I apologize. I tried adhering to the format of the code you shared, which I mistakenly assumed to being intentional. Let me fix that:

    using System;
    using SharpDX;
    using SharpDX.Direct2D1;
    using SharpDX.DXGI;
    using SharpDX.Windows;
    using AlphaMode = SharpDX.Direct2D1.AlphaMode;
    using Factory = SharpDX.Direct2D1.Factory;
    
    namespace OverlayDirectXSample
    {
    public class OverlayForm : RenderForm
    {
    private RenderTarget _renderTarget;
    private SolidColorBrush _brush;
    private Factory _factory;

    public OverlayForm()
    {
        // Set the overlay's window style
        TopMost = true;
        WindowState = FormWindowState.Maximized;
        FormBorderStyle = FormBorderStyle.None;
        
        // Initialize DirectX
        InitializeDirectX();
    }

    private void InitializeDirectX()
    {
        _factory = new Factory();

        var renderProperties = new HwndRenderTargetProperties
        {
            Hwnd = Handle,
            PixelSize = new Size2(ClientSize.Width, ClientSize.Height),
            PresentOptions = PresentOptions.None
        };

        var properties = new RenderTargetProperties(RenderTargetType.Hardware,
            new PixelFormat(Format.B8G8R8A8_UNorm, AlphaMode.Premultiplied), 0, 0, RenderTargetUsage.None,
            FeatureLevel.Level_DEFAULT);

        _renderTarget = new WindowRenderTarget(_factory, properties, renderProperties);
        _brush = new SolidColorBrush(_renderTarget, Color.Red);
    }

    protected override void OnPaint(PaintEventArgs e)
    {
        base.OnPaint(e);

        // Clear the render target and set the overlay color
        _renderTarget.BeginDraw();
        _renderTarget.Clear(Color.Transparent);
        _renderTarget.FillRectangle(new RectangleF(100, 100, 200, 200), _brush);
        _renderTarget.EndDraw();
    }

    protected override void Dispose(bool disposing)
    {
        base.Dispose(disposing);
        _brush?.Dispose();
        _renderTarget?.Dispose();
        _factory?.Dispose();
    }
    }
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