^{(N)}|>\epsilon |Y_t^{(N)}=x\right)=0.& (19)\end{array}$

Let us examine the probability

${\mathbb{P}}^{(N)}\left(|Y_{t+h}^{(N)}-Y_t^{(N)}|>\epsilon |Y_t^{(N)}=0\right).$

Since the chain Y^(N) takes values in the set {0, 1}, the condition $|Y_{t_h}^{(N)}-Y_t^{(N)}|>\epsilon$ means that $Y_{t+h}^{(N)}=1-Y_t^{(N)}$. Thus, we can write

$\begin{array}{r}{\mathbb{P}}^{(N)}\left(|Y_{t+h}^{(N)}-Y_t^{(N)}|>\epsilon |Y_t^{(N)}=0\right)={\mathbb{P}}^{(N)}\left(Y_{t+h}^{(N)}=1|Y_t^{(N)}=0\right)\\ ={\mathbb{P}}^{(N)}\left(Y_h^{(N)}=1|Y_0^{(N)}=0\right),\end{array}$

where the last equality follows from homogeneity.

Similarly,

$\begin{array}{r}{\mathbb{P}}^{(N)}\left(|Y_{t+h}^{(N)}-Y_t^{(N)}|>\epsilon |Y_t^{(N)}=1\right)={\mathbb{P}}^{(N)}(Y_{t+h}^{(N)}=0|Y_t^{(N)}=1)\\ ={\mathbb{P}}^{(N)}\left(Y_h^{(N)}=0|Y_0^{(N)}=1\right).\end{array}$

To evaluate the latter probabilities, we will need a general form of n-step transition probability for a 2 × 2 Markov chain, which has the form

$\begin{array}{c}{\mathbb{P}}^{(N)}\left(Y_m^{(N)}=1|Y_0^{(N)}=0\right)={\pi }_1^{(N)}-{\pi }_1^{(N)}{(a^{(N)}-1)}^m,\\ {\mathbb{P}}^{(N)}(Y_m^{(N)}=0|Y_0^{(N)}=1)={\pi }_0^{(N)}-{\pi }_0^{(N)}{(a^{(N)}-1)}^m,\end{array}$

where $a^{(N)}=e^{-{\lambda }_0\frac{T}{N}}+e^{-{\lambda }_1\frac{T}{N}}$ (note that a^(N) ∈ (0, 2)), and

${\pi }^{(N)}=({\pi }_0^{(N)},{\pi }_1^{(N)})=\left(\frac{1-e^{-{\lambda }_1\frac{T}{N}}}{2-a^{(N)}},\frac{1-e^{-{\lambda }_0\frac{T}{N}}}{2-a^{(N)}}\right)$

is an invariant distribution for the chain Y^(N) (see Appendix, Equation A1). For a fixed h ∈ [0, T], recall the notation

$h^{(N)}=\lfloor \frac{hN}{T}\rfloor .$

Now, we can rewrite the left-hand side of Equation (19) as

$\begin{array}{l}{\displaystyle \underset{h\to 0}{lim}}{\displaystyle \underset{N\to \infty }{lim sup}}{\displaystyle \underset{x\in \left\{0,1\right\},0\le s-t\le h}{sup}}{\mathbb{P}}^{(N)}\left(|Y_s^{(N)}-Y_t^{(N)}|>\epsilon |Y_t^{(N)}=x\right)\le \\ \le {\displaystyle \underset{h\to 0}{lim}}{\displaystyle \underset{N\to \infty }{lim sup}}{\displaystyle \underset{x\in \left\{0,1\right\},0\le s\le h}{sup}}{\mathbb{P}}^{(N)}\left(Y_s^{(N)}=1-x|Y_0^{(N)}=x\right)\\ \le {\displaystyle \underset{h\to 0}{lim}}{\displaystyle \underset{N\to \infty }{lim sup}}{\displaystyle \underset{x\in \left\{0,1\right\},0\le s\le h}{sup}}\left({\pi }_x^{(N)}-{\pi }_x^{(N)}{(a^{(N)}-1)}^{s^{(N)}}\right)\\ ={\displaystyle \underset{h\to 0}{lim}}{\displaystyle \underset{N\to \infty }{lim sup}}{\displaystyle \underset{x\in \left\{0,1\right\}}{max}}\left({\pi }_x^{(N)}-{\pi }_x^{(N)}{(a^{(N)}-1)}^{h^{(N)}}\right)\\ \le {\displaystyle \underset{h\to 0}{lim}}{\displaystyle \underset{N\to \infty }{lim sup}}\left(1-{(a^{(N)}-1)}^{\frac{hN}{T}}\right)\\ ={\displaystyle \underset{h\to 0}{lim}}(1-e^{-({\lambda }_0+{\lambda }_1)h})=0.\end{array}$

                □

Theorem 4. Processes ${(U_t^{(1,N)})}_{t\in \left[0,T\right]}$ converge to ${(U_t^{(1)})}_{t\in \left[0,T\right]}$ in the uniform topology.

Proof. Using the Taylor formula for logarithm, we get the following representation: for x > 0,

$log(1+x)=x+\rho (x)x,$

where |ρ(x)| ≤ h(N) when $x\in \left(0,\frac{C}{N}\right)$ for some constant C, and h(N) → 0, N → ∞. For any fixed $t\in \left[\frac{(k-1)T}{N},\frac{kT}{N}\right)$ we have

$\begin{array}{l}{U}_t^{(1,N)}={\displaystyle \sum _{j=0}^{k-1}}log\left(1+R_j^{(1,N)}\right)={\displaystyle \sum _{j=0}^{k-1}}\left(R_j^{(1,N)}+\rho \left(R_j^{(1,N)}\right)R_j^{(1,N)}\right)\\ ={\displaystyle \sum _{j=0}^{k-1}}\left(\frac{{\delta }_{0}T}{N}{Y}_j^{(N)}+\frac{{\delta }_{1}T}{N}\left(1-Y_j^{(N)}\right)\right)+{\displaystyle \sum _{j=0}^{k-1}}\rho \left(R_j^{(1,N)}\right)R_j^{(1,N)}\\ ={\displaystyle {\int }_0^{\frac{(k-1)T}{N}}}\left({\delta }_0{Y}_s^{(N)}+{\delta }_1\left(1-Y_s^{(N)}\right)\right)ds+{\displaystyle \sum _{j=0}^{k-1}}\rho \left(R_j^{(1,N)}\right)R_j^{(1,N)}\\ ={\displaystyle {\int }_0^t}\left({\delta }_0{Y}_s^{(N)}+{\delta }_1\left(1-Y_s^{(N)}\right)\right)ds\\ -{\displaystyle {\int }_{\frac{(k-1)T}{N}}^t}\left({\delta }_0{Y}_s^{(N)}+{\delta }_1\left(1-Y_s^{(N)}\right)\right)ds\\ +{\displaystyle \sum _{j=0}^{k-1}}\rho \left(R_j^{(1,N)}\right)R_j^{(1,N)}.\end{array}$

Next, we have a.s.

$\begin{array}{c}\left|{\displaystyle \sum _{j=0}^{k-1}}\rho \left(R_j^{(1,N)}\right)R_j^{(1,N)}\right|\le h(N)T max\left\{{\delta }_0,{\delta }_1\right\}\to 0,N\to \infty ,\\ \left|{\displaystyle {\int }_{\frac{(k-1)T}{N}}^t}\left({\delta }_0{Y}_s^{(N)}+{\delta }_1\left(1-Y_s^{(N)}\right)\right)ds\right|\le \frac{max\left\{{\delta }_0,{\delta }_1\right\}T}{N}\to 0,N\to \infty .\end{array}$

Using Theorem 3, we may conclude that for each fixed t ∈ [0, T],

$\begin{array}{l}{\displaystyle {\int }_0^t}\left({\delta }_0{Y}_s^{(N)}+{\delta }_1\left(1-Y_s^{(N)}\right)\right)ds{\to }^d{\displaystyle {\int }_0^t}({\delta }_0{Y}_s+{\delta }_1(1-Y_s))ds,N\to \infty .\end{array}$

We can now use the Slutsky theorem, and conclude that

$U_t^{(1,N)}{\to }^d{U}_t^{(1)},$

where by → ^d we denote a weak convergence in distribution. Let us now consider a linear combination of the form

${\displaystyle \sum _{j=0}^m}{\alpha }_j{U}_{t_j}^{(1,N)},{\alpha }_j\in ℝ,m\ge 0,0\le t_0<\dots <t_m\le T.$

Using the properties of the Riemann integral and Slutsky theorem we can apply similar reasoning to conclude that

${\displaystyle \sum _{j=0}^m}{\alpha }_j{U}_{t_j}^{(1,N)}{\to }^d{\displaystyle \sum _{j=0}^m}{\alpha }_j{U}_{t_j}^{(1)},N\to \infty ,$

which implies weak convergence of finite-dimensional distributions of the process ${(R_t^{(1,N)})}_{t\in \left[0,T\right]}$ to that of ${(U_t^{(1)})}_{t\in \left[0,T\right]}$.

Let us consider the modulus of continuity of the sequences of the processes $({\int }_0^t{({\delta }_0{Y}_s^{(N)}+{\delta }_1(1-Y_s^{(N)})ds)}_{t\in \left[0,T\right]}$. Obviously, for all 0 ≤ u < t ≤ T

$|{\displaystyle {\int }_u^t}({\delta }_0{Y}_s^{(N)}+{\delta }_1(1-Y_s^{(N)})ds|\le (t-u)max\left\{{\delta }_0,{\delta }_1\right\}.$

The latter inequality implies that the family of processes$({\int }_0^t{({\delta }_0{Y}_s^{(N)}+{\delta }_1(1-Y_s^{(N)})ds)}_{t\in \left[0,T\right]}$ is tight in the uniform topology. The statement of the theorem follows from this fact, together with the convergence of finite-dimensional distributions.

                □

4 Weak convergence to a geometric Brownian motion with Markov switching drift rate in the multiplicative scheme of series

Conditions of weak convergence of the sequence of processes

$U^{(2,N)}:=\left\{U_t^{(2,N)},t\in \left[0,T\right]\right\},N\ge 1,$

created in Equation 10, to the process $U_2(t)=\sigma W_t-\frac{{\sigma }^2}{2}t$, are classical. They can be deduced from the respective results contained in the books [20] and [18]. However, for the reader's convenience, we describe them briefly, basing them on the Skorokhod theorem about weak convergence of sums of independent random variables to the continuous process with independent increments (see, e.g., Theorem 1, pages 452–453 from Gikhman and Skokohod [21]). So, we consider the scheme of series of the form $U_t^{(2,N)}=0,t\in [0,\frac{T}{N}),$ $U_T^{(2,N)}={\displaystyle \sum _{i=1}^N}log(1+R_i^{(2,N)}),$ and

$U_t^{(2,N)}={\displaystyle \sum _{i=1}^{\left[\frac{Nt}{T}\right]}}log\left(1+R_i^{(2,N)}\right),t\in \left[\frac{T}{N},T\right).$

We can simplify these records by putting $\sum _{i=1}^0$ and

$U_t^{(2,N)}={\displaystyle \sum _{i=1}^{\left[\frac{Nt}{T}\right]}}log\left(1+R_i^{(2,N)}\right),t\in \left[0,T\right].$

Assume that there exist two real-valued sequences {α_N, β_N, N ≥ 1} such that $-1<{\alpha }_N<R_i^{(2,N)}<{\beta }_N$ with probability 1 and α_n, β_N → 0 as N → ∞. Then

$U_t^{(2,N)}={\displaystyle \sum _{i=1}^{\left[\frac{Nt}{T}\right]}}\left(R_i^{(2,N)}-\frac{1}{2}{\left(R_i^{(2,N)}\right)}^2\right)+{\text{Δ}}_N(t),$

where $\left|{\text{Δ}}_N(t)\right|\le \text{Δ}({\alpha }_N,{\beta }_N){\displaystyle \sum _{i=1}^N}{(R_i^{(2,N)})}^2,$ and real-valued positive sequence Δ(α_N, β_N) → 0 as N → ∞. Recall that we already assumed that $(R_i^{(2,N)},1\le i\le N)$ are mutually independent.

Theorem 5. Assume that the following conditions hold:

(i) $\text{𝔼}{R}_i^{(2,N)}=0,1\le i\le N,N\ge 1$.

(ii) For any t ∈ [0, T]

${\displaystyle \sum _{i=1}^{t^{(N)}}}\text{𝔼}{\left[R_i^{(2,N)}\right]}^2\to {\sigma }^{2}t.$

Then the sequence ${\mathbb{P}}_T^{(2,N)}$ of measures corresponding to processes $\left\{U_t^{(2,N)},t\in \left[0,T\right]\right\}$ weakly converges to the measure ${\mathbb{P}}_T^{(2)}$ corresponding to process $\left\{\sigma W_t-\frac{{\sigma }^2}{2}t,t\in \left[0,T\right]\right\}$.

Proof. Conditions (i) and (ii) mentioned in Theorem 5, together with Theorem 5.53 from Föllmer et al. [20], imply that for any 0 ≤ s < t ≤ T, the distribution of the increment $U_t^{(2,N)}-U_s^{(2,N)}$ weakly converges to $\sigma (W_t-W_s)-\frac{{\sigma }^2}{2}(t-s).$ Moreover, these conditions, together with restrictions on the values of $R_i^{(2,N)}$, support Lindeberg's condition in Theorem 1, pages 452–453 from Gikhman and Skokohod [21], whence the proof follows.    □

5 Incompletenesses of the market with switching

This section explores the incompleteness of the continuous-time market with drift Markov switching, as described by Equation 1. Although this topic is not directly related to the convergence problem studied in the previous sections, it is of interest to the financial applications of the model.

In this section, we assume that (_Xt)t≥0 represents the discounted asset price in an arbitrage-free market, which consists of this risky asset and a risk-free asset. Since the risky asset price involves two independent sources of randomness, the financial market is incomplete. To demonstrate the incompleteness explicitly, let us construct a MMM and separately a class of martingale measures especially related to the Markov process. First, fix the interval [0, T] and attempt to construct an equivalent martingale measure ℚ ~ ℙ, whose Radon-Nikodym derivative restricted to the interval [0, T] has the form

$\begin{array}{ll}\frac{d{\mathbb{Q}}_T}{d{\mathbb{P}}_T}=exp\left(\sigma {\displaystyle {\int }_0^T}\phi (u)d{W}_u-\frac{{\sigma }^2}{2}{\displaystyle {\int }_0^T}{\phi }^2(u)du\right),& (20)\end{array}$

where φ(u) is a ${\text{F}}_u$-adapted stochastic process satisfying condition $\text{𝔼}(\frac{d{\mathbb{Q}}_T}{d{\mathbb{P}}_T})=1$ (in this case ℚ_T is indeed a probability measure). Moreover, recall the notion of the MMM from Föllmer and Schweizer [22]:

Definition 1. (Föllmer and Schweizer [22]) Let the discounted asset price in a financial market be given by the real-valued semimartingale of the form

$S=S_0+M+A,$

where S₀ > 0 is a constant, M is a local ℙ-martingale, A is a process of locally bounded variation, ℙ is the initial probability measure, and M₀ = A₀ = 0. The minimal martingale measure (MMM) for S is an equivalent probability measure $\widehat{\mathbb{P}}$ that is characterized by the properties that it transforms S into a local martingale and preserves the martingale property for any local ℙ-martingale that is strongly orthogonal to M.

According to Föllmer and Schweizer [22], assume additionally that M is a ℙ-square-integrable martingale, and A has a form

$A_t={\displaystyle {\int }_0^t}{\lambda }_{s}d{\langle M\rangle }_s,t\in \left[0,T\right],$

where ${\int }_0^T{\lambda }_s^{2}d{\langle M\rangle }_s<\infty$ a.s., 〈M〉 is the quadratic characteristics of M (see, e.g., Liptser and Shiryayev [17] for detail). Moreover, if

$d{S}_t=S_t({\rho }_{t}dt+{\sigma }_{t}d{W}_t),$

and σ is a strictly positive adapted process on [0, T], then ${\lambda }_s={\rho }_s{\sigma }_s^{-2},s\in \left[0,T\right]$, and

${\displaystyle {\int }_0^t}{\lambda }_s^2{\langle M\rangle }_s={\displaystyle {\int }_0^t}{\rho }_s^2{\sigma }_s^{-2}ds,t\in \left[0,T\right].$

If the MMM $\widehat{\mathbb{P}}$ exists, then its Radon-Nikodym derivative restricted to the interval [0, T] is given by the stochastic exponent of the form

$\begin{array}{c}\frac{d{\widehat{\mathbb{P}}}_T}{d{\mathbb{P}}_T}=E\left(-\int \lambda dM\right)=exp\left\{-{\displaystyle {\int }_0^T}{\lambda }_{s}d{M}_s-\frac{1}{2}{\displaystyle {\int }_0^T}{\lambda }_s^{2}d{\langle M\rangle }_s\right\}\times \\ \times {\displaystyle \prod _{0\le s\le T}}(1-\lambda \text{Δ}{M}_s)exp\left(\lambda \text{Δ}{M}_s-\frac{1}{2}{\lambda }^2{(\text{Δ}M)}_s^2\right).\end{array}$

Lemma 2. The equivalent martingale measure for the market is described by Equation (1), which has the form Equation (20), is unique, and the function φ equals

$\begin{array}{ll}\phi (u)=-\left(\frac{{\delta }_0}{{\sigma }^2}{Y}_u+\frac{{\delta }_1}{{\sigma }^2}(1-Y_u)\right),u\in \left[0,T\right].& (21)\end{array}$

and $\mathbb{Q}=\widehat{\mathbb{P}}$ is a MMM in this market.

Proof. For all t ∈ [0, T] the following equality holds

$\begin{array}{ll}{\text{𝔼}}_{\mathbb{Q}}\left[X_t-X_s|{\text{F}}_s\right]=\frac{\text{𝔼}\left[\frac{d{\mathbb{Q}}_T}{d{\mathbb{P}}_T}(X_t-X_s)|{\text{F}}_s\right]}{\text{𝔼}\left[\frac{d{\mathbb{Q}}_T}{d{\mathbb{P}}_T}|{\text{F}}_s\right]}.& (22)\end{array}$

Assume that

$\begin{array}{ll}\text{𝔼}exp\left(\sigma {\displaystyle {\int }_0^T}\phi (u)d{W}_u-\frac{{\sigma }^2}{2}{\displaystyle {\int }_0^T}{\phi }^2(u)du\right)=1.& (23)\end{array}$

Then the process $exp(\sigma {\int }_0^t\phi (u)d{W}_u-\frac{{\sigma }^2}{2}{\int }_0^t{\phi }^2(u)du),t\in \left[0,T\right]$ is a martingale, in particular,

$\begin{array}{c}\text{𝔼}\left(exp\left(\sigma {\displaystyle {\int }_0^t}\phi (u)d{W}_u-\frac{{\sigma }^2}{2}{\displaystyle {\int }_0^t}{\phi }^2(u)du\right)|{\text{F}}_s\right)\\ =exp\left(\sigma {\displaystyle {\int }_0^s}\phi (u)d{W}_u-{\sigma }^2{\displaystyle {\int }_0^s}{\phi }^2(u)du\right),\end{array}$

therefore,

$\begin{array}{l}\text{𝔼}\left[\frac{d{\mathbb{Q}}_T}{d{\mathbb{P}}_T}(X_t-X_s)|{\text{F}}_s\right]=X_s\text{𝔼}[exp(\sigma {\displaystyle {\int }_0^t}\phi (u)d{W}_u-\frac{{\sigma }^2}{2}{\displaystyle {\int }_0^t}{\phi }^2(u)du)\\ \times (exp({\displaystyle {\int }_s^t}({\delta }_0{Y}_u+{\delta }_1(1-Y_u))du+\sigma (W_t-W_s)-\frac{{\sigma }^2}{2}(t-s))-1)|{\text{F}}_s]=0& (24)\end{array}$

if and only if

$\begin{array}{c}\text{𝔼}[exp({\displaystyle {\int }_s^t}(\sigma \phi (u)\\ +\sigma )d{W}_u+{\displaystyle {\int }_s^t}({\delta }_0{Y}_u+{\delta }_1(1-Y_u)-\frac{{\sigma }^2}{2}{\phi }^2(u)-\frac{{\sigma }^2}{2})du)|{\text{F}}_s]=1,\end{array}$

which in turn is true if and only if

${\delta }_0{Y}_u+{\delta }_1(1-Y_u)-\frac{{\sigma }^2}{2}{\phi }^2(u)-\frac{{\sigma }^2}{2}=-\frac{{\sigma }^2}{2}{(\phi (u)+1)}^2,$

whence φ(u) satisfies equality (Equation 21). According to Föllmer and Schweizer [22], measure ℚ is a MMM for this market.

Indeed, in our case, $M_t=\sigma {\int }_0^t{X}_{s}d{W}_s$ and $A_t={\int }_0^t({\delta }_0{Y}_s+{\delta }_1(1-Y_s))X_{s}ds$. Obviously, M is a continuous square-integrable martingale, ${\lambda }_t={\sigma }^{-2}({\delta }_0{Y}_t+{\delta }_1(1-Y_t))$, and for MMM

$\begin{array}{l}\frac{d{\widehat{\mathbb{P}}}_T}{d{\mathbb{P}}_T}=E(-{\displaystyle {\int }_0^T}\lambda dM)\\ =exp\{-{\sigma }^{-1}{\displaystyle {\int }_0^T}({\delta }_0{Y}_s+{\delta }_1(1-Y_s))d{W}_s\\ -\frac{{\sigma }^{-2}}{2}{{{\displaystyle \int }}^{\text{}}}_0^T{\left({\delta }_0{Y}_s+{\delta }_1\left(1-Y_s\right)\right)}^{2}ds\},\end{array}$

therefore, ${\widehat{\mathbb{P}}}_T={\mathbb{Q}}_T$ from (Equation 20) with Equations 21-24 in hand. Moreover, equality (Equation 23) holds. So, the lemma is proved.    □

Nevertheless, there can be other equivalent martingale measures. To construct a wide class of equivalent martingale measures, let us consider the following objects: First, we shall use the standard definition of the Feller process (see e.g., Chung [23], p. 50) and the following definition of the left quasi-continuous process, taken from Chung [23] and Liptser and Shiryayev [17].

Definition 2. Let us have a stochastic basis with filtration and an adapted process U = {U_t, t ≥ 0}. Process U is left quasi-continuous, if for any stopping time τ and any sequence of stopping times τ_n↑τ, $U_{\tau }=\underset{{\tau }_n\uparrow \tau }{lim}{U}_{{\tau }_n}$ P-a.s. on the set {τ < ∞}.

Now we summarize the following facts from Liptser and Shiryayev [17] and Gushchin [24], simplifying them for our situation (in general, these properties can be formulated in a local version, but our processes under consideration are integrable). We consider càdlàg processes, which have a.s. continuous trajectories from the right and with left limits at all points.

(i) For any adapted process A of integrable variation, there exists a predictable process A^π of integrable variation (dual predictable projection, or compensator of A) such that the process M = A − A^π is a martingale.

(ii) If process A is left quasi-continuous, then process A^π is continuous.

(iii) The left quasi-continuity of the adapted process A of integrable variation is equivalent to any of the following properties:

(a) for any predictable stopping moment τ Δ_τA𝟙_{τ < ∞} = 0, where Δ_tA = A_t − A_t−, the jump at point t, which is correctly defined for càdlàg processes.

(b) for any bounded stopping moment τ and for any sequence of non-decreasing stopping times τ_n ↑ τ

$\text{𝔼}{A}_{{\tau }_n}\to \text{𝔼}{A}_{\tau },n\to \infty .$

Now we are in a position to construct a wide class of equivalent martingale measures for our market with Markov switching, but we decide to operate only with the Markov process Y. It should be noted that Y has bounded variation |Y| on [0, T] with finite moments of any order (variation |Y| on [0, t] is simply a number of jumps N_t, which, according to Corollary 1, has a finite exponential moment). Therefore, Y is a process of integrable variation and admits a dual predictable projection Y^π of integrable variation.

Lemma 3. Process Y is left quasi-continuous.

Proof. The desired property follows directly from Theorem 4 (Section 2.4, page 70) in Chung [23], once we establish that Y is a Feller process.

Recall that time-homogenous Markov process has values in some compact space E is called Feller if the following two conditions hold true:

(i) for all f ∈ C(E)

${\displaystyle \underset{t\downarrow 0}{lim}}{\displaystyle {\int }_E}{P}_t(·,dy)f(y)=f(·),$

(ii) for every fixed t and f ∈ C(E)

${\displaystyle {\int }_E}{P}_t(·,dy)f(y)\in C(E),$

where C(E) is a space of all functions continuous on E and P_t(x, A) is a transition probability on the time interval [0, t].

In our case, E = {0, 1}, so every finite function on E is continuous, and (ii) follows immediately.

Since the matrix 𝔸 defined in Equation (2) is a generator of the process Y, we have by definition

$\text{𝔸}f(·)={\displaystyle \underset{t\downarrow 0}{lim}}\frac{{\displaystyle {\int }_E}{P}_t(·,dy)f(y)-f(·)}{t},$

for all continuous functions f on {0, 1}, which implies (i).    □

Now, according to Gushchin [24], any left quasi-continuous process of integrable variation has a continuous integrable dual predictable projection (compensator). Therefore, we can consider the dual predictable projection Y^π of Y, which is a continuous process of integrable variation, and let $M_t=Y_t-Y_t^{\pi }.$ Then M is a martingale. Therefore, according to Liptser and Shiryayev [17], M admits a decomposition M = M^c+M^d, where M^c is a continuous local martingale, and M^d is a purely discontinuous local martingale where pure discontinuity means that common quadratic variation [M^c, M^d] is a zero process.

Lemma 4. M is a purely discontinuous martingale with a finite a.s. number of jumps on any fixed interval [0, T].

Proof. Pure discontinuity immediately follows from the fact that both the purely jump process Y and the continuous compensator of Y^π have zero common quadratic variations [Y, B] and [Y^π, B] with any continuous process B. The lemma is proven.    □

Therefore, if we create a stochastic exponent $E(M)$, it will have the form

$\begin{array}{l}{E}_t(M)={\displaystyle \prod _{0\le u\le t}}(1+\text{Δ}{M}_u)exp\left\{-\text{Δ}{M}_u\right\}={\displaystyle \prod _{0\le u\le t}}(1+\text{Δ}{Y}_u)exp\left\{-\Delta Y_u\right\},\end{array}$

where Δ(·)_s stands for the jump of the respective process at point s, and these jumps are correctly defined for càdlàg processes. However, the problem with this stochastic exponent is that the jumps of M can equal −1. To avoid this difficulty, let us consider any strictly positive continuous process ψ_t, 0 ≤ t ≤ T adapted to σ_{0, t}(Y) such that $\psi (t)\le (\frac{|{\lambda }_1-{\lambda }_0|T}{4}\wedge \frac{1}{2})$, consider stochastic integral $M_t^{(\psi )}={\int }_0^t{\psi }_{s}d{M}_s$, which is in fact a sum of a finite number of terms, and construct stochastic exponent $E_t(M^{(\psi )}).$ Introduce the following notations: $E_{s,t}(M^{(\psi )})=E_t(M^{(\psi )}){(E_s(M^{(\psi )}))}^{-1},0<s\le t$, and

$M_t^{(\phi )}=\sigma {\displaystyle {\int }_0^t}\phi (u)d{W}_u,{\langle M^{(\phi )}\rangle }_t={\sigma }^2{\displaystyle {\int }_0^t}{\phi }^2(u)du,$

where φ is defined in Equation (21),

$\begin{array}{l}{E}_t(M^{(\phi )})=exp\left\{M_t^{(\phi )}-\frac{1}{2}{\langle M^{(\phi )}\rangle }_t\right\},E_{s,t}(M^{(\phi )})\\ =E_t(M^{(\phi )}){(E_s(M^{(\phi )}))}^{-1},0<s\le t.\end{array}$

Theorem 6. Probability measures ℚ^{φ, ψ}, for which its Radon-Nikodym derivative restricted on the interval [0, T] has the form

$\frac{d{\mathbb{Q}}_T^{\phi ,\psi }}{d{\mathbb{P}}_T}=E_T(M^{(\psi )})E_T(M^{(\phi )}),$

is a probability equivalent martingale measure for the market defined by Equation (1).

Proof. First, notice that for any s > 0

$\begin{array}{c}(1+\text{Δ}{M}_s^{(\psi )})exp\left\{-\text{Δ}{M}_s^{(\psi )}\right\}\\ \le (1+\frac{|{\lambda }_1-{\lambda }_0|T}{4}){\text{𝟙}}_{\text{Δ}{Y}_s=1}+e^{\frac{|{\lambda }_1-{\lambda }_0|T}{4}}{1}_{\text{Δ}{Y}_s=-1},\end{array}$

therefore, according to Corollary 1, $E_t(M^{(\psi )})$ does not exceed $exp(\frac{|{\lambda }_1-{\lambda }_0|T}{4}{N}_T)$, and so, it is integrable. It means that being a local martingale and stochastic exponent, and also being an integrable, $E_t(M^{(\psi )}),t\in \left[0,T\right]$ is a martingale. In particular, $\text{𝔼}{E}_T(M^{(\psi )})=1$ and $E_T(M^{(\psi )})$ define a probability measure ℙ^(ψ) on $(\text{Ω},\text{F})$, equivalent to measure ℙ. Now, for any 0 ≤ s ≤ t ≤ T, introduce the σ-fields σ_{s, t}(Y) = σ{Y_u, s ≤ u ≤ t} generated by the process Y on the respective intervals. Then

$\begin{array}{ll}\text{𝔼}(E_T(M^{(\psi )})E_T(M^{(\phi )}))=\text{𝔼}(E_T(M^{(\psi )})\text{𝔼}(E_T(M^{(\phi )})|{\sigma }_{0,T}(Y))).& (25)\end{array}$

Denote x = x_t, t ∈ [0, T] some bounded, measurable, and non-random function. Then, taking into account the independence of W and Y, we can write that

$\begin{array}{c}\text{𝔼}(E_T(M^{(\phi )})|{\sigma }_{0,T}(Y))\\ =\left(\text{𝔼}exp\left({\displaystyle {\int }_0^T}{x}_{t}d{W}_t-\frac{1}{2}{\displaystyle {\int }_0^T}{x}_t^{2}dt\right){|}_{{\phi }_t=x_t,t\in \left[0,T\right]}\right)=1.& (26)\end{array}$

Therefore,

$\text{𝔼}(E_T(M^{(\psi )})E_T(M^{(\phi )}))=\text{𝔼}(E_T(M^{(\psi )}))=1,$

whence ℚ^{φ, ψ} is a probability measure and $\frac{d{\mathbb{Q}}_t^{\phi ,\psi }}{d{\mathbb{P}}_t},t\in \left[0,T\right]$ is a martingale. Now we shall use the independence of W and Y again in order to prove that ℚ is an equivalent martingale measure. Indeed, similarly to the proof of Lemma 2,

$\begin{array}{l}{\text{𝔼}}_{{\mathbb{Q}}^{\phi ,\psi }}\left[X_t-X_s|{\text{F}}_s\right]=\frac{\text{𝔼}\left[\frac{d{\mathbb{Q}}_T^{\phi ,\psi }}{d{\mathbb{P}}_T}(X_t-X_s)|{\text{F}}_s\right]}{\text{𝔼}\left[\frac{d{\mathbb{Q}}_T^{\phi ,\psi }}{d{\mathbb{P}}_T}|{\text{F}}_s\right]}\\ =X_s\text{𝔼}[E_{s,t}(M^{(\phi )})E_{s,t}(M^{(\psi )})\times \\ \times (exp\{{\displaystyle {\int }_s^t}({\delta }_0{Y}_u+{\delta }_1(1-Y_u))du+\sigma (W_t-W_s)-\frac{{\sigma }^2}{2}(t-s)\}-1)|{\text{F}}_s]\\ =X_s\text{𝔼}[E_{s,t}(M^{(\psi )})(exp\left\{{\displaystyle {\int }_s^t}\sigma ({\phi }_u+1)d{W}_u-\frac{{\sigma }^2}{2}{\displaystyle {\int }_s^t}{({\phi }_u+1)}^{2}du\right\}\\ -exp\left\{\sigma {\displaystyle {\int }_s^t}\phi (u)d{W}_u-\frac{1}{2}{\sigma }^2{\displaystyle {\int }_s^t}{\phi }^2(u)du\right\})|{\text{F}}_s]=:G(s,t).\end{array}$

Consider the σ-field

$H_s^t={\text{F}}_s\vee {\sigma }_{s,t}(Y),$

the smallest σ-field containing ${\text{F}}_s$ and σ_{s, t}(Y). Then $E_{s,t}(M^{(\psi )})$ is $H_s^t$-measurable, and, similarly to Equations (25, 26),

$\begin{array}{l}G(s,t)=X_s\text{𝔼}[E_{s,t}(M^{(\psi )})\text{𝔼}(exp\left\{{\displaystyle {\int }_0^t}\sigma ({\phi }_u+1)d{W}_u-\frac{{\sigma }^2}{2}{\displaystyle {\int }_0^t}{({\phi }_u+1)}^{2}du\right\}\\ -exp\left\{\sigma {\displaystyle {\int }_s^t}\phi (u)d{W}_u-\frac{1}{2}{\sigma }^2{\displaystyle {\int }_s^t}{\phi }^2(u)du\right\})|H_s^t)|{\text{F}}_s],\end{array}$

where

$\begin{array}{c}\text{𝔼}(exp\left\{{\displaystyle {\int }_0^t}\sigma ({\phi }_u+1)d{W}_u-\frac{{\sigma }^2}{2}{\displaystyle {\int }_0^t}{({\phi }_u+1)}^{2}du\right\}\\ -exp\left\{\sigma {\displaystyle {\int }_s^t}\phi (u)d{W}_u-\frac{{\sigma }^2}{2}{\displaystyle {\int }_s^t}{\phi }^2(u)du\right\}|H_s^t)\\ =\text{𝔼}(exp\left\{{\displaystyle {\int }_s^t}\sigma (x_u+1)d{W}_u-\frac{{\sigma }^2}{2}{\displaystyle {\int }_s^t}{(x_u+1)}^{2}du\right\}\\ -exp\left\{\sigma {\displaystyle {\int }_s^t}x(u)d{W}_u-\frac{{\sigma }^2}{2}{\displaystyle {\int }_s^t}{x}^2(u)du\right\}){|}_{{\phi }_t=x_t,t\in \left[0,T\right]}=1-1=0.\end{array}$

It means that ℚ^{φ, ψ} is an equivalent martingale measure for the market defined by Equation (1), and the theorem is proved.    □

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author/s.

Author contributions

VG: Conceptualization, Formal analysis, Investigation, Writing – original draft. YM: Conceptualization, Formal analysis, Investigation, Methodology, Project administration, Supervision, Validation, Writing – review & editing. KK: Conceptualization, Methodology, Writing – review & editing.

Funding

The author(s) declare financial support was received for the research, authorship, and/or publication of this article. YM was supported by the Swedish Foundation for Strategic Research (Grant No. UKR24-0004), the by Japan Science and Technology Agency CREST JPMJCR2115, and ToppForsk (Project No. 274410) of the Research Council of Norway with the title STORM: Stochastics for Time-Space Risk Models.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher's note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

References

1. Quandt RE. The estimation of the parameters of a linear regression system obeying two separate regimes. J Am Stat Assoc. (1958) 53:873–80. doi: 10.1080/01621459.1958.10501484

Crossref Full Text | Google Scholar

2. Quandt RE, Goldfeld SM. A Markov model for switching regressions. J Econom. (1973) 1:3–15. doi: 10.1016/0304-4076(73)90002-X

Crossref Full Text | Google Scholar

3. Hamilton JD. A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica. (1989) 57:357–84. doi: 10.2307/1912559

Crossref Full Text | Google Scholar

4. Ang A, Timmermann A. Regime changes and financial markets. Annu. Rev. Financ. Econ. (2012) 04:313–37. doi: 10.1146/annurev-financial-110311-101808

Crossref Full Text | Google Scholar

5. Sotomayor LR, Cdenillas A. Explicit solutions of consumption-investment problems in financial markets with regime switching. Math. Finance. (2009) 19:251–79. doi: 10.1111/j.1467-9965.2009.00366.x

Crossref Full Text | Google Scholar

6. Dai M, Zhang Q, Zhu QJ. Trend following trading under a regime switching model. SIAM J Financ Math. (2010) 1:780–810. doi: 10.1137/090770552

Crossref Full Text | Google Scholar

7. Dai M, Yang Z, Zhang Q, Zhu QJ. Optimal trend following trading rules. Math Oper Res. (2016) 41:626–42. doi: 10.1287/moor.2015.0743

PubMed Abstract | Crossref Full Text | Google Scholar

8. Maheu JM, McCurdy TH, Song Y. Components of bull and bear markets: bull corrections and bear rallies. J Bus Econ Stat. (2012) 30:391–403. doi: 10.1080/07350015.2012.680412

PubMed Abstract | Crossref Full Text | Google Scholar

9. Décamps JP, Mariotti T, Villeneuve S. Investment timing under incomplete information. Math Oper Res. (2005) 30:472–500. doi: 10.1287/moor.1040.0132

PubMed Abstract | Crossref Full Text | Google Scholar

10. Klein M. Comment on “Investment timing under incomplete information”. Math Oper Res. (2009) 34:249–54. doi: 10.1287/moor.1080.0349

PubMed Abstract | Crossref Full Text | Google Scholar

11. Aingworth DD, Das SR, Motwani R. A simple approach for pricing equity options with Markov switching state variables. Quant Finance. (2006) 6:95–105. doi: 10.1080/14697680500511215

Crossref Full Text | Google Scholar

12. Elliott RJ, Siu TK, Chan L. Option pricing for Garch models with Markov switching. Int J Theor Appl Finance. (2006) 09:825–41. doi: 10.1142/S0219024906003846

Crossref Full Text | Google Scholar

13. Bae GI, Kim WC, Mulvey JM. Dynamic asset allocation for varied financial markets under regime switching framework. Eur J Oper Res. (2014) 234:450–8. doi: 10.1016/j.ejor.2013.03.032

Crossref Full Text | Google Scholar

14. Ekström E, Lu B. Optimal selling of an asset under incomplete information. Int J Stoch Anal. (2011) 17:543590. doi: 10.1155/2011/543590

Crossref Full Text | Google Scholar

15. Ekström E, Lindberg C. Optimal closing of a momentum trade. J Appl Probab. (2013) 2:374–87. doi: 10.1239/jap/1371648947

Crossref Full Text | Google Scholar

16. Henderson V. Kladívko K, Monoyios M, Reisinger C. Executive stock option exercise with full and partial information on a drift change point. SIAM J Financ Math. (2020) 11:1007–62. doi: 10.1137/18M1222909

Crossref Full Text | Google Scholar

17. Liptser R, Shiryayev A. Theory of Martingales (Mathematics and its Applications, 49). London: Kluwer Academic Publisher. (1989). doi: 10.1007/978-94-009-2438-3

Crossref Full Text | Google Scholar

18. Mishura Y, Ralchenko K. Discrete-Time Approximations and Limit Theorems In Applications to Financial Markets, Vol 2. Berlin: De Gruyter. (2021). doi: 10.1515/9783110654240

Crossref Full Text | Google Scholar

19. Gikhman II, Skokohod AV. The Theory of Stochastic Processes I. Cham: Springer (2004). doi: 10.1007/978-3-642-61921-2

Crossref Full Text | Google Scholar

20. Föllmer H, Schied A. Stochastic Finance: An Introduction in Discrete Time. Berlin: De Gruyter (2011). doi: 10.1515/9783110218053

Crossref Full Text | Google Scholar

21. Gikhman II, Skokohod AV. Introduction to the Theory of Random Processes. Philadelphia, PA: WB Saunders Company (1969).

Google Scholar

22. Föllmer H, Schweizer M. Minimal martingale measure. Encycl Quant Finance. (2010) 3:1200–4. doi: 10.1002/9780470061602.eqf04015

PubMed Abstract | Crossref Full Text | Google Scholar

23. Chung KL. Lectures from Markov Processes to Brownian Motion. Berlin: Springer Science (1982). doi: 10.1007/978-1-4757-1776-1

Crossref Full Text | Google Scholar

24. Gushchin A. Stochastic Calculus for Quantitative Finance. Amsterdam: Elsevier (2015).

Google Scholar

Appendix

In this appendix, we present, for the reader's convenience, a direct formula for the n-step transition probability of a 2 × 2 discrete-time Markov chain. Consider a transition probability matrix of the form

$\begin{array}{l}P=\left(\begin{array}{cc}\alpha & 1-\alpha \\ 1-\beta & \beta \end{array}\right),\end{array}$

for some α, β∈(0, 1). Transition probability P admits a unique invariant probability measure

$\pi =({\pi }_0,{\pi }_1)=\left(\frac{1-\beta }{2-\alpha -\beta },\frac{1-\alpha }{2-\alpha -\beta }\right).$

Let us find an eigendecomposition of P. Clearly, 1 is an eigenvalue, and the corresponding eigenvector is (1, 1). The second eigenvalue is λ = α + β − 1, and the corresponding eigenvector is v = (1 − α, β − 1). Thus, we have a decomposition

$\begin{array}{l}{P}^n=\left(\begin{array}{cc}1& 1-\alpha \\ 1& \beta -1\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& {(\alpha +\beta -1)}^n\end{array}\right)\left(\begin{array}{cc}\frac{1-\beta }{2-\alpha -\beta }& \frac{1-\alpha }{2-\alpha -\beta }\\ \frac{1}{2-\alpha -\beta }& -\frac{1}{2-\alpha -\beta }\end{array}\right)\\ \hspace{1em}=\left(\begin{array}{cc}{\pi }_0+{\pi }_1{(\alpha +\beta -1)}^n& {\pi }_1-{\pi }_1{(\alpha +\beta -1)}^n\\ {\pi }_0-{\pi }_0{(\alpha +\beta -1)}^n& {\pi }_1+{\pi }_0{(\alpha +\beta -1)}^n\end{array}\right).& (\mathrm{A1})\end{array}$